hsiang-ping huang. webwork assignment number 5. due …...u of u math 1060-1 spring 2009 hsiang-ping...

U of U Math 1060-1

Spring 2009

Hsiang-Ping Huang.

WeBWorK assignment number 5.

due 02/18/2009 at 11:00pm MST.This set covers sections 4.3 and 4.4 of the textbook plus the

other 3 trig functions: cotangent, secant, and cosecant from 4.2.

Do let me know if you have any questions. Use the feedbackbutton, it tells me more than an ordinary e-mail.

1. (10 pts) set5/s5p1.pgYou may think this first problem a little odd. All it asks is thatuse your calculator to evaluate the trigonometric functions ofsome angles. The reason for including this problem is the ob-servation that a substantial number of mistakes in solving trigproblems are due to having your calculator in the wrong mode,and not noticing this fact. Whenever you use your calculatorcheck its mode!

In this problem you will need to enter your answers as decimalapproximations (with at least 4 digits). Usually WeBWorK willlet you enter expressions like sin(...), but note that in that caseWeBWorK always assumes your angles are measured in radians.

sin(15◦) =sin(15) =cos(1.4◦) =cos(1.4) =

Correct Answers:

• 0.258819045102521• 0.650287840157117• 0.999701489781183• 0.169967142900241

2. (10 pts) set5/s5p2a.pgA question like this is unlikely to arise in an application, but wedo need to learn to keep straight the different ways of measuringangles.

Suppose you have a triangle in which one angle is π

6 , and anotheris 40◦. Then the third angle is degrees, or radians.

Correct Answers:

• 110• 1.91986217719376

3. (10 pts) set5/s5p2.pgThe next few problems deal with the definition of the trigono-metric functions in a right triangle. We use a standard notation:a, b, and c denote the sides of the triangles and their lengths, andA, B, and C denote the angles opposite a, b, and c, respectively,as indicated in the nearby figure.

Thus a and b are the lengths of the two short sides, and c isthe hypotenuse. Use the Pythagorean Theorem to compute anymissing length. Assume also that A denotes the angle oppositea, B the angle opposite b, and, of course, C the right angle.

I recommend that in this problem you enter values of the trigfunctions as fractions.

Suppose a = 5 and b = 12.Then

c = ,sin(A) = ,cos(A) = ,tan(A) = .csc(A) = ,sec(A) = , andcot(A) = .

Correct Answers:

• 13• 0.384615384615385• 0.923076923076923• 0.416666666666667• 2.6• 1.08333333333333• 2.4

4. (10 pts) set5/s5p3.pgUsing the same notation as in the preceding problem, supposeb = 3 and c = 5.

Thena = ,sin(A) = ,

1

cos(A) = ,tan(A) = ,csc(A) = ,sec(A) = , andcot(A) = .

Correct Answers:• 4• 0.8• 0.6• 1.33333333333333• 1.25• 1.66666666666667• 0.75

5. (10 pts) set5/s5p4.pgUsing the same notation as in the preceding problem, supposea = 20 and c = 29.

Thenb = ,sin(B) = ,cos(A) = ,tan(A) = .sec(B) = ,csc(A) = , andcot(B) = .

Correct Answers:• 21• 0.724137931034483• 0.724137931034483• 0.952380952380952• 1.45• 1.45• 0.952380952380952

6. (10 pts) set5/s5p5.pgYou are hiking along the edge of the Green River (which is run-ning straight in the area of interest). Straight across from youon the opposite shore there is a particularly noticeable boulderat the edge of the river. You walk 120 feet along the shore ofthe river, and now the line from you to the boulder makes anangle of 35◦ with the edge of the river. You whip out your trustyscientific calculator and deduce that the width of the river at thispoint is feet.

Hint: Draw a picture. ”Straight across” means that the line fromyou to the boulder forms a right angle with the shore of the river.

Correct Answers:• 84.024904

7. (10 pts) set5/s5p6.pgYou are flying a kite on a line that is 350 feet long. Let’s sup-pose the line is perfectly straight (it never really is) and it makesan angle of 65 degrees with the horizontal direction. The kite isflying at an altitude of feet.Hint: Draw a picture. Look for a right triangle.


8. (10 pts) set5/s5p7.pgSuppose you have an isosceles triangle, and each of the equalsides has a lenght of 1 foot. Suppose the angle formed by thosetwo sides is 45◦. Then the area of the triangle issquare feet.

Hint: Draw the triangle. Draw the height. It divides the tri-angle into two congruent right triangles. Compute the missinglengths of the sides of that triangle using the trigonometric func-tions. Then compute the area of one of those two triangles, andmultiply with 2.

Correct Answers:

• 0.353553

9. (10 pts) set5/s5p8.pgThis is a generalization of the previous problem. The ideas areexactly the same.

Suppose you have an isosceles triangle, and each of the equalsides has a length of 1 foot. Suppose the angle formed by thosetwo sides is t. Then the area of the triangle issquare feet. Of course, your answer will involve t. Use sin(...)and cos(...) to enter the sin or cos of something.


Correct Answers:

• sin(t/2)*cos(t/2)

10. (10 pts) set5/s5p9.pgThis is a generalization of the previous problem. The ideas areexactly the same.

Suppose you have an isosceles triangle, and each of the equalsides has a length of r feet. Suppose the angle formed by thosetwo sides is t. Then the area of the triangle issquare feet.


Correct Answers:

• r**2*sin(t/2)*cos(t/2)

11. (10 pts) set5/s5p10.pgSuppose you inscribe a regular octagon into a circle of radius r.Then the area of that octagon is .

Hint: Use the result from the preceding problem.Correct Answers:

• 8*r**2*sin(0.785398163397448/2)*cos(0.785398163397448/2)

2

12. (10 pts) set5/s5p11.pgRecall the definition: Let θ be an angle in standard position.Its reference angle is the acute angle θ′ formed by the terminalside of θ and the horizontal axis.

Below, express θ′ in the same units (degrees or radians) asθ. You can enter arithmetic expressions like 210−180 or3.5−pi.

The reference angle of 100◦ is ◦.The reference angle of 350◦ is ◦.The reference angle of 4 is .Hint: Draw the angle. The Figure on page 383 of the textbookmay be helpful

Correct Answers:

• 80• 10• 0.858407346410207

13. (10 pts) set5/s5p12.pg

Below, express the reference angle θ′ in the same units (de-grees or radians) as θ. You can enter arithmetic expressions like210−180 or 3.5−pi.

The reference angle of 30◦ is ◦.

The reference angle of −30◦ is ◦.The reference angle of 1,000,000◦ is ◦.The reference angle of 100 is .Hint: Draw the angle. The Figure on page 383 of the textbookmay be helpful. To see the angle 1,000,000◦ subtract a suit-able multiple of 360◦. To see the angle 100, subtract a suitablemultiple of 2π.

Correct Answers:• 30• 30• 80• 0.53096491487338

14. (10 pts) set5/s5p13.pg

You approach a hill on top of which there is a tall radio an-tenna. You know from your map that your horizontal distancefrom the bottom of the radio antenna is 600 feet. The angle ofelevation to the bottom of the antenna is 10◦, and the angle ofelevation to the top of the antenna is 25◦. You figure that theheight of the hill is feet, and the height of the antenna is

feet. (Enter your answers rounded to the nearest foot.)Hint: Draw a picture. Figure out the height of the hill. Figureout the combined height of the antenna and the hill. Computethe difference.

Correct Answers:• 105.796188425079• 173.98840646792

Generated by the WeBWorK system c©WeBWorK Team, Department of Mathematics, University of Rochester

3

U of U Math 1060-1

Spring 2009

Hsiang-Ping Huang.


due 01/21/2009 at 11:00pm MST.The main purpose of this first WeBWorK set is to help you

get familiar with WeBWorK. If you have used WeBWorK beforeyou will find this set very easy. In that case please bear with usuntil we have everybody up to the same page! The last two prob-lems introduce a major subject of this class: the matheamtics ofright triangles.

Here are some hints on how to use WeBWorK effectively:- After first logging into WeBWorK change your pass-

word.- Find out how to print a hard copy on the computer system

that you are going to use. Contact me if you have any problems.Print a hard copy of this assignment. Note, however, that the on-line versions of these problems may have links to the web pagesthat are of course absent on the hard copy.

- Get to work on this set right away and answer these ques-tions well before the deadline. Not only will this give you thechance to figure out what’s wrong if an answer is not accepted,you also will avoid the likely rush and congestion prior to thedeadline.

- The primary purpose of the WeBWorK assignments in thisclass is to give you the opportunity to learn by having instantfeedback on your active solution of relevant problems. Makethe best of it!

Ken Bromberg, JWB 303.

1. (10 pts) set1/s1p1.pgThis first question is just an exercise in entering answers intoWeBWorK. It also gives you an opportunity to experimentwith entering different arithmetic and algebraic expressions intoWeBWorK and seeing what WeBWorK really thinks you are do-ing (as opposed to what you believe it should think).

Notice the buttons on this page and try them out before movingto the next problem. Use the ”Back” Button on your browser toget back here when needed.

- ”Prob. List” gets you back to the list of all problems in thisset.

- ”Next” gets you to the next question in this set.- ”Submit Answer” submits your answer as you might ex-

pect, but there may be other ways to do so. Specifically, in thisproblem, there is only one question. In that case you can sub-mit your answer by typing it into the answer window and thenpressing ”Return” (or ”Enter”) on your keyboard. But even in

this case, you can also type the answer and click on the ”sub-mit” button. There is no harm in submitting an answer even ifyou are not quite sure that it’s correct, since if it is not you havean unlimited number of additional tries. On the other hand, itis usually more efficient to print your own private problems set,work out the answers in a quiet environment like your home, andthen sit down in front of a computer and enter your answers. Ifsome are wrong you can try to fix them right at the computer, oryou may want to go back and work on them quietly elsewherebefore returning to the computer.

- Pressing on the ”Preview Answer” Button makes WeB-WorK display what it thinks you entered in the answer window.After using ”Preview” you can modify your answer and use a”Preview Again” button.

- ”typeset” denotes the ordinary display mode on your work-station. ”formatted text” is a little faster than ”typeset”. Theoption of plain ”text” is not useful.

- ”Logout” terminates this WeBWorK session for you. Youcan of course log back in and continue.

- ”Feedback” enables you to send a message to your instruc-tor. If you use this way of sending e-mail the recipients receiveinformation about your WeBWorK state, in addition to your ac-tual message.

- The ”Help” Button transports you to an official WeBWorKhelp page that has more information than this first problem.

- ”Problem Sets” transports you back to the page where youcan select a certain problem set. When you do this particularproblem in this first set, there is only one set, but eventuallythere will be 13 of them.

- For all problems in this course you will be able to see theAnswers to the problems after the due date. Go to a problem,click on ”show correct answers”, and then click on ”submit an-swer”. You can also download and print a hard copy with theanswers showing. These answers are the precise strings againstwhich WeBWorK compares your answer. If the answer is analgebraic expression your answer needs to be equivalent to theWeBWorK answer, but it may be in a different form. For ex-ample if WeBWorK thinks the answer is 2 ∗ a, it is OK for youto type a + a instead. If WeBWorK expects a numerical an-swer then you can usually enter it as an arithmetic expression(like 1/7 instead of .142857), and usually WeBWorK will ex-pect your answer to be within one tenth of one percent of whatit thinks the answer is.

- Most of the problems (including this one) in this course willalso have solutions attached that you can see after the due dateby clicking on ”show solutions” followed by ”submit answers”.The solutions are text typed by your instructor that gives moreinformation than the ”answers”, and in particular often explainshow the answers can be obtained.

Now for the meat of this problem. Notice that the answer win-dow is extra large so you can try the things suggested above.

Type the number 3 here:

1

.Try entering other expressions and use the preview buttonto see what WeBWorK thinks you entered. Return to thisproblem to try out things when you get stuck somewhereelse.

Here are some good examples to try. Check them all out usingthe Preview button. (In later questions on this set you will getto use what you learn here.) Never mind that you may have al-ready answered the correct answer 3. Once you get credit for ananswer it won’t be taken away by trying other answers.

a/2b versus a/2/b versus a/(2b)

a/b+c versus a/(b+c)

a+b**2 versus (a+b)**2

sqrt a+b versus sqrt(a+b)

4/3 pi r**2 versus (4/3) pi r**2 (In other words, if you are notsure use parentheses freely.)

Note: WeBWorK will not usually let you enter algebraic ex-pressions when the answer is a number, and it will only let youuse certain variables when the answer is in fact an algebraic ex-pression. So the above window, and the opportunity for exper-imentation that it offers is unique. Make good use of it!

Presumably this has been your first encounter with WeBWorK.Come back here to try things out and to refresh your memory ifyou get stuck somewhere down the line.

Correct Answers:

• 3

2. (10 pts) set1/s1p2.pgThe purpose of this exercise is to illustrate further the use of thebuttons on this page and to show you the most common way inwhich WeBWorK processes partially correct problems. Try en-tering incorrect answers in the answer fields below, to see whathappens. (This time WeBWorK will reject algebraic expressionssince I told it to expect a numerical answer.)

Type the number 4 here: .

Type the number 5 here: .

Correct Answers:

• 4• 5

3. (10 pts) set1/s1p3.pg

In the first few problems, now that you are familiar with thebasic mechanics of WeBWorK, you will be asked to evaluatesome arithmetic expressions and enter the answer as a number

into WeBWorK. You may of course use a calculator. In laterproblems you will be able to enter the answer as an arithmeticexpression, but at present your answer must be a number suchas 4, -4, or 17.5.

Evaluate the expression9(8+9) = .(Remember that by convention a missing arithmetic operatormeans multiplication .)

