Pre Calculus Worksheet 5.5 Solve each triangle using the Law of Sines. 1. ABC where 16 , 103 , 12B C c 2. WXY where 81 , 59 , 92X Y w 3. How can you tell, without using the Law of Sines, that a triangle cannot be formed by using the measurements

89 , 104 , 28A B a ? Explain. 4. Use PQR at the right to answer the questions below. a) Draw the altitude to PR and label it QT.

b) Write an expression for the length of QT. Do not find the actual length.

c) Find the area of PQR .

5. To find the length of the span of a proposed ski lift from A to B, a surveyor measures the angle DAB to be 25 and then

walks off a distance of 1000 feet to C and measures the angle ACB to be 15. What is the distance from A to B ?

6. An emergency dispatcher must determine the position of a caller reporting a fire. Based on the caller’s cell phone records, she is located in the area shown. Overcome by the desire to solve for any missing lengths, the dispatcher momentarily forgets about the fire and wants to know what the unknown side lengths are in the triangle.

1000 ft

B

A C

25 D

P R

Q

17cm

10cm

60.1

67.5 52.4

Tower 2

Tower 1

Tower 3 4.6 mi

7. US 41, a highway whose primary directions are north-south, is being constructed along the west coast of Florida. Near Naples, a bay obstructs the straight path of the road. Since the cost of a bridge is prohibitive, engineers decide to go around the bay. The illustration shows the path that they decide on and the measurements taken. What is the length of highway needed to go around the bay? 8. A buoy is anchored offshore to mark a sandbar. The straight shoreline at that location runs north and south. From two

observation points on the shore 2.4 miles apart, the bearings to the buoy are 134 and 22. a) What is the distance from the buoy to each of the observation points? b) How far is the buoy from the shore? 9. An adventurer who is stuck on the top of a cliff is trying to decide whether or not he can jump to the next ledge. In his free time he is able to determine the angle from the bottom of the tree to the edge of the cliff and the angle from the top of the tree to the edge of the cliff as shown in the diagram below. While he was climbing the tree to measure that second angle he figured out that the tree is 13 feet tall. Find the distance d the adventurer will have to jump in order to make it to the ledge (assuming he doesn’t trip over the roots of the tree and fall to the bottom of the cliff).

38

125

d

Pelican Bay

US 41

1/8 mi

1/8 mi 2 mi

140

135

Pre Calculus Worksheet 5.6 Solve each triangle using the Law of Cosines. 1. ABC where 131 , 13, 8B a c 2. WXY where 28, 17, 30x y w

Determine whether to use SohCahToa, Law of Sines or Law of Cosines to solve triangle DEF. Explain.

3. 15, 31 , 42f D E 4. 32, 42, 13d e f 5. 28 , 98 , 6D E d 6. 3, 5.5, 40f e D

7. 11, 22 , 68d D E 8. 27, 20, 119f e E

9. Give two reasons why you cannot have a triangle with sides 13 cm, 9 cm and 4 cm. 10. Find the area of the triangle in question 6. 11. Find the area of the triangle in question 4.

12. In attempting to fly from city A to city B, an aircraft followed a course that was 10 in error, as indicated in the figure.

After flying a distance of 50 miles, the pilot corrected the course by turning at point C and flies the remaining distance. If the straightline distance from A to B is 119 miles, and the average speed of the aircraft was 250 miles per hour, how much time was lost due to the error?

C A

B

10

13. US 41, a highway whose primary directions are north-south, is being constructed along the west coast of Florida. Near Naples, a bay obstructs the straight path of the road. Since the cost of a bridge is prohibitive, engineers decide to go around the bay. The illustration shows the path that they decide on and the measurements taken. What is the length of highway needed to go around the bay?

14. An airplane flies north from Ft. Myers to Sarasota a distance of 150 miles, and then changes his bearing to 50 and

flies to Orlando, a distance of 100 miles. a) How far is it from Ft. Myers to Orlando? b) What bearing is needed for the pilot to return from Orlando to Ft. Myers? 15. Solve the following equation for P: ( )2 2 2 2 cosp h d hd P= + -

17. A streetlight is designed as shown below. Determine the angle in the design.

Pelican Bay

US 41

1/8 mi

1/8 mi 2 mi

140

135

3

4.5 2

θ

Pre Calculus

Worksheet 5.5 (ambiguous case)

1. Explain why given SSA is called the ambiguous case.

State whether the given measurements determine zero, one or two triangles ABC.

2. 120 , 18, 9C a c 3. 36 , 17, 16C a c 4. 82 , 17, 15B b c

5. For any questions 2–4 that have TWO triangles, solve BOTH triangles.

Solve the triangle WXY with the given parts below. IF there are two triangles, SOLVE BOTH!!

6. 103 , 46, 61Y w y 7. 57 , 11, 10X w x

8. On Spring Break, Bob and his friends decide to go 4-wheeling off road in his new Jeep. The Jeep has a winch (a lifting

device with a cable) that is used to pull the Jeep in case it gets stuck. After driving too fast over a hill, Bob finds himself

stuck in the middle of a shallow stream. While wading through the stream to attach the cable to a tree a hill on the other

side of the stream (see diagram), Bob ponders the mathematics of his situation…He wonders what the angle of elevation

of the cable was before his Jeep was pulled to the edge of the stream.

100 ft

65 ft

37°

Pre Calculus Worksheet: Fundamental Identities Day 1 Use the indicated strategy from your notes to simplify each expression. Each section may use the indicated strategy and those strategies before. Strategy I: Changing to sines and cosines

1. tan sec cscx x x 2. csc cot

sec

Strategy II: Pythagorean or Co-Function Identities

3. 2 2 2 2sin cos csc cotx x x x 4. 2sin tan cos cos

Strategy III: Factoring the GCF

5. 2

sin

sin sin cos

x

x x x 6.

3cot cot

cos

w w

w

 

 

 

 

Strategy IV: Even-Odd Identities

7. 2cot( ) cotx x 8. 2tan( ) csc 1

 

We can use Identities to simplify trigonometric expression, but there is also a visual explanation for why we can simplify. Before we get into proving identities, try this exploration to further your understanding… 9. Use your graphing calculator, set in radian mode, to complete the following:

a. Graph the function 3 2sin cos siny x x x .

Note: you will need to enter the exponents using parenthesis around the trig functions, such as 3sin( )x .

b. Select Zoom 7: ZTrig for a good window.

c. Explain what you see and sketch a graph.

d. Write an Identity for the expression you graphed and what it equals. PROVE your identity!

For the remainder of this lesson, we will PROVE identities. Remember to indicate where you are starting and to show all steps that lead you to the other side. If you are stuck, think through Strategies I through IV and keep in mind what you are trying to prove!! Use extra paper to show your work!!!

Verify (Prove) each identity.

10. tan2 x = sin2 x + sin2 x tan2 x 11. sin2 x cos2 x

sec2 x cos2 x

12. tan csc

1sec

x x

x

13. (sin x)(tan x cos x – cot x cos x) = 1 – 2cos²x

14. cos = cot sin2 csc 