Correct Answers:

• 153


Evaluate the expression7(2−5) = .

Correct Answers:

• -21


Evaluate the expression2/(3+4) = .Enter you answer as a decimal number listing at least 4 decimaldigits. (WeBWorK will reject your answer if it differs by morethan one tenth of 1 percent from what it thinks the answer is.)

Correct Answers:

• 0.285714285714286


Evaluate the expression12− (10−9) = .

Correct Answers:

• 11



Correct Answers:

• 8

8. (10 pts) set1/s1p8.pgThis problem illustrates the standard rules of arithmetic prece-dence:

Multiplication and Division precede Subtraction and Addition.Among operations with the same level of precedence, evalua-tion proceeds from left to right.However, expressions in parentheses are evaluated first.

2

Evaluate the expression6×4−2×3 =

Evaluate the expression6× (4−2)×3 =

Evaluate the expression6× (4−2×3) =

Correct Answers:

• 18• 36• -12

9. (10 pts) set1/s1p9.pgThis problem provides more illustrations of the use of parenthe-ses.

Evaluate the expression7−7−3−9 =

Evaluate the expression7− (7−3)−9 =

Evaluate the expression7− (7−3−9) =

Evaluate the expression7− (7− (3−9)) =

Correct Answers:

• -12• -6• 12• -6

10. (10 pts) set1/s1p10.pgThe key idea in Algebra is to use variables in addition to num-bers. Sometimes we need to replace variables with specificnumbers. That’s called evaluating an algebraic expression. Forexample, if a = 2 then 3a = 6, and we say that we evaluated theexpression 3a at a = 2. We’ll do this sort of thing all semesterlong, and in this problem you get your first experience withevaluating algebraic expressions. Again, the emphasis in theseexercises in on understanding the rules of arithmetic precedence.

Let a = 12, b = 3, c = 13.Then a−b/c =and (a−b)/c =As usual, enter your answers as decimal numbers with at

least 4 digits.Correct Answers:

• 11.7692307692308• 0.692307692307692

11. (10 pts) set1/s1p11.pgLet r = 9.

Then 4/π∗ r =and 4/(π∗ r) =Correct Answers:

• 11.4591559026165• 0.141471060526129

12. (10 pts) set1/s1p12.pgLet a = 3, b = 6, c = 7, d = 8.

Then a−b/c−d = ,

(a−b)/(c−d) = ,a− (b/c−d) = , anda−b/(c−d) = .

Correct Answers:• -5.85714285714286• 3• 10.1428571428571• 9

13. (10 pts) set1/s1p13.pgThe next three problems are like the preceding three, except thatyou need to get all answers correct before WeBWorK will giveyou credit. This will be true for many problems in this class.The purpose of insisting on all answers being correct is to en-courage you to think about the whole context of the problemrather than the individual pieces.

Let a = 2.1, b = 3.7, c = 5.9.Then a−b/c =and (a−b)/c =Correct Answers:

• 1.4728813559322• -0.271186440677966

14. (10 pts) set1/s1p14.pgLet r = 6.7.


• 8.53070494972559• 0.190035752945547

15. (10 pts) set1/s1p15.pgLet a = 4.5, b = 5.5, c = 8.1, d = 9.3.

Thena−b/c−d = ,

(a−b)/(c−d) = ,a− (b/c−d) = , anda−b/(c−d) = .

Correct Answers:• -5.47901234567901• 0.833333333333334• 13.120987654321• 9.08333333333334

3

16. (10 pts) set1/s1p16.pgIn this and the following problems you will practice enteringalgebraic expressions into WeBWorK. Remember the Rules ofArithmetic Precedence which apply universally to arithmetic ex-pressions (involving only numbers) and also agebraic expres-sions (involving also variables).

- Multiplication and Division come before Addition and Sub-traction.

- When carrying out operations of the same level of prece-dence (e.g., a mixture of additions and subtractions) we workfrom left to right.

- Expressions in parentheses are evaluated first.You can have superfluous parentheses, so use parentheses

freely to make your meaning clear, particularly if you are notcertain. Most of the difficulties students have with WeBWorKare due to not appreciating the precise rules that govern the in-terpretation of what you enter. This is not just a matter of WeB-WorK understanding what you are trying to say, The rules areused universally all over the world. Appreciating and apply-ing them properly is also crucial, for example, in computer pro-gramming. Make sure you understand what’s going in theseproblems. If you enter a wrong expression use the Preview But-ton to see what WeBWorK thinks you have entered.

We start simply. Enter here the expression a+b.Correct Answers:

• a+b

17. (10 pts) set1/s1p17.pgEnter here the expression

a+12+b

Enter here the expressiona+bc+d

If WeBWorK rejects your answer use the preview button tosee what it thinks you are trying to tell it.

Correct Answers:• (a+1)/(2+b)• (a+b)/(c+d)


11a + 1

b

Enter here the expressiona+b+11+ 1

a+b

Correct Answers:• 1/(1/a+1/b)• (a+b+1)/(1+1/(a+b))

19. (10 pts) set1/s1p19.pg

Enter here the expressionab + c

def + g

h.

Correct Answers:• (a/b+c/d)/(e/f+g/h)

20. (10 pts) set1/s1p20.pg

To enter square roots you can use the function sqrt . For ex-ample, to enter the square root of 2 you can type sqrt(2) .

Enter here the expression√

aLater this semester we will obtain answers that are functions

other than the square root evaluated at certain numbers.Correct Answers:

• sqrt{a}


√a+b

Enter here the expressiona√

a+bEnter here the expression

a+b√a+b

Correct Answers:• sqrt(a+b)• a/sqrt(a+b)• (a+b)/sqrt(a+b)

22. (10 pts) set1/s1p22.pg

Enter here the expression√x2 + y2

Enter here the expression

x√

x2 + y2

Enter here the expressionx+ y√x2 + y2

Correct Answers:• sqrt(x**2+y**2)• x*sqrt(x**2+y**2)• (x+y)/sqrt(x**2+y**2)

4

23. (10 pts) set1/s1p23.pgYou are walking in a city. You are 6 feet tall and you cast ashadow that’s 8 feet long. You notice that one of the buildingscasts a shadow that’s 160 feet long. The height of that buildingis feet.

Note: this is the first problem in this class to come with a Hint.After you submit an answer for the first time a new button la-beled “Hint” will appear below. (Try it right now and for themoment don’t worry about your answer being correct.) Clickingthat button followed by submitting an answer will show the hint.Many problems in this class will have hints. Most problems willalso have solutions that you can see after the set closes. (Clickon ”Show Solution” and once again submit an answer—but re-member that this works only after the deadline for this set.) Af-ter the deadline you can also see the “answers” of a questionwhich are the numbers (or words or choices) that WeBWorK

expects you to enter.Hint: Draw a couple of similar triangles. Assign a variable tothe unknown height, write down an equation involving that vari-able, and solve the equation for the variable.

Correct Answers:

• 120

24. (10 pts) set1/s1p24.pgYou are approaching the island of Hawaii in a small boat. Thehighest point on Hawaii is Mauna Loa at 13,677 feet. You seeit just barely above the horizon. The radius of Earth is 3,963miles. Ignoring atmospheric effects, you figure that you are

miles in a straight line from the top of Mauna Loa.Hint: Draw the triangle whose vertices are the top of MaunaLoa, your boat, and the center of the earth. Apply thePythagorean Theorem. A mile has 5,280 feet.

Correct Answers:

• 143.300383809674


5

U of U Math 1060-1

Spring 2009

Hsiang-Ping Huang.


due 01/28/2009 at 11:00pm MST.The purpose of this set is to remind you of some of the pre-

requisites for this class. If you are having difficulties with any ofthese questions (except for the word problems) then you shouldreview the relevant material (Appendix A and Chapter 1 in thetextbook).

This set covers the following topics:- Common pitfalls in algebra, 1–5.- Solving Linear Equations, 6–8.- Evaluating a function, 9–13.- Cartesian Coordinates, 14-15. Some basic terms, and the

concept of distance.- Quadratic Equations, 16–20.- Circles, 21–25. Some word problems involving circles (and

entering algebraic expressions).Remember that all problems have Answers and many have

Hints or Solutions. To see the answers (or solutions) click on”show answers” or ”show solutions” and submit an answer af-ter the set closes. When there is a hint it will become availableafter you submit an answer for the first time (even before the setcloses). Click on ”Show Hint” and submit your answer again.

1. (10 pts) set2/s2p1.pgThe first few problems in this set are meant to help you rec-ognize and avoid common pitfalls in doing algebra. For eachstatement below enter a T (true) if the statement is true and anF (false) otherwise. In this problem WeBWorK will tell you foreach answer if it’s correct, but don’t just try out the possibilities,make sure you understand why the statement is true or false.(In the following problems you will have to get everything rightbefore you get credit.)

For all real numbers a and b,

a+b = b+a.

For all positive real numbers a and b,√a2 +b2 = a+b.

For all positive real numbers a

a0 = 1.

Correct Answers:• T

• F• T


For each statement below enter a T (true) the statement istrue and an F (false) otherwise. In this problem you need to geteverything correct before receiving credit. Whenever there is adivision below, we assume that the divisor is non-zero.

For all real numbers a, b, and c

a(b+ c) = ab+ac

For all real numbers a, b, c and dab

+cd

=a+ cb+d

For all real numbers a, b, c and dab

+cd

=ad +bc

bd

For all real numbers a, b, and ca+b

c=

ac

+bc.

For all real numbers a, b, and ca

b+ c=

ab

+ac.

Correct Answers:

• T• F• T• T• F


For each statement below enter a T (true) the statement istrue and an F (false) otherwise. In this problem you need to geteverything correct before receiving credit. Whenever there is adivision below,we assume that the divisor is non-zero.

For all real numbers a, b, c, and da−bc−d

=b−ad− c

.

For all real numbers a, b, and cab+ac

a= b+ c.

For all real numbers a, b, and cabc

=bac

.

1


=abc

.


=bac

.


=cab

.

Correct Answers:

• T• T• F• T• F• T


For each statement below enter a T (true) the statement istrue and an F (false) otherwise. In this problem you need to geteverything correct before receiving credit. Whenever there isa division below,we assume that the divisor is non-zero. Wealso assume that where it matters the base of any power is pos-itive.


abac = abc.


abac = ab+c.

For all real numbers a and b

a−b =1ab .


a−b =−ab.


ab = ab.

Correct Answers:

• F• T• T• F• F


For each statement below enter a T (true) the statement istrue and an F (false) otherwise. In this problem you need to geteverything correct before receiving credit. Whenever there is adivision below,we assume that the divisor is non-zero.

For all real numbers a, b, and x

a−b(x−1) = a−bx−b.


a−b(x−1) = a−bx+b.


3+bx3

= 1+bx.


(x− r)2 = x2− r2.


(x− r)2 = x2−2rx+ r2.


(x− r)(x+ r) = x2− r2.

Correct Answers:

• F• T• F• F• T• T


The solution of the equation

x(x+2) = (x−1)(x+4)

is x = .

Hint: Apply the Distributive Law on both sides.Correct Answers:

• 4

2



1x−4

=2

x+9

is x = .Hint: Take reciprocals on both sides.

Correct Answers:

• 17



xx−2

=x−3x+3

is x = .(You may enter your answer as a decimal number or as a frac-tion.)Hint: Multiply with both denominators on both sides.

Correct Answers:

• 0.75

9. (10 pts) set2/s2p9.pgFor the next few problems you need to understand what it meansto evaluate a function. You simply replace the variable with thenumber at which you evaluate the function. For example, theanswer to the first question below is 17 since

17 = 6∗2+5.

Let the function f be defined by

f (x) = 6x+5.

Then f (2) = and f (3) =Correct Answers:

• 17• 23

10. (10 pts) set2/s2p10.pg


f (x) =−2x−6.

Then f (−5) = and f (−4) =Correct Answers:

• 4• 2

11. (10 pts) set2/s2p11.pg


f (x) = 9x+9.

Then f (6)+ f (7) = and f (6+7) =Correct Answers:

• 135• 126

12. (10 pts) set2/s2p12.pg


f (x) = 2x+8.

Then f ( f (3)) = and f ( f (4)) =Hint: As always, first figure out what’s inside the parentheses.

Correct Answers:• 36• 40

13. (10 pts) set2/s2p13.pg


f (x) = 3x+2.

Then f ( f (1)) = ,f 2(1) = ,and f 2( f (1)) = .Hint: As always, first figure out what’s inside the parentheses.

Correct Answers:• 17• 25• 289

14. (10 pts) set2/s2p14.pgI recommend that instead of decimal numbers you enter an ex-pression using sqrt() to indicate a square root. For example in-stead of 1.4142 you would enter sqrt(2) .

The distance between the points (2,6) and (6,1) is.

The distance between the points (3,2) and (7,8) is.

The distance between the points (−7,5) and (8,6) is.

The distance between the points (−8,−4) and (−2,−3) is.

Hint: Don’t try to memorize a formula. Draw a picture showingthe two points in a rectangular coordinate system and apply thePythagorean Theorem.

Correct Answers:• 6.40312423743285• 7.21110255092798• 15.0332963783729• 6.08276253029822

3

15. (10 pts) set2/s2p15.pgEach of the following phrases describes a point in the Cartesian(rectangular) coordinate system. Enter the coordinates x and yof the point.

The point is located 5 units to the left of the y axis and 2 unitsabove the x axis.x= and y=The point is located 10 units to the right of the y axis and 4 unitsbelow the x axis.x= and y=The coordinates of the point have equal absolute value, the pointis in the second quadrant, and the distance of the point from theorigin is 2.x= and y=Hint: use ”sqrt(z)” to denote the square root of a number z.The point is the origin.x= and y=The point is on the positive x axis 10 units from the origin.x= and y=Hint: If you get stuck draw a picture.

Correct Answers:

• -5• 2• 10• -4• -1.4142135623731• 1.4142135623731• 0• 0• 10• 0

16. (10 pts) set2/s2p16.pgThe next few problems will ask you to solve quadratic equa-tions. Some such equations can be solved by factoring, but mostrequire that you complete the square or apply the quadratic for-mula. As you know, I recommend that you apply the quadraticformula.

When asked to enter numbers in increasing sequence rememberthat a number b is larger than a number a if a is to the left of bon the number line. So, for example, 1 is larger than −10000.When the answers involve square roots, like 1 +

√5, you can

compute and enter decimal approximations, but it’s easier to en-ter an expression like 1+sqrt(5).

The equationx2 +2x−3 = 0

has two solutions. Enter the smaller here and the largerhere .

Correct Answers:

• -3• 1

17. (10 pts) set2/s2p17.pgThe equation

3x2 +2x−4 = 0

has two solutions. Enter the smaller here and the largerhere .

Correct Answers:

• -1.535183758488• 0.86851709182133

18. (10 pts) set2/s2p18.pgQuadratic equations do not always occur in standard form.Sometimes they have to be converted to standard form usingour basic principle of doing the same thing on both sides to getwhere we want to go.

The equation

x+1x

= 2

has only one solution. It isx = .Hint:When you see the solution you will say ”of course”. As a firststep, multiply with x on both sides.

Correct Answers:

• 1


1− xx

= x

has two real solutions. Enter the smaller here andthe larger here .Hint:Begin by multiplying with x on both sides.

Correct Answers:

• -1.61803398874989• 0.618033988749895


x−12x+1

=x+1x−1

has two real solutions. Enter the smaller one here and thelarger one here .Hint:Begin by multiplying with 2x+1 and x−1 on both sides.

Correct Answers:

• -5• 0

4

21. (10 pts) set2/s2p21.pgYou are on a pleasure cruise through the universe and you crashin the ocean of an unknown planet. Your spaceship floats onthe water and its top is 21 feet above the surface of the water.You swim away from the spaceship until you see its top on thehorizon. Your laser range meter tells you that your eyes are 2.9miles away from the top of the space ship. (You are a capable–if reckless and curious–swimmer.) The radius of the planet is

miles.Hint: This is like the Mauna Loa problem except that the un-known is different.

Correct Answers:

• 1057.25515422078

22. (10 pts) set2/s2p22.pgYou are going to drive 480 miles today at an average speed of60 miles per hours. Thus you are going to drive for a total ofhours. The diameter of the wheels on your car (including tires)is 25 inches. Thus each wheel is going to turn a total oftimes today.

Correct Answers:

• 8• 387227.75356324

23. (10 pts) set2/s2p23.pg

The essence of algebra is the manipulation of algebraic ex-pressions. One application of algebraic expressions is the state-ment of formulas that describe general facts. The next few ques-tions will ask you to enter some simple algebraic expressions.

Just enter them like you might type them on paper, without quo-tation marks, using the basic four operations, a double aster-isk to denote exponentiation, and ”pi” (without the quotationmarks) to denote π.

Enter here an algebraic expression that gives thearea of a rectangle with a length of a and a width of b. (Use anasterisk to denote multiplication.)

Correct Answers:• a*b

24. (10 pts) set2/s2p24.pg

Enter here an algebraic expression that gives thearea of a circle with a radius r.

Correct Answers:• pi*r*r

25. (10 pts) set2/s2p25.pg

Suppose you inscribe a square in a circle of radius r. Thusall four corners of the square lie on the circle. The area of thatsquare is .

Draw a picture and apply the Pythagorean Theorem.Correct Answers:

• 2*r*r


5

U of U Math 1060-1

Spring 2009

Hsiang-Ping Huang.


due 02/04/2009 at 11:00pm MST.This set concludes the review of prerequisites for Trigonom-

etry.It covers the following topics:- Problems 1–6. The interplay between the graph of a func-

tion and simple algebraic modifications of that function. In par-ticular, adding a number to the function value shifts the graphup or down, and adding a number to the value of x shifts the thegraph left or right. Multiplying the function value with a certainfunctor rescales the graph vertically. It is easy in WeBWorK toshow you a graph and then ask for a statement about the corre-sponding algebra. It would be nice if we could give you somealgebra and ask you to draw the graph. I haven’t figured outhow to do this effectively, but if you like you can practice thissort of thing using a graphing calculator or the java applet foundat http://gcalc.net.

- Problems 7–10. A circle is the set of all points of a givendistance from a given point. The link with algebra is given viathe distance formula. These problems reinforce the link betweengeometry and algebra.

Note that all shifts and scales in problems 1–10 are integer,so that you can easily read them off the given graphs. Some ofthe answers are negative.

- Problem 11–15. To study Trigonometry we have to be thor-oughly familiar with triangles, particularly right ones, and thePythagorean Theorem. Problems 11-15 all require that Theo-rem. One theme is the idea of inscribing and circumscribingpolygons to circles.

- Problems 16–18. These problems reinforce familiaritywith the concepts of even and odd functions. The graph of aneven function is symmetric about the y axis, and the graph ofan odd function is symmetric about the origin. Many functionsare neither even nor odd, but the trigonometric functions we willstudy later in this semester all are in fact even or odd.

- Problem 19. The last problem reviews the concept of an in-verse function. We will study trigonometric functions and theirinverses in this course.

- Problems 20-21 A couple of simple word problems involv-ing the Pythagorean Theorem.


1. (10 pts) set3/s3p1.pgConsider the function

f (x) = x2.

Its graph is the standard parabola shown in this Figure:

The graph in this Figure

has been obtained by shifting the first graph one unit to the right,and one unit down. (Don’t get confused by the fact that thescales on the two axes are different.) It is the graph of the func-tiong(x) =Hint: Ask what happens to the vertex of the graph.

Correct Answers:

• (x-1)**2-1

1

2. (10 pts) set3/s3p2.pgSimilarly, the graph in this Figure

is the graph of the functiong(x) =Hint: Ask what happens to the vertex of the graph.

Correct Answers:

• (x+1)**2+2


The graph in this Figure

is the graph of the functiong(x) =Hint: Ask what happens to the vertex of the graph.

Correct Answers:

• (x-3)**2-2

4. (10 pts) set3/s3p4.pgThe graph in this Figure

is the graph of the functiong(x) =Hint: Ask what happens to the vertex of the graph, and what itmeans algebraically when we flip the parabola upside down.

Correct Answers:

• -(x-1)**2+1

5. (10 pts) set3/s3p5.pgConsider again the function

f (x) = x2.

The Figure

shows the graphs of f and also the graphs of

g1(x) = 2x2 and g2(x) =x2

2,

2

which are obtained by scaling the graph of f vertically by a fac-tor 2 or 1/2, respectively.

The graph in this Figure is obtained from that of f by a com-bination of scaling and shifting:

It is the graph of the functiong(x) =Hint: Figure out the scale by going 1 unit to the right from thevertex, and see what happens to the function value. Considerthe location of the vertex.

Correct Answers:

• 2*(x+1)**2-1

6. (10 pts) set3/s3p6.pgThe graph in this Figure is obtained from that of f (x) = x2 by acombination of scaling and shifting:

It is the graph of the functiong(x) =Hint: Figure out the scale by going one unit to the right from

the vertex, and see what happens to the function value. Considerthe location of the vertex.

Correct Answers:

• -3*(x-1)**2+4

7. (10 pts) set3/s3p7.pgConsider the circle with the equation

(x+2)2 +(y−1)2 = 25.

Its center is( , )and its radius is r = .Hint: The circle with radius r and center (xC,yC) is the set ofall points that are at a distance r from the center. It is the graphof the equation

(x− xC)2 +(y− yC)2 = r2.

Correct Answers:

• -2• 1• 5

8. (10 pts) set3/s3p8.pgConsider the circle with the equation

(x−3)2 +(y−5)2 = 49.

Its center is( , )and its radius is r = .Hint: The circle with radius r and center (xC,yC) is the set ofall points that are at a distance r from the center. It is the graphof the equation

(x− xC)2 +(y− yC)2 = r2.

Correct Answers:

• 3• 5• 7

3

9. (10 pts) set3/s3p9.pgThe Figure

shows the unit circle , i.e., the circle around the origin with ra-dius r = 1. It is defined by the equation

x2 + y2 = 1.

As with any equation, other circles can be obtained by scalingand shifting. The circle in this Figure

is the graph of the equation(x− )2+(y- )2 = 2.Hint: Obtain from the Figure the center and the radius of thecircle, and then write down the standard equation of the circle.Notice that some of your results may be negative.

Correct Answers:• 2• -3• 2

10. (10 pts) set3/s3p10/prob.pgMatch the equations with their graphs.

1. x2 + y2 = 12. (x−1)2 + y2 = 13. x2 +(y−1)2 = 14. (x−1)2 +(y−1)2 = 1

A B C D

(Click on image for a larger view )Hint: Think of the definition of a circle and the distance for-mula.

Correct Answers:

• A• D• C• B

11. (10 pts) set3/s3p11.pgAn equilateral triangle is one where all sides have equal length.Suppose you have an equilateral triangle where each side is 2feet long. Its area is square feet.Actually, we might do a problem like this more generally. Sup-pose your equilateral triangle has sides of length s feet. Then itsarea is A = square feet.

(The following similar problems will not have the special caseto begin with. But if you like work a specific problem first ongraph paper, and measure the distances involved to see if yourfiguring is reasonable.)Hint: Draw the right triangle and its height. The area of thetriangle equals of base times height. You know the base. Usethe Pythagorean Theorem to figure out the height. Use sqrt(...)to enter the square root of something.

Correct Answers:

• 1.73205080756888• sqrt(3)/4*s**2

12. (10 pts) set3/s3p12.pgAn isosceles triangle is one where two sides have the samelength and the third side has a possibly different length. Sup-pose you have an isosceles triangle where two sides have lengtha and the base has length b. Then the height of that triangle ish = .The area of your triangle is A = .Hint: Draw the right triangle and its height. Apply thePythagorean Theorem. Use sqrt(...) to enter the square root ofsomething. The area of a triangle is one half times base timesheight.

Correct Answers:

• sqrt(a**2-b**2/4)

4

• b/2*sqrt(a**2-b**2/4)

13. (10 pts) set3/s3p13.pgAssume you inscribe a regular hexagon inside a circle of radiusr, as shown by the green hexagon in this Figure:

(The yellow hexagon circumscribes the circle.)The area of the inscribed hexagon is A = .

Hint: A hexagon is a polygon with six sides. A regular hexagonis a hexagon where all sides and all interior angles are equal.A regular hexagon inscribed in a circle is the largest regularhexagon that fits in the circle. Think of the hexagon as con-sisting of 6 equilateral triangles. Use the formula for the area ofan equilateral triangle that you obtained in an earlier problem.

Correct Answers:

• 3*sqrt(3)/2*rˆ2

14. (10 pts) set3/s3p14.pg

Suppose you have an equilateral triangle with a height of hfeet. Then its area isA = square feet.Hint: Draw the right triangle and its height. The area of thetriangle equals of base times height. You know the height. Usethe Pythagorean Theorem to figure out the base. Use sqrt(...) toenter the square root of something.

Correct Answers:

• hˆ2/sqrt{3}

15. (10 pts) set3/s3p15.pg

Suppose you circumscribe a regular hexagon around a circleof radius r. Thus the circle is inscribed the hexagon. The areaof that hexagon isA = square feet.Hint: Use the formula for the area of an equilateral trianglewith a given height, as obtained in an earlier problem.

Correct Answers:

• 6*rˆ2/sqrt{3}

16. (10 pts) set3/s3p16.pg

A function f is even if it satisfies f (x) = f (−x) for all x inits domain. An example of an even function is f (x) = x2 since(x2) = (−x)2.

f is odd if it satisfies f (x) =− f (−x) for all x in its domain. Anexample of an odd function is f (x) = x3 since x3 =−(−x)3.

Functions may be neither even nor odd, for example the functionf (x) = x2 + x3 is in that category.

For each function below enter the letter E if the function iseven, the letter O (not the digit 0!) if it’s odd, and the letter N ifit’s neither even nor odd.

f (x) = x4.f (x) = x5.f (x) = x4 + x5.

Correct Answers:

• E• O• N

17. (10 pts) set3/s3p17.pg

For each function below enter the letter E if the function iseven, the letter O if it’s odd, and the letter N if it’s neither evennor odd.

f (x) = x4 + x2.f (x) = x5 + x3.f (x) = x4 + x5.f (x) = |x|.f (x) = |x|3.f (x) = x2

x2+1 .

Hint: If in doubt, evaluate the functions at pairs of numbers like1 and −1.

Correct Answers:

• E• O• N• E• E• E

18. (10 pts) set3/s3p18.pgSuppose f is a function that’s defined for all real numbers x.Assume it’s neither even nor odd. Below you will find otherfunctions defined in terms of f

For each function enter the letter E if the function is even, theletter O if it’s odd, and the letter N if it’s neither even nor odd.

g(x) = f (x)+ f (−x).5

g(x) = f (x)− f (−x).g(x) = f (−x).

Correct Answers:• E• O• N

19. (10 pts) set3/s3p19.pgA function f is the inverse of a function g if

f (g(x)) = x

for all x in the domain of g. For example, f (x) = x+12 is the

inverse of the function g(x) = 2x−1 since

f (g(x)) = f (2x−1) =(2x−1)+1

2= x.

Functions do not have an inverse if several values of x giverise to the same value of f (x). For example the function f (x) =x2 does not have an inverse.

To compute the inverse of a function g you solve the equa-tion y = g(x) for x and then reverse the role of x and y. In theabove example we solve y = 2x− 1 for x which gives x = y+1

2 .Interchanging x and y gives the inverse function of g which isf (x) = x+1

2 .

Consider now the function

g(x) = 3x+4.

Its inverse function isf (x) = .Hint: Proceed as in the example.

Correct Answers:• (x-4)/3

20. (10 pts) set3/s3p20.pgYou build a box that is 4 feet long, 5 feet wide, and 8 feet high.The distance from the top left front corner to the top right backcorner is feet.The distance from the bottom left front corner to the top rightback corner is feet.Hint: Draw a picture and apply the Pythagorean Theorem.

Correct Answers:• 6.40312423743285• 10.2469507659596

21. (10 pts) set3/s3p21.pgYou have a 5 foot long ladder, and you lean it against a wall suchthat the bottom of the ladder is 3 feet away from the wall. Thetop of the ladder touches the wall at a height of feet.Hint: Draw a picture and apply the Pythagorean Theorem.

Correct Answers:• 4


6

U of U Math 1060-1

Spring 2009

Hsiang-Ping Huang.


due 02/11/2009 at 11:00pm MST.This set covers sections 4.1 and 4.2 of the textbook.The first midterm is coming up on Wednesday, February

11. It will cover sections 4.1 and 4.2 plus the review material.The best way to study is to go over the homework assignments,including this assignment.

Make sure to turn off the caps lock button before typingin the answers. X is not the same as x.

For your convenience, following is a list of the most basicproperties of the trigonometric functions. As you know, I ad-vise against memorizing too many formulas. However, to un-derstand trigonometry you need to work with it so thoroughlythat you can’t help remembering the definitions and propertiesbelow, just like you can find the way to your kitchen withoutever having memorized it. The key part is the definition of sinand cos as coordinates of a point on the unit circle. Everythingflows from there.

- DefinitionLet t be an angle in standard position. Thus its vertex is at

the origin, its initial side is on the x-axis, and its terminal sideintersects the unit circle (of radius 1 around the origin) at a point(x,y). Then:

cos t = x, sin t = y, tan t =sin tcos t

=yx.

Note: cos, sin, and tan are functions, but the parenthesesaround their arguments are usually omitted whenever this causesno confusion. Of course, tan t is undefined when cos t = 0. Oth-erwise the trigonometric functions are defined for all real num-bers t.

- Key IdentityWe will use the following key identity all the time. It follows

straight from the definition of sin and cos as the coordinates ofa point on the unit circle. For all real number t:

cos2 t + sin2 t = 1.

Here we use the standard notation: cos2 t = (cos t)2, and sim-ilarly for the sin function.

- Periodicity

cos(t +2π) = cos t, sin(t +2π) = sin t, tan(t +π) = tan t.

- Sign reversal

cos(t +π) =−cos t, sin(t +π) =−sin t.

- SymmetryThe cos function is even, sin and tan are odd, i.e.,

cos t = cos(−t), sin t =−sin(−t) tan t =− tan(−t).

- Right triangleLet t be one of the acute angles of a right triangle. Then

sin t = oppositehypotenuse

cos t = adjacenthypotenuse

tan t = oppositeadjacent


1. (10 pts) set4/s4p1.pgWe need to be able to convert radians to degrees and vice versa.Below you can enter radians as decimal approximations, but Irecommend that you enter arithmetic expressions like pi/2 forπ

2 . I trust that you can compute a decimal value on your calcu-lator if required. However, if you do enter a decimal approxi-mation compute and enter at least four digits. WeBWorK willconsider your value correct if it is within one tenth of one per-cent of the answer that it has been told.

In this problem WeBWorK will tell you for each answerwhether it’s right or wrong, in the next problem you will have toget everything right before getting credit.

90◦ = rad60◦ = rad327◦ = rad−375◦ = radHint: Everything follows from the fact that

360◦ = 2π rad.

Correct Answers:

• 1.5707963267949• 1.0471975511966• 5.70722665402146• -6.54498469497874

2. (10 pts) set4/s4p2.pgFill in the following equations:

45◦ = rad234◦ = rad−596◦ = radHint: Everything follows from the fact that

360◦ = 2π rad.

Correct Answers:1

• 0.785398163397448• 4.08407044966673• -10.4021623418862

3. (10 pts) set4/s4p3.pgIn this problem you are asked to convert radians to degrees.

πrad = ◦.1.5rad = ◦.−4πrad = ◦.Hint: Everything follows from the fact that

360◦ = 2π rad.

Correct Answers:

• 180• 85.9436692696235• -720

4. (10 pts) set4/s4p4.pgIn this problem you are asked to convert radians to degrees.

6πrad = ◦.−0.7rad = ◦.(√

2)

πrad = ◦.

Hint: Everything follows from the fact that

360◦ = 2π rad.

Correct Answers:

• 1080• -40.1070456591576• 254.558441227157

5. (10 pts) set4/s4p7.pgThe diameter of the wheels on your car (including the tires) is25 inches. You are going to drive 289 miles today. Each of yourwheels is going to turn by an angle of degrees.

Hint: A mile has 5280 feet, a foot has 12 inches, and the cir-cumference of a circle with a diameter d equals πd.

Correct Answers:

• 83931625


You travel again in your trusty car with the 25 inch wheels.You installed a nifty gadget that measures how often yourwheels turn each hour. It shows that right now they turn 53000times per hour. You figure you are going at a speed ofmiles per hour. (Round your answer to the nearest tenth of amile per hour.)

Hint: Multiply the circumference of the wheel with the numberof rotations per hour. Convert inches to miles.

Correct Answers:

• 65.7

7. (10 pts) set4/s4p8.pgPositions on earth are defined by their longitude (how far eastor west you are) and latitude (how far north or south you are).Both latitude and longitude are angles.

A nautical mile is one minute of degree of longitude alongthe equator. (A nautical mile is a little more than a regular mile.)Traveling at x knots means traveling at x nautical miles per hour.

Suppose you fly in a Lockheed SR-71 (the fastest jet everbuilt) along the equator at a speed of 1,800 knots (roughly thetop speed of the SR-71).

Ignoring issues of refueling and such, it takes you hoursto fly once around the world. You leave after breakfast and areback in time for a late dinner. Some airplane!Hint: Flying once around the equator you cover 360 degreesof longitude.

Correct Answers:

• 12


This problem outlines how Eratosthenes of Cyrene approx-imated the diameter of the earth in approximately 200BC. Byobserving the shadow of the sun at noon he recognized that hishome town of Alexandria was approximately 7.2◦ north of thetown of Syene. He paid somebody to walk and measure thedistance between Syene and Alexandria. Suppose it is 787 km.Based on these figures, the circumference of the earth iskilometers.Hint: Observe that 7.2◦ is to 360 degrees as walking 787 km isto walking the entire circumference.

Correct Answers:

• 39350

9. (10 pts) set4/s4p11.pgRecall than an angle is acute if it’s between 0 and π

2 radians,right if it equals π

2 radians, and obtuse if is between π

2 and π

radians. For each angle below enter R if it is a right angle, A ifit’s an acute angle, and O (upper case o, not the digit 0) if it isan obtuse angle.

90◦

60◦

130◦

1rad2radπ

2 radCorrect Answers:

• R• A• O• A• O• R

2

10. (10 pts) set4/s4p12.pgConsider a regular 12 hour clock, and measure time in hoursstarting from noon. Let m denote the angle that the minute handmakes with the top position (12). Similarly, let h be the anglethat the hour hand makes with the vertical position. In this case,suppose we measure the angle clockwise, so that the angles in-volved are positive.

At 2:30pm we have

h = degrees.At that time the minute hand has completed two and half rota-tions, and so the anglem = degrees. (Enter a number between 720 and 1080.)In general, at time t (expressed in hours so that e.g. 2:30 is 2.5hours)h = degrees, andm = degrees.

There is a time t between 2 and 3 pm at which the minutehand is on top of the hour hand. At that time the hour handforms an angle x of degrees with the vertical.Hint: Set up and solve an equation for the time where the twohands point in the same direction, and then multiply the timewith 30 to get the required angle.

Correct Answers:

• 75• 900• 30t• 360t• 65.4545454545455

11. (10 pts) set4/s4p13.pgThe remaining problems in this set deal with the definitions ofthe basic trigonometric functions, sin, cos, and tan.

You can answer some of these questions simply by keying thingsinto your calculator. However, the purpose of these problems isto help you get familiar with the definitions of the basic trigono-metric functions, and to improve your ability to work with thosedefinitions. All the questions can be answered straight from thedefinitions of the trigonometric functions, perhaps after drawinga simple picture, without the aid of a calculator, and I recom-mend that you don’t use one. Use ’pi’ to enter the value of π

and use sqrt(...) to enter the square root of something.

A line drawn from the origin and forming the angle of t = 7π

6with the positive x-axis intersects the unit circle at the point(−√

32 ,− 1

2

). Complete the following equations:

t = degrees.

cos t = .

sin t = .

tan t = .Correct Answers:

• 210• -0.866025403784439• -0.5• 0.577350269189626

12. (10 pts) set4/s4p14.pg

A line drawn from the origin and forming the angle t with thex-axis intersects the unit circle at the point

(13 , 2

√2

3

). Complete

the following equations:

cos t = .sin t = .tan t = .

Correct Answers:• 0.333333333333333• 0.942809041582063• 2.82842712474619

13. (10 pts) set4/s4p15.pg

A line drawn from the origin and forming the angle t with thex-axis intersects the unit circle at the point

(√5

3 ,− 23

). Complete

the following equations:

cos t = .sin t = .tan t = .

Correct Answers:• 0.74535599249993• -0.666666666666667• -0.894427190999916

14. (10 pts) set4/s4p16.pg

Let t be the angle between 0 and π

2 such that

sin t =14.

Then

cos t = .sin(−t) = .cos(−t) = .tan t = .

Correct Answers:• 0.968245836551854• -0.25• 0.968245836551854• 0.258198889747161

3

15. (10 pts) set4/s4p17.pg

Let again t be the angle between 0 and π

2 such that

sin t =14.

Then

cos(t +π) = .sin(t−π) = .cos(t +2π) = .tan(t +π) = .

Correct Answers:• -0.968245836551854• -0.25• 0.968245836551854• 0.258198889747161

16. (10 pts) set4/s4p18.pg

Here are the values of the trigonometric functions at somebasic angles. You need not memorize those, but you should beable to figure them out, for example by drawing a suitable pic-ture involving a triangle or the unit circle.

cos(0) = .sin(0) = .tan(0) = .cos(π

2 ) = .sin(π

2 ) = .Correct Answers:

• 1• 0• 0• 0• 1

17. (10 pts) set4/s4p19.pg

The angle π

4 equals degrees.

Here are some more basic trig values:

cos(

π

4

)= .

sin(

π

4

)= .

tan(

π

4

)= .

Correct Answers:

• 45• 0.707106781186547• 0.707106781186547• 1

18. (10 pts) set4/s4p20.pg

The angle π

3 equals degrees.

More basic values:

cos(

π

3

)= .

sin(

π

3

)= .

tan(

π

3

)= .

Correct Answers:

• 60• 0.5• 0.866025403784439• 1.73205080756888

19. (10 pts) set4/s4p21.pg

The angle π

6 equals degrees.

And more basic trig values:

cos(

π

6

)= .

sin(

π

6

)= .

tan(

π

6

)= .

Correct Answers:

• 30• 0.866025403784439• 0.5• 0.577350269189626


4

U of U Math 1060-1

Spring 2009

Hsiang-Ping Huang.


due 02/25/2009 at 11:00pm MST.This set covers section 4.5 of the textbook. It is mostly about

the graphs of the sine and the cosine functions.In all of the problems in this set angles are measured in radi-

ans.Do let me know if you have any questions. Use the feedback

button, it tells me more than an ordinary e-mail.

1. (10 pts) set6/q4.pg

For each function below enter the letter E if the function iseven, the letter O if it’s odd, and the letter N if it’s neither evennor odd.

f (x) = sinx.f (x) = cosx.f (x) = tanx.f (x) = (sinx)(cosx).f (x) = sin2 x.

Note: Remember that this notation means the square of sinx,i.e.,

sin2 x = (sinx)2.

f (x) = cos2 xf (x) = sinx+ cosx

Correct Answers:• O• E• O• O• E• E• N


This is a little more tricky, but there are only two functions.Try to answer the question without guessing. For each functionbelow enter the letter E if the function is even, the letter O if it’sodd, and the letter N if it’s neither even nor odd.

f (x) = sin(cos(x)).f (x) = sin |x|.

Correct Answers:• E• E


Let

f (x) = 3sin(5x−4)+2.

The amplitude of this function is ,its period is ,its phase shift is ,and its vertical translation is ,

Correct Answers:

• 3• 1.25663706143592• 0.8• 2


Let

f (x) = 7sin(3πx−2)−3.


Correct Answers:

• 7• 0.666666666666667• 0.212206590789194• -3

5. (10 pts) set6/q3.pgLet

f (x) = 12sin(4x−2)+3.


Correct Answers:

• 12• 1.5707963267949• 0.5• 3

1

6. (10 pts) set6/s6p1.pgOne of the interesting aspects of trigonometric functions is thattheir representation is not unique. For example, we saw in classthat

cosx =−cos(x+π).There are many more so called trigonometric identities aboutwhich we will learn later in this course.

If functions can be represented differently you may be wor-ried that you get the right answer, but WeBWorK won’t rec-ognize it because it thinks of a different representation of theanswer than you do.

Remember how WeBWorK tests algebraic expressions thatyou enter: It evaluates your expression, and the expression thatit has been given, at several random values of the variables, andcompares the results. If they agree it deems your expression cor-rect, no matter how you wrote it. So you need not worry aboutthe precise form in which you enter an answer!

The purpose of this exercise is to illustrate that capability ofWeBWorK, and to reassure you.

Enter here

the expressioncosx

Write it as cos(x), or just cosx, and you’ll see that WeBWorKwill accept your answer.

Now enter your answer also as-−cos(x−π) — written as−cos(x−pi) (cut and paste this

expression),- cos(x+100π) — written as cos(x+100∗pi), andsin

(x+ π

2

)— written as sin(x+pi/2).

Finally, here’s a little hocus-pocus: enter the expression

2cos2( x

2

)−1

(written as 2∗cos(x/2)∗∗2−1 ) and notice that as far as WeB-WorK (and mathematics) is concerned, this is the same as cosx.If you believe this is a trick question, and ww will simply acceptanything in this question, try it! Enter anything other than theexpressions listed here. If you are enterprising, try to come upwith some other expressions that do work. If you do send me anemail!

WeBWorK will accept any and all of these expressions, be-cause they are all equal (and the functions defined by them havethe same graph).

Correct Answers:• cos(x)

7. (10 pts) set6/s6p2.pgIn the next few problems all the answers are expressions of theform

asin(bx− c)+d or acos(bx− c)+dwhere the parameters a, b, c and d are simple numbers, usuallysmall integers or small integer fractions such as 1/2, and some-times a simple multiple of π.

Often they are very simple, for example the choice

a = b = 1, c = d = 0

in the first expression just gives sinx. In any case, you are notasked to enter values of the parameters, your answers should bealgebraic expression involving the trigonometric functions.

The graphs may show up more clearly in a browser than on theprinted hard copy of this assignment.

The figure

shows the graph of the functionf (x) = .The figure

shows the graph of the functionf (x) = .

Correct Answers:

• sin(x)• cos(x)

8. (10 pts) set6/s6p3.pgThe figure



Correct Answers:

• -cos(x)• -sin(x)

2




Correct Answers:• sin(2x)• sin(3.14159265358979*x)




Correct Answers:• 2*sin(x)• 3*sin(x)

11. (10 pts) set6/s6p6.pgIn this problem the functions you see are horizontally shiftedsine function. The figure



Correct Answers:

• sin(x-1)• sin(x-2)



Correct Answers:

• sin(x)+1



Correct Answers:

• cos(3.14159265358979*x)+1



Correct Answers:

• 1/2*cos(2*x)+1/2


3

U of U Math 1060-1

Spring 2009

Hsiang-Ping Huang.


due 03/04/2009 at 11:00pm MST.

This homework covers sections 4.6 and 4.7 of the textbook.Section 4.6 is about the graph of the tangent function and 4.7 ison inverse trig functions, i.e., arcsin, arccos, and arctan. Work-ing with these functions can get pretty subtle and sophisticated,but for our purposes we need them basically just for comput-ing angles in triangles, given some trigonometric function of anangle.

Some of the questions in this set expect you to enter a num-ber, but ususally you can use expression like asin(x), acos(x),and atan(x) in your webwork answers. You can set your calcu-lator so that these functions return angles measured in degreesor in radians, but in webwork (and virtually all programminglanguages) these functions return angles measured in radi-ans!


1. (10 pts) set7/s6p10.pgWe start with a couple of examples of graphs related to the tanfunction.

The figure

shows the graph of the functionf (x) = . The figure


Correct Answers:

• tan(x)• -tan(x)



Correct Answers:

• tan(2*x)

1



Correct Answers:

• tan(3.14159265358979*x)

4. (10 pts) set7/s6p13.pgA room measures 12 feet in the North-South direction, 15 feetin the East-West Direction, and its ceiling is 10 feet high. Lett denote the the angle that the line from the bottom Southeastcorner to the top Northwest corner of the room forms with thefloor. Thentan t = .Hint: This is much like the box problem on set 3. Considerthe right triangle formed by the line from the bottom Southeastcorner to the bottom Northwest corner, the line from the bottomSoutheast Corner to the top Northwest corner, and the line fromthe bottom Northwest corner to the top Northwest corner.

Correct Answers:

• 0.520579206295354

5. (10 pts) set7/s1.pgThis and the following three problems cover some subtleties ofthe inverse trig functions. In these problem, WeBWorK willnot accept as answers mathematical expressions that involvethe trigonometric functions or their inverses. You need to en-ter numbers.

In this problem let’s measure angles in degrees. As discussed inclass, the inverse sin function returns an angle between -90 and+90 degrees. This gives perhaps surprising results when youevaluate the sine somewhere and then the inverse sine at the re-sult. You can just key the problems below and in the followingproblems into your calculator, but try to get them without a cal-culator. If you yield to temptation and do use your calculator, atleast think about the answers when they are unexpected.

Complete the following equations:arcsin(sin(37◦)) = ◦,arcsin(sin(−25◦)) = ◦,arcsin(sin(100◦)) = ◦,

Correct Answers:

• 37• -25• 80

6. (10 pts) set7/s2.pgThis is like the previous problem, except that angles are mea-sured in radians.Complete the following equations:arcsin(sin(1)) = ,arcsin(sin(−1.1)) = ,arcsin(sin(2)) = .

Correct Answers:

• 1• -1.1• 1.14159265358979

7. (10 pts) set7/s3.pg

Complete the following equations:arccos(cos(37◦)) = ◦,arccos(cos(−25◦)) = ◦,arccos(cos(100◦)) = ◦,

Correct Answers:

• 37• 25• 100


Complete the following equations:arctan(tan(37◦)) = ◦,arctan(tan(−25◦)) = ◦,arctan(tan(100◦)) = ◦,

Correct Answers:

• 37• -25• -80

9. (10 pts) set7/graphs/prob.pgMatch the functions with their graphs.

1. f(x) = cos(x)2. f(x) = sin(x)3. f(x) = tan(x)4. f(x) = arcsin(x)5. f(x) = arccos(x)6. f(x) = arctan(x)

2

A B C D E F

(Click on image for a larger view . The small images maynot show up properly on a hard copy, but they will be fine in abrowser.)

Correct Answers:• D• B• A• C• F• E

10. (10 pts) set7/q1.pgSuppose you have a right triangle whose two short sides areof length 3 and 4 respectively. Then the smaller of the twoacute angles is degrees and the larger acute angleis degrees. If you enter your angles as decimalapproximations compute at least one digit beyond the decimalpoint.Hint: Compute the length of the hypotenuse, use it to computesines of the angles in question, and then apply the inverse sinefunction. Set your calculator to degree mode. If you use the wwasin function make sure you convert to degrees.

Correct Answers:• 36.869897645844• 53.130102354156

11. (10 pts) set7/q2.pgSuppose you have a right triangle whose hypotenuse has length8. One of the other sides has length 5. Then the smaller of thetwo acute angles is degrees and the larger acuteangle is degreesHint: Compute the length of the missing short side, use it tocompute sines of the angles in question, and then apply the in-verse sine function. Set your calculator to degree mode. If youuse the ww asin function make sure you convert to degrees.

Correct Answers:• 38.6821874534894• 51.3178125465106

12. (10 pts) set7/q3.pgYou are driving along the highway and see a sign indicating thatyou are about to enter a downhill slope of 10%. You realize thatyou are going to travel downward at an angle of degreesto the horizontal. (Enter your answer as a mathematical expres-sion or with at least two digits beyond the decimal point.)Hint: A descent of 10% means that you are going to descend by1 foot for every ten feet that you travel in a horizontal direction.The angle of your descent is the acute angle that the highwaymakes with a horizontal line. Draw a picture and use an inversetrig function.


13. (10 pts) set7/q4.pgYou are about to land your Cessna airplane in Salt Lake City.You are approaching the runway at a ground speed of 78 milesper hour and you are sinking at 380 feet per minute. (The groundspeed is the speed of the point on the ground directly underneathyour plane. You can also think of it as the horizontal componentof your current velocity.) You are going to hit the runway at anangle of degrees. (Enter your answer as a mathe-matical expression, or with at least three digits beyond the deci-mal point.)Hint: Draw a triangle showing your descent and the horizontaldistance you cover in one minute.


14. (10 pts) set7/q6.pgYou just took off in your trusty Lockheed SR-71. It’s a finewindless day. Your instruments indicate a true air speed (speedrelative to the surrounding air) of 400 miles per hour and a climbrate (vertical speed) of 10,000 feet per minute. You figure thatyour flight path makes an angle of degrees withthe horizontal. (Enter a mathematical expression or a decimalapproximation with at least one digit beyond the decimal point.)Hint: Draw a triangle showing your ascent and the distancealong your flight path that you travel in one minute. Apply aninverse trig function.


15. (10 pts) set7/q7.pgGeographic directions are described in degrees, counting clock-wise from a line going due north. Thus, for example, due northis zero degrees, east is 90 degrees, and southwest is 225 degrees.

All distances in this question are measured horizontally as youwould measure them on a map. You can also think of flyingat the altitude of the peak in question, instead of driving alongthe highway. The direction in which you see the peak is calledits bearing . So a bearing of 135 degrees means it’s southeastof you. Enter your answers as mathematical expressions (rec-ommended) or as decimal approximations with at least 4 digitstotal.

You are driving at a constant speed of sixty miles per hour alonga straight road going north. You see a prominent peak at a bear-ing of 45 degrees, and you know that that peak is 10 mileseast of the road. At this time your distance from the peak is

miles. Five minutes later you see the peak at a bearingof degrees. After another five minutes the peak isdue east of you. At that precise spot there is a historical markerthat tells you about the peak. 7 minutes after you pass the markerthe peak is at a bearing of degrees. You sit backin your car and reflect on the pleasant fact that the trigonometry

3

class you are taking makes it possible for you to figure out thatt minutes after you pass the historical marker the bearing of thepeak is degrees. (Enter a mathematical expressioninvolving the variable t.)Hint: Draw a picture. Think of the Pythagorean Theorem. 60miles per hour means a mile a minute. For the last two partsof this problem think of the angle you need to add to 90 de-grees to get the bearing of the peak. You will need to use an

inverse trig function. Remember that in WeBWorK the inversetrig functions return an angle measured in radians. You willneed to convert it to degrees.

Correct Answers:

• 14.142135623731• 63.434948822922• 124.992020198559• 90+atan(t/10)/3.14159265358979*180


4

U of U Math 1060-1

Spring 2009

Hsiang-Ping Huang.


due 03/12/2009 at 12:00am MDT.

The second midterm is coming up on Wednesday, March 11.It will cover homework assignments 5 through 8 which is sec-tions 4.3 to 4.8 in the book. The best way to study is to go overthe homework assignments, including this assignment.

This homework covers section 4.8, applications of trigonom-etry. So it’s a collection of word problems! It’s important that ifyou enter decimal approximations as your answer then useat least four digits! Even though that accuracy may not makesense in the context of the problems you are solving, ww mayotherwise reject your answer as being too inaccurate. Actually,I strongly recommend entering your answers as mathematicalexpressions involving the trig functions and their inverses. Thatway I can see what you are doing and give you a pointer if yousend me an email. Remember that ww knows only about radi-ans, so if the problem asks for degrees, convert!

When your answer turns out to be incorrect, look at the num-ber that ww displays next to that dreaded word. Is this numberin the right ballpark? If not, check your equations again. Anddraw a picture.

As in an earlier problem, in this set we measure bearings ona scale from 0 to 360 degrees, with 0 (or 360) degrees beingdue north, 90 degrees being due east, 225 degrees being south-west, etc. (The textbook uses a different convention which is notwidely used. However, you can easily translate back and forthbetween the two conventions.)

There aren’t as many problems as usual, but most likely youwill find them a little more complicated than most of the wwproblems you have solved so far. In all cases it helps to drawa picture (I can’t emphasize this enough), and to think care-fully about what each part of the problem means in terms of thepicture.

Later in the semester we will learn some advanced tools thatwill make it much easier to solve problems like some of those inthis set. At this stage, however, just use what you’ve learned upto now. These are basically the definitions of the trigonometricfunctions and their inverses, their relation to a right triangle, anddon’t forget the Pythagorean Theorem!


1. (10 pts) set8/s1.pgYou are building a (rectangular) pool in your backyard. It mea-sures 30m in the north-south direction and 25 meters east-west.(Some pool!) The bottom of the pool is a slanted plane so thatit’s 1m deep along the southern edge and 10m deep along thenorthern edge. (As I said, some pool!)To build the pool you compute the angle of depression of aline from the southeast corner to the northeast corner. It is

degrees,Hint: A similar pool is shown in Figure 4.71 on page 423 of thetextbook.

Correct Answers:

• 16.6992442339936

2. (10 pts) set8/s2.pgConsider the same pool as described in the previous question.You are artistically inclined and plan to have a bright red line atthe bottom of your pool from the southeast corner to the north-west corner. You figure that the angle of depression of that lineis degrees,Hint: This is like the preceding problem. First figure out thedistance between the southeast and northwest corners at the sur-face of the pool.

Correct Answers:

• 12.9781411911717

3. (10 pts) set8/s3.pgThe Great Pyramid of Cheops has a square base with a sidelength of 756 feet. Its height is 482 feet. If you walk straight upfrom the center of the north side to the top of the pyramid youhave to climb an angle of degrees.You decide to simplify your life and walk up along one of theridges. Thus you have to climb only at an angle of de-grees.On your way up the ridge you walk a distance of feet.Hint: For the first two parts draw right triangles and use an in-verse trig function. For the third part just use the PythagoreanTheorem.

Correct Answers:

• 51.8953094143326• 42.0395484835217• 719.786079331908

4. (10 pts) set8/s4.pgThe diameter of the moon is approximately 2,160 miles. Its av-erage distance from earth is approximately 236,000 miles. Con-sider the angle θ at the tip of the isosceles triangle formed byyour eyes, and the North and South extremities of the moon. Itis approximately minutes of degree. (We say that themoon subtends the angle θ.)The diameter of the sun is approximately 864,000 miles. Its av-erage distance from earth is approximately 92,900,000 miles. Itsubtends an angle of minutes of degree.Hint: Draw the height in that isosceles triangle and compute the

1

angle at the top of the triangle. Multiply with 2. A degree has60 minutes of degree.

Correct Answers:• 31.4639033463449• 31.9719246602531

5. (10 pts) set8/s5.pgThe diameter of the star Betelgeuse in the constellation of Orionis approximately 200 million miles. It’s distance from earth isapproximately 3× 1015 (3 quadrillion) miles (about 507 lightyears). It subtends the angle seconds of degree. A pennyhas a diameter of 0.75 inches. You figure that Betelgeuse sub-tends the same angle as a penny at a distance of miles.Hint: One degree is 3600 seconds of degree. Proceed as in thepreceding problem, the validity of the mathematics is indepen-dent of the scale of things. For the second part of the problemthink in terms of similar triangles.

Correct Answers:• 0.0137509870831398• 177.556818181818


You are on a stationary fishing boat, and you send out yourlife boat to retrieve some gear. The boat moves one nautical mileeast, and then 1.5 nautical miles northeast. Then it stops. Youcan see the boat at a bearing of degrees and at a distanceof nautical miles.Hint: Draw a picture. Consider two right triangles. Apply thePythagorean Theorem and an inverse trig function.

Correct Answers:• 62.7642760796095• 2.31761091289277

7. (10 pts) set8/s7.pgYou are relaxing in your stationary yacht. You see a fishing boatdue north. Your laser range finder tells you that it is 2 nauti-cal miles away. The boat is moving in a straight line in a veryroughly southeasterly direction. You do know that it is moving

at 4 knots (because, let’s presume, that’s what fishing boats dowhen reeling in their gear). After one hour your laser range me-ter measures your distance to the boat as 3 nautical miles. Un-fortunately your compass is no longer working. Undaunted, andfortified by your Math 1060 training, you compute your bearingto the boat as degrees.Hint: Moving at one knot means moving at one nautical mileper hour. Draw a picture. Consider two right triangles. Applythe Pythagorean Theorem and an inverse trig function.



You are hiking north along the west shore of a river that’s flow-ing due north. You notice a tree on the far shore at a bearingof 30 degrees. You walk on for another 100 feet and you arestopped by an unclimbable cliff. You contemplate swimmingacross the river and wonder how wide it is. The tree on the otherside now appears at a bearing of 45 degrees. Remembering yourtrig class, you figure out that the river is feet wide.Hint: Again, consider two right triangles. One of them isisosceles, which simplifies things.



You are driving south along the shore of the Mississippi River.At this point it is flowing due South. You notice a prominentradio tower right on the far shore, at a bearing of 115 degrees.You drive on for 2 miles and notice that now the radio towerappears at a bearing of 55 degrees. You figure that at this pointthe Mississippi river is miles wide.Hint: Draw two right triangles, set up some equations, andsolve them for the width of the river.



2

U of U Math 1060-1

Spring 2009

Hsiang-Ping Huang.


due 03/25/2009 at 11:00pm MDT.This homework covers sections 5.1 and 5.2, a first introduc-

tion to trigonometric identities.For your convenience, here is a list of some basic identities,

together with an indication of where they come from. You willencounter many more identities in this course, but you shouldnot attempt to memorize these identities as abstract formulas.Instead, you should understand the definition and properties ofthe trig functions well enough that you can’t help being aware ofthe identity, or at least you should be able to derive them whenrequired.

- Since the sin and cos of an angle u are the coordinates of apoint on the unit circle we have

sin2 u+ cos2 u = 1

for all angles u. The ratio of the sin and cos is called the tangent:

tan(u) =sinucosu

.

This is simply the definition of the tangent.- The reciprocals of the three trigonometric function above

are called cosecant, secant, and cotangent, respectively. We areignoring them in this class, but you need to be aware of their ex-istence. If you encounter them in an application (outside of thisclass) you can derive everything you need to know about themfrom their definition.

- Since rotating a point (x,y) halfway around the unit circlereplaces it with (−x,−y) we have

sin(u+π) =−sinu, cos(u+π) =−cosu,

andtan(u+π) = tan(u).

Note that the sin and cos are 2π periodic and the tan is π peri-odic.

- Suppose the point (x,y) has been obtained by rotating thepoint (1,0) by the angle u. If instead we rotate by the angle −uwe obtain the point (−y,x). Thus

cos(−u) = cosu, sin(−u) =−sinu,

andtan(−u) =− tanu.

In other words, the sin and tan are odd functions, the cos is even.- Rotating the point (x,y) counterclockwise a quarter of the

way around the unit circle replaces it with the point (−y,x). To

see this, it is easier to think of rotating the coordinate systemclockwise by 90 degrees. Hence:

cos(

u+π

2

)=−sinu, and sin

(u+

π

2

)= cosu.

The tan is a little more tricky:

tan(

u+π

2

)=

−1tanu

.

Note that:Unless otherwise stated, angles in this homework set aremeasured in radians.


Suppose sin(u) = 35 and cos(u) is positive. Then

cos(u) =tan(u) =sin(−u) =cos(−u) =tan(−u) =sin(u+π) =cos(u+π) =tan(u+π) =sin

(u+ π

2

)=

cos(u+ π

2

)=

tan(u+ π

2

)=

Correct Answers:

• 0.8• 0.75• -0.6• 0.8• -0.75• -0.6• -0.8• 0.75• 0.8• -0.6• -1.33333333333333


Suppose sin(u) = 35 and cos(u) is negative. Then


(u+ π

2

)=

cos(u+ π

2

)=

tan(u+ π

2

)=

Correct Answers:1

• -0.8• -0.75• -0.6• -0.8• 0.75• -0.6• 0.8• -0.75• -0.8• -0.6• 1.33333333333333


Suppose sin(u) = 1213 and cos(u) is negative. Note that cosu is a

rational number. I recommend you enter fractions below.


(u+ π

2

)=

cos(u+ π

2

)=

tan(u+ π

2

)=

Correct Answers:

• -0.384615384615385• -2.4• -0.923076923076923• -0.384615384615385• 2.4• -0.923076923076923• 0.384615384615385• -2.4• -0.384615384615385• -0.923076923076923• 0.416666666666667


Suppose cosu = 513 and sinu is negative. Here are some

small variations on the previous problems:

sin(u) =sin(u−π) =cos(u−π) =sin

(u− π

2

)=

cos(u− π

2

)=

Correct Answers:

• -0.923076923076923• 0.923076923076923• -0.384615384615385• -0.384615384615385

• -0.923076923076923


Suppose cosu = 35 and sinu is positive.

sin(u) =sin(u−π) =cos(u−π) =sin

(u− π

2

)=

cos(u− π

2

)=

Correct Answers:

• 0.8• -0.8• -0.6• -0.6• 0.8

6. (10 pts) set9/s6.pgThe subject of this homework set is trigonometric identities.WeBWorK has the usually useful feature that it will not dis-tinguish between any equivalent expressions. For example if Iasked you to express sin2 x in terms of cosx, and told ww thatthe correct answer is 1− cos2 x, then WeBWorK would be justas happy if you typed sin2x, and so bypassed the entire problem!

To finesse this feature we have to do something contrived. Ineach of the next few problems I will tell you to use a differentvariable for certain things.

Here is an easy warm up question and illustration:

Use C to denote cosx, and express sin2 x in terms of C:sin2(x) = .Since sin2 x + cos2 x = 1, in this case you would of course en-ter 1−C2 as your answer. You can type it, for example, as1−C∗∗2 or 1−C ˆ 2. If you feel adventurous you can alsotype (1−C)(1+C), but you must use C instead of cosx!

Note that mathematics and WeBWorK are case sensitive, soyou must use upper case C in this problem. For the fun of it youmight check the cryptic error message you get when you use alower case c or some other variable!

Computers are fun!Let’s see if you got all this. Use S to denote sinx, and express

cos2 x in terms of S: cos2(x) = .Correct Answers:

• 1-C*C• 1-S*S

2


Use C to denote cosx, and express1+ tan2(x) = in terms of C.Similarly, express(1− sinx)(1+ sinx) = in terms of C.Hint: For the first part use the definition of the tangent function.For the second part use the third binomial formula:

(a+b)(a−b) = a2−b2.

Correct Answers:

• 1/(C*C)• C*C

8. (10 pts) set9/s8.pgSometimes, however, things are simpler than they appear. Us-ing the results of the preceding problem, simplify as much aspossible:(1− sinx)(1+ sinx)(1+ tan2(x)) =This is a little more tricky: Simplify as much as possible:sin4 x− cos4 x− sin2 x+ cos2 x =

Correct Answers:

• 1• 0


Use C to denote cosx, and expresscos(x+20,000π) = in terms of C.

Similarly, expresscos(x+20,001π) = in terms of C.

Correct Answers:

• C• -C

10. (10 pts) set9/s10.pgUse T to denote tanx, and expresstan(x+20,000π) = in terms of T .Similarly, expresstan(x+20,001π) = in terms of T .

Correct Answers:

• T• T

11. (10 pts) set9/s12.pgConsider two fire towers in the Canadian wilderness. Tower Bis 40 miles due East of tower A. The guards notice a fire thatappears at a bearing of 70 degrees from Tower A, and a bearingof 310 degrees from Tower B.

The fire is at a distance miles from the tower B (thecloser of the two towers).

Hint: Draw a picture. This problem is similar to several prob-lems in set 8.

Correct Answers:

• 15.7972337453879


3

U of U Math 1060-1

Spring 2009

Hsiang-Ping Huang.


due 04/01/2009 at 11:00pm MDT.This homework set covers section 5.3 of the textbook.Unless otherwise stated, angles are measured in radians.1. (10 pts) set10/p1.pg

The smallest positive number for which tan(3x) = 1 is x =. The next larger such number is x = .

Hint: Apply an inverse trig function and use the periodicity ofthe tan function.

Correct Answers:• 0.261799387799149• 1.30899693899575

2. (10 pts) set10/p2.pgThe smallest positive number for which 3tanx = 1 is x = .The next larger such number is x = .Hint: This is the like the preceding problem, except that youreverse the sequence of applying the inverse trig function anddividing by 3.

Correct Answers:• 0.321750554396642• 3.46334320798644

3. (10 pts) set10/p3.pgThe smallest positive number for which 3sinx = 1 is x = .The next larger such number is x = .Hint: Divide by 3 and solve the resulting equation. Apply aninverse trig function and exploit the symmetry of the sin func-tion.

Correct Answers:• 0.339836909454122• 2.80175574413567

4. (10 pts) set10/p4.pgThe smallest positive number for which 3sin(2x−1) = 1 is x =

.Hint: Isolate the sin, apply an inverse trig function, and solvethe resulting linear equation.


5. (10 pts) set10/p5.pgThe smallest positive number for which 3sin(2x−6) = 1 is x =

.Hint: Proceed as in the preceding problem to find some solutionof the equation. Then look for the smallest positive.

Correct Answers:

• 0.0283258011372678

6. (10 pts) set10/p6.pg

The smallest positive number for which 3sin(2x− 5) = 1 isx = .Hint: This is much like the preceding problem.

Correct Answers:

• 0.759285218478043

7. (10 pts) set10/p7.pg

The smallest positive number for which 4cos2 x−9cosx+2 = 0is x = .Hint: Solve the quadratic equation for cosx and then solve forx.

Correct Answers:

• 1.31811607165282

8. (10 pts) set10/p8.pg

If you are puzzled by the notation in this problem read text-book section 1.7 on the composition of functions. The actualsolution is quite straightforward.

Let

f (x) = 2x−1

The smallest positive number for which

f(

cos(

f (x)))

= 0

is x = .Hint: Begin by using the definition of f to remove f from theequation. Then solve a linear equation for cos(2x− 1). Thensolve for x.

Correct Answers:

• 1.0235987755983


1

U of U Math 1060-1

Spring 2009

Hsiang-Ping Huang.


due 04/08/2009 at 11:00pm MDT.This homework covers sections 5.4-5.5.1. (10 pts) set10/p9.pg

Let u and v be angles in the first quadrant, and let

sinu =13

and sinv =14.

Then cosu = ,cosv = ,sin(u+ v) = .Hint: Apply a suitable sum formula.

Correct Answers:• 0.942809041582063• 0.968245836551854• 0.558450872579467

2. (10 pts) set10/p10.pgSuppose you wish to express sin(3t) in terms of sin t and cos t.Apply the sum formula to sin(3t) = sin(t +2t) to obtain an ex-pression that contains sin(2t) = sin(t + t) and cos(2t) = cos(t +t). Apply the sum formulas to those two expressions. Enter theresulting expression for sin(3t) here

,using S to denote sint and C to denote cos t. For example, if youranswer was 3sin t cos t you’d simply enter 3∗S∗C. Rememberthat WeBWorK (and mathematics) is case sensitive! You maybe able to find an equivalent expression for sin(3t) in the liter-ature, but WeBWorK will not recognize the equivalence. Youhave to follow the procedure outlined above.

Hint: Apply suitable sum formulas twice.Correct Answers:

• 3*S*C**2-S**3

3. (10 pts) set10/q1.pgA ship is moving due west at 8 knots. You are in a speed boat√

2 nautical miles directly southeast of the ship. (Thus yourbearing as seen from the ship is 135 degrees.) You need to catchup with the ship, and you can move at a speed of 16 knots. Soyou take off at a bearing of degrees, and you reachthe ship in minutes. Enter your answers as deci-mal expression with at least four digits, or enter mathematicalexpressions.Hint: Draw two right triangles.

Correct Answers:

• 294.295188945365• 9.11437827766148

4. (10 pts) set10/q2.pgThis is like the preceding problem, except that the numbers area little different. A ship is moving due west at 10 knots. Youare in a speed boat at a distance of 7 nautical miles from theship. Your bearing as seen from the ship is 106 degrees. Youneed to catch up with the ship. So you take off at a speed of14 knots and a bearing of degrees, and you reachthe ship in minutes. Enter your answers as dec-imal expression with at least four digits, or enter mathematicalexpressions.

Correct Answers:

• 274.645207503537• 102.106342229926

5. (10 pts) set11/p1.pgThe smallest positive number for which

sin(2x)− sinx = 0

is x = .Hint: Apply a multiple angle or difference to product rule.

Correct Answers:

• 1.0471975511966


sin(x+1) = sin(x)

is x = .Hint: Apply a sum to product formula.

Correct Answers:

• 1.0707963267949


sin t + cos(2t) = 0

is t = .Hint: Apply a multiple angle formula and solve a quadraticequation. Or, draw the graph and guess the answer.

Correct Answers:

• 1.5707963267949


sin t− cos(2t) = 0

is t = .Hint: This is like the preceding problem.

Correct Answers:

• 0.523598775598299

1


2

U of U Math 1060-1

Spring 2009

Hsiang-Ping Huang.


due 04/15/2009 at 11:00pm MDT.The second midterm is coming up on Wednesday, April 15.

It will cover homework assignments 9 through 12 which is sec-tions 5.1 to 6.1 in the book. The best way to study is to go overthe homework assignments, including this assignment.

This problem set covers the Law of Sines, and more applica-tions of trigonometry. This is section 6.1.

1. (10 pts) set11/p5.pgIn this and the next few problems we consider general (not nec-essarily or usually right) triangles and use the notation describedon p. 410 of the textbook, and indicated in this Figure:

The angles are labeled A, B, and C, and the sides opposite theseangles have lengths a, b, and c, respectively.Suppose you are given a triangle where:

A = 36◦, B = 41◦ and c = 35.

ThenC = ,a = , andb = .Note: In this problem angles are measured in degrees.

Correct Answers:

• 103• 21.1136246624849• 23.5660626742765

2. (10 pts) set11/p6.pgSuppose you are given a triangle where:

A = 0.7, B = 0.6 and c = 2.

ThenC = ,a = , andb = .Note: In this problem angles are measured in radians.

Correct Answers:

• 1.84159265358979• 1.33716405918707• 1.17199455505754

3. (10 pts) set11/p7.pgSuppose you are given a triangle where:

A = 30◦, a = 12, and b = 10.

ThenB = ,C = , andc = .Note: In this problem angles are measured in degrees.

Correct Answers:

• 24.6243183521641• 125.375681647836• 19.5689661524801

4. (10 pts) set11/p8.pgThere are two triangles for which

A = 30◦, a = 7, and b = 10.

The larger one hasc = , and the smaller one hasc = .

Correct Answers:

• 13.5592335234107• 3.76127455227803

5. (10 pts) set11/p9.pgFor each of the following combinations of A, a, and b indicateif there are 0, 1, or 2 triangles with these data. (Enter the digit0, 1, or 2.)

A = 30◦, b = 10, a = 12 :A = 30◦, b = 10, a = 8 :A = 30◦, b = 10, a = 3 :

Correct Answers:

• 1• 2• 0

1

6. (10 pts) set11/p10.pgSuppose you are given a triangle where: A = 45◦ and b = 10.Then there is one and only one value of a that is less than b andfor which there is a unique triangle with the data A, a, and b.That value of a is .Hint: Look for a right triangle.

Correct Answers:

• 7.07106781186548

7. (10 pts) set11/p11.pgSuppose A = 30◦ and b = 10. Then there is one value of a that

is less than b and for which there is a unique triangle with thedata A, a, and b. That value of a is .


8. (10 pts) set12/area.pgSuppose you are given a triangle where:

a = 28, b = 30, and C = 30◦.

Then the area of the triangle is



2

U of U Math 1060-1

Spring 2009

Hsiang-Ping Huang.

WeBWorK assignment number 12a.

due 04/22/2009 at 11:00pm MDT.

This problem set covers the Law of Sines, the Law ofCosines, and more applications of trigonometry. These are sec-tions 6.1 and 6.2.

1. (10 pts) set12a/p1.pgIn this and the following problems we consider general (not nec-essarily or usually right) triangles and use the notation describedon p. 410 of the textbook, and indicated in this Figure:

The angles are labeled A, B, and C, and the sides opposite theseangles have lengths a, b, and c, respectively. In abstract prob-lems the lengths aren’t specified in any particular units, but weassume that the units are same for all lengths involved.Suppose you are given a triangle with

a = 4, b = 7, c = 10.

ThenA = degrees,B = degrees, andC = degrees.Enter your answers with two digits beyond the decimal point.

Correct Answers:

• 18.1948723387668• 33.1229402077438• 128.682187453489

2. (10 pts) set12a/p2.pgSuppose you are given a triangle with

A = 60◦, b = 6, c = 9.

Thena = ,B = degrees, andC = degrees.Enter your answers with two digits beyond the decimal point.

Correct Answers:

• 7.93725393319377• 40.8933946491309• 79.1066053508691


A = 60◦, a = 5, b = 3.

ThenB = degrees,C = degrees, andc = .Enter your answers with two digits beyond the decimal point.

Correct Answers:

• 31.3064462486731• 88.6935537513269• 5.77200187265877


a = 8, b = 3, c = 8.

ThenA = degrees,B = degrees, andC = degrees.Enter your answers with two digits beyond the decimal point.

Correct Answers:

• 79.1930771251397• 21.6138457497207• 79.1930771251397


A = 46◦, B = 39◦, c = 5.

Thena = , andb = .Enter your answers with at least 3 digits beyond the decimalpoint.

Correct Answers:

• 3.61043780757204• 3.15862146353182

1

6. (10 pts) set12a/p6.pgYou leave your friend behind on the shore and you travel 3 milesdue east in your boat. Then you travel 2 miles northeast. Thenyou travel 1 mile due north. Your friend can see you at a distanceof miles

and at a bearing of degrees.

Enter your answers with at least 3 digits beyond the decimalpoint.

Hint: Carefully draw a picture. Use the Laws of Sines andCosines.

Correct Answers:

• 5.03127304953575• 61.3249499368952

7. (10 pts) set12a/p7.pgOrienteering is a sport in which competitors use a map and acompass to find their way through unfamiliar terrain. Your taskto is to travel along a triangular course. The first leg is due west,and 6,000 meters long. The other two legs are north of the firstleg. The second leg is 4,000 m long, and the third leg is 5,000meters long. Your compass bearing (the direction in which youare moving) during the second leg is degrees

During the third leg of your course your bearing isdegrees.

Enter your answers with at least 2 digits beyond the decimalpoint.

Hint: Carefully draw a picture. Use the Law of Cosines.Correct Answers:

• 34.2288663278126• 131.409622109271

8. (10 pts) set12a/p8.pgYou have a cone shaped bag. At the bottom of the bag is an or-ange with a radius of 2 inches. On top of the orange is a melonwith a radius of 6 inches. It touches the orange and fits snuglyin the bag, touching it in a ring around the orange. Its top is atthe same level as the top of the bag. All of this is illustrated inthis crude figure:

The height of the cone is inches, and its radius isinches.

Hint: Draw a picture. Note that the radii of the orange and themelon meet the cone perpendicularly where it touches the fruit.

Correct Answers:

• 18• 10.3923048454133

9. (10 pts) set12a/p10.pg

Calculating the necessary aircraft heading to counter a windvelocity and proceed along a desired bearing to a destination isa classic problem in aircraft navigation. It makes good use ofthe law of sines and the law of cosines.

Suppose you wish to fly in a certain direction relative to theground. The wind is blowing at 50mph at an angle of 40 degreesto that direction. Your plane is flying at 100mph with respect tothe surrounding air. The situation is illustrated in this Figure(where your desired direction of travel is due East):

Then you head into the wind at an angle ofdegrees (enter your value of α), and your ground

speed is miles per hour (enter your value of x).Hint: Apply the Law of Sines to get the angle α and the law ofcosines to get the ground speed.

Correct Answers:

• 18.7472372510375• 132.996784946447

10. (10 pts) set12a/p11.pgThis is like the preceding problem, except it’s more general.Suppose the speed of the wind is w, the speed of the plane isp, and the ground speed is g. Let A denote the angle your planeneeds to make with the wind, and B the angle that the windmakes with your desired direction of travel, as illustrated in thisFigure:

2

For simplicity assume that the angles are measured in ra-dians so that you can use the WeBWorK trig functions andtheir inverses without conversion.

Then A = .

Your answer will be a mathematical expression in B, p and w.Once you have the angle A you can compute your ground speedusing either the Law of Sines or the Law of Cosines.Hint: Apply the Law of Sines to get the angle.

Correct Answers:

• asin(w*sin(B)/p)

11. (10 pts) set12a/e3.pgThe focus of this problem will be on solving triangles. A tri-angle has three sides and three angles. The awkward phrase”solving a triangle” means given three of those data, find theother three. (It may also include finding other quantities, likefinding some or all of the heights, or finding the area.)In this problem, as usual, the angles are labeled A, B, and C,and the sides opposite these angles have lengths a, b, and c, re-spectively. In abstract problems the lengths aren’t specified inany particular units, but we assume that the units are same forall lengths involved. We can distinguish a number of cases forthe three given data. For each cases we proceed differently. Thecases are:

- abc. Three sides and no angles are given. In that case anytwo sides have to add up to more than the third side. (For exam-ple, there is no triangle for which two sides are one foot long,and the third has a length of one mile.) If so there is a uniquetriangle with those three sides. You can find the angles using theLaw of Cosines. For example, suppose

a = 3, b = 4, c = 6.

ThenB = degrees. (You can compute A and C similarly.)

- Axy One angle and two sides are given. There are two sub-cases:

- abC The two sides make up the angle. There is a uniquetriangle. Find the opposite side using the law of cosines. Thenfind the other angles using the either the law of sines or the lawof cosines. For example, suppose

a = 3, b = 4, C = 40◦.

Then c = .- abA One of the sides is opposite the angle. Apply the law

of sines. There are several subcases:- a is too small, there is no solution.- a is just large enough so that there is a unique right triangle.- a < b but it’s large enough so that there are two triangles.- a = b There is one isosceles triangle.- a > b There is a unique triangle.For example, suppose A = 40◦ and b = 5. Then there will be

no triangle if a < . If in fact a equals that critical valuethen c = .

If a is greater than that critical value, but less than 5, for ex-ample, if a = 4, then there are two triangles, with two possiblevalues of B. The smaller of these two values is

B = degreesand the larger is degrees.- ABx. If we know two angles then we know all three angles.

Given two angles and one side we can use the Law of Sines tofind all sides, and there is unique angle. Of course the given twoangles need to add to less than 180 degrees, otherwise there isno triangle. For example, if

A = 30◦, B = 80◦, c = 3,

then a= .- ABC. If we only know three angles (adding to 180 degrees)

and no sides then there are infinitely many triangles with thoseangles. Those triangles are called similar . They have the prop-erty that ratios of corresponding sides are equal.

Correct Answers:• 36.3360575146139• 2.57195127581075• 3.2139380484327• 3.83022221559489• 53.4641490143885• 126.535850985612• 1.59626665871387


3

U of U Math 1060-1

Spring 2009

Hsiang-Ping Huang.


due 04/29/2009 at 11:00pm MDT.

We are approaching the end of a long road. The purpose ofthis set is to help you review the semester and to prepare for thefinal exam. Recall that the final exam will take place

Monday, May 4, 2009, 10:30-12:30

in our regular classroom.As you know, I believe one can learn math only by working

a good number of reasonably complicated problems. We havebeen doing this all semester long. On the other hand, exams arean artificial situation full of stress and distraction, and so theyshould contain only, or at least mostly, simple questions. Theproblems in this set and on the final itself will indeed all be sim-ple! You want to make sure that you can solve questions likethose on this home work set, confidently, easily, accurately, andreasonably quickly. Your goal is not just to answer these ques-tions and get the credit for your answers, but to understand thesubject matter surrounding each question.

As an exercise, a day or two before the exam, you may wantto erase your memory of this set, set aside a couple of hours,print a fresh hard copy of this set, and redo the problems fromscratch as if you were actually taking the exam.

Note that unless otherwise stated angles are measured inradians. To express angles in degrees the word ”degrees” maybe used or a circular superscript as in 30◦. If that word or super-script are missing then we presume that the angle is measuredin radians. (Sometime the word ”radians” is added for clarity.)

In the problems involving triangles, unless otherwise stated,we use our standard notation, with A, B, and C denoting theangles (and also the corresponding vertices), and a, b and c de-noting the lengths of the sides opposite A, B, and C, respectively.

When lengths are given without units then we assume thatthey all have the same unit (without specifying it).

There are no hints, solutions, or formulas listed for this set.You should be able to work with your notes and textbook. Theactual exam will be closed notes and books as usual, and cal-culators are not allowed. I will give you the angle sum anglesfor both sine and cosine, the laws of sine and cosine and thequadratic formula.

1. (10 pts) set13/p1.pgSuppose you have an isosceles right triangle. Then two of theangles are equal, and each have the value degrees =

radians. (In the second box, enter an exact expressionusing p to denote π.)

Correct Answers:• 45• p/4

2. (10 pts) set13/p2.pg

Let A = 80◦ = radians. (Enter an exact expressionusing p to denote π.)

Let B = π

5 = degrees.Correct Answers:

• 80/180*p• 36

3. (10 pts) set13/p3.pgThe domain of the inverse sin function is the interval from

to and its range is the interval from to.

Correct Answers:• -1• 1• -1.5707963267949• 1.5707963267949

4. (10 pts) set13/p4.pgSuppose u is an angle in the second quadrant and

sinu =23

Thencosu = andtanu = .Indeed,u = .

Correct Answers:• -0.74535599249993• -0.894427190999916• 2.41186499736283

5. (10 pts) set13/p5.pgThe graph shown in this picture

has amplitude and period .1

Correct Answers:

• 3• 2

6. (10 pts) set13/p6.pgThis figure

shows the graph of the functionf (x) =Your answer may be of the form f (x) = asin(bx + c) where a,b, and c are small integers or π.

Correct Answers:

• 2*sin(3.14159265358979*x+2)

7. (10 pts) set13/p7.pgThe equation

cos(3x) =12

has two solutions between 0 and 120 degrees. The smaller isdegrees and the larger is degrees.

Correct Answers:

• 20• 100

8. (10 pts) set13/p8.pgThe smallest positive solution of the

3sin(2x−1)−1 = 0

is .Correct Answers:

• 0.669918454727061

9. (10 pts) set13/p9.pgConsider a right triangle whose two short sides have lengths of 5and 12 feet, respectively. Then the smallest angle in that triangleis degrees.


10. (10 pts) set13/p10.pg

Suppose we have a triangle with A = 40◦, B = 60◦ and c = 4.Then a = .


11. (10 pts) set13/p11.pg

Suppose we have a triangle with A = 70◦, b = 4 and c = 5.Then a = .


12. (10 pts) set13/p12.pg

Suppose we have a triangle with A = 65◦, b = 4. There is aminimum value of a for which we obtain a triangle with thosedata. That value is

a = .Correct Answers:

• 3.6252311481466

13. (10 pts) set13/p13.pgLet u be an angle in the second quadrant with

sinu =23.

Then cosu = andsin(2u) = .Correct Answers:

• -0.74535599249993• -0.993807989999906

14. (10 pts) set13/p15.pgYou are on level ground in the late afternoon. The Sun is atangle of elevation of 20 degrees. A tree casts a 200 feet longshadow. The height of the tree is feet.



2

Hsiang-Ping Huang math1060spring2009-1WeBWorK assignment number 0-Demo is due : 07/27/2009 at 07:00pm MDT.The primary purpose of WeBWorK is to let you know that you are getting the correct answer or to alert you if you are making

some kind of mistake. Usually you can attempt a problem as many times as you want before the due date. However, if you arehaving trouble figuring out your error, you should consult the book, or ask a fellow student, one of the TA’s or your professor forhelp. Don’t spend a lot of time guessing – it’s not very efficient or effective.

Give 4 or 5 significant digits for (floating point) numerical answers. For most problems when entering numerical answers,you can if you wish enter elementary expressions such as 2∧ 3 instead of 8, sin(3 ∗ pi/2)instead of -1, e∧ (ln(2)) instead of 2,(2+ tan(3))∗ (4− sin(5))∧6−7/8 instead of 27620.3413, etc. Here’s the list of the functions which WeBWorK understands.

You can use the Feedback button on each problem page to send e-mail to the professors.

1. (1 pt) 1210Library/set0 WebWork Demo/1210s0p5.pgThis problem demonstrates a WeBWorK True/False question.

Enter a T or an F in each answer space below to indicatewhether the corresponding statement is true or false.

1. 6−1 ≤ 62. −8 <−83. −4 ≤−44. −1 <−9

Notice that if one of your answers is wrong then, in this prob-lem, WeBWorK will tell you which parts are wrong and whichparts are right. This is the behavior for most problems, but fortrue/false or multiple choice questions WeBWorK will usuallyonly tell you whether or not all the answers are correct. It won’ttell you which ones are wrong. In those cases you also will needto get all the answers correct before getting credit. The idea isencourage you think rather than to just try guessing.

Correct Answers:

• T• F• T• F

2. (1 pt) 1210Library/set0 WebWork Demo/1210s0p6.pgThis problem demonstrates a WeBWorK Matching question.

Match the statements defined below with the letters labelingtheir equivalent expressions.

1. The distance from x to 6 is more than 32. x is less than or equal to 63. x is any real number4. x is greater than 65. x is less than 6

A. x ≤ 6B. x < 6C. −∞ < x < ∞

D. |x−6|> 3E. 6 < x

For matching questions, as for true/false questions (but not forthis demonstration problem) you will usually need to get all an-swers correct to obtain credit, and if some answers are false wwwill not tell you whioch or how many.

Correct Answers:• D• A• C• E• B

3. (1 pt) 1210Library/set0 WebWork Demo/1210s0p7.pgThis problem demonstrates a WeBWorK question that requiresyou to enter a number or a fraction.

Evaluate the expression |116−309||−8| . Give you answer in deci-

mal notation correct to three decimal places or give your answeras a fraction.

Now that you have finished you can use the ”Prob. List”button at the top of the page to return to the problem list page.You’ll see that the problems you have done have been labeledas correct or incorrect, so you can go back and do problemsyou skipped or couldn’t get right the first time. Once you havedone a problem correctly it is ALWAYS listed as correct evenif you go back and do it incorrectly later. This means you canuse WeBWorK to review course material without any danger ofchanging your score.


4. (1 pt) 1210Library/set0 WebWork Demo/m1p165.pgEnter here the expression 1

a + 1b .

Enter here the expression 1a+b .

Correct Answers:• 1/a+1/b• 1/(a+b)

5. (1 pt) 1210Library/set0 WebWork Demo/s1p3.pg

In the first few problems, now that you are familiar with thebasic mechanics of WeBWorK, you will be asked to evaluatesome arithmetic expressions and enter the answer as a numberinto WeBWorK. You may of course use a calculator. In later

1

problems you will be able to enter the answer as an arithmeticexpression, but at present your answer must be a number suchas 4, -4, or 17.5.

Evaluate the expression6(3 + 4) = . (Remember that by conventiona missing arithmetic operator means multiplication .)

Correct Answers:

• 42


Evaluate the expression8(2−4) = .

Correct Answers:

• -16


Evaluate the expression8/(2+2) = .Enter your answer as a decimal number listing at least 4 decimaldigits. (WeBWorK will reject your answer if it differs by morethan one tenth of 1 percent from what it thinks the answer is.)

Correct Answers:

• 2



Correct Answers:

• 15



Correct Answers:

• 7

10. (1 pt) 1210Library/set0 WebWork Demo/s1p8.pgThis problem illustrates the standard rules of arithmetic prece-dence:

Multiplication and Division precede Subtraction and Addition.Among operations with the same level of precedence, evalua-tion proceeds from left to right.However, expressions in parentheses are evaluated first.

Evaluate the expression5×5−4×3 =

Evaluate the expression5× (5−4)×3 =

Evaluate the expression5× (5−4×3) =

Correct Answers:

• 13• 15• -35

11. (1 pt) 1210Library/set0 WebWork Demo/s1p9.pgThis problem provides more illustrations of the use of parenthe-ses.

Evaluate the expression8−6−3−4 =

Evaluate the expression8− (6−3)−4 =

Evaluate the expression8− (6−3−4) =

Evaluate the expression8− (6− (3−4)) =

Correct Answers:

• -5• 1• 9• 1

12. (1 pt) 1210Library/set0 WebWork Demo/s1p10.pgThe key idea in Algebra is to use variables in addition to num-bers. Sometimes we need to replace variables with specificnumbers. That’s called evaluating an algebraic expression.For example, if a = 2 then 3a = 6, and we say that we evaluatedthe expression 3a at a = 2. We’ll do this sort of thing all semes-ter long, and in this problem you get your first experience withevaluating algebraic expressions. Again, the emphasis in theseexercises is on understanding the rules of arithmetic precedence.

Let a = 17, b = 8, c = 13.Then a−b/c =and (a−b)/c =As usual, enter your answers as decimal numbers with at

least 4 digits.Correct Answers:

• 16.3846153846154• 0.692307692307692

2

13. (1 pt) 1210Library/set0 WebWork Demo/s1p11.pgLet r = 9.


• 11.4591559026165• 0.141471060526129

14. (1 pt) 1210Library/set0 WebWork Demo/s1p16.pgIn this and the following problems you will practice en-tering algebraic expressions into WeBWorK. Remember theRules of Arithmetic Precedence and use parentheses to makeyour meaning clear. Most of the difficulties students have withWeBWorK are due to not appreciating the precise rules that gov-ern the interpretation of what you enter. This is not just a matterof WeBWorK understanding what you are trying to say, Therules are used universally all over the world. Appreciating andapplying them properly is also crucial, for example, in computerprogramming. Make sure you understand what’s going in theseproblems. If you enter a wrong expression use the Preview But-ton to see what WeBWorK thinks you have entered.

We start simply. Enter here the expression a+b.Correct Answers:

• a+b

15. (1 pt) 1210Library/set0 WebWork Demo/s1p17.pgEnter here the expression

a+12+b


a+bc+d

If WeBWorK rejects your answer use the preview button tosee what it thinks you are trying to tell it.

Correct Answers:

• (a+1)/(2+b)• (a+b)/(c+d)


11a + 1

b


a+b+11+ 1

a+b

Correct Answers:

• 1/(1/a+1/b)• (a+b+1)/(1+1/(a+b))


The square x2 of a number x simply means the product of xwith itself. For example, 32 = 3∗3 = 9. You can enter a numbersuch as 32 as 3**2 . (An expression such as 32 or x2 is calleda power . We will learn a great deal more about powers duringthis semester.)

Enter here the expression x2

The square root√

x of a number x is a number whose squareequals x. For example

√25 = 5 since 52 = 5∗5 = 25.

To enter square roots you can use the function sqrt . Forexample, to enter the square root of 2 you can type sqrt (2).

Enter here the expression√

aCorrect Answers:

• x**2• sqrt{a}


√a+b


a√a+b


a+b√a+b

Correct Answers:

• sqrt(a+b)• a/sqrt(a+b)• (a+b)/sqrt(a+b)



−b+√

b2−4ac2a

Note: this is an expression that gives the solution of a quadraticequation by the quadratic formula . We will learn much moreabout it later in the semester.

Correct Answers:

• (-b+sqrt(b**2-4*a*c))/(2a)

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20. (1 pt) 1210Library/set0 WebWork Demo/s1p25.pgConsider again the formula for the solution of a quadratic equa-tion:

x =−b+

√b2−4ac

2aFor each of the WeBWorK phrases below enter a T (true) if

the phrase describes x, correctly, and a F (false) otherwise.

You need to get all answers correct before obtaining credit.

-b+sqrt(b**2-4*a*c)/2a

(-b+sqrt(b**2-4*a*c))/2a

(-b+sqrt(b**2-4*a*c))/(2a)

Correct Answers:

• F• F• T


For each of the WeBWorK phrases below enter a T (true) ifthe two given phrases describe the same algebraic expressionand an F (false) otherwise. One way you can decide whether thephrases are equivalent is to substitute specific values for a, b,etc. If you get two different results the two phrases are certainlynot equivalent. If you get the same values there is small chancethis happened accidentally for just that choice of particular val-ues. In any case, pay close attention to when these phrases areequivalent and when they are not, it will help you tremendouslywith future WeBWorK assignments.

a+b b+a

a+b+ c a+(b+ c)

a−b− c a− (b− c)

Correct Answers:

• T• T• F


More of the same.

a+b2 (a+b)2

a2 +b2 (a+b)2

a∗b∗ c a∗ (b∗ c)

a/b/c a/(b/c)

Correct Answers:

• F• F• T• F


In mathematics, lower and upper case letters mean differentthings. The letter a is not the same as the letter A. Keep that inmind when answering the questions below, using T or F as inthe preceding questions.

a a

a A

a+A A+a

Correct Answers:

• T• F• T


Much of this course will center around the manipulation ofalgebraic expressions, often with the goal of solving an equa-tion. This exercise is the first step in this direction. Again, indi-cated with T or F if the two expressions are equivalent.

a∗ (b+ c) a∗b+a∗ c

1/(a+b) 1/a+1/b

1/a/a 1/(a∗a)

Correct Answers:

• T• F• T


The reason why Mathematics is required for so many sub-jects is that it can be used to solve problems outside of mathe-matics, the dreaded word problems . There will be many wordproblems in this class, usually leading to a mathematical prob-lem of the kind we are discussing at the time. Students don’t

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like word problems because they involve the extra layer of con-verting the word problem to a math problem. But keep in mindthat math classes are the only kind of classes you take wheresome problems are not word problems!

This first word problem of this course can be solved by derivingand solving an equation, but it can also be solved essentially byguessing and modifying the answer until it fits, without any al-gebraic manipulation. We will revisit it in the future in a morecomplicated setting.

You buy a pot and its lid for a total of $ 11. The sales per-son tells you that the pot by itself costs $ 10 more than the lid.The price of the pot is $ and the price of the lid is $

.Correct Answers:

• 10.5• 0.5


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hsiang-ping huang. webwork assignment number 5. due …...u of u math 1060-1 spring 2009 hsiang-ping...

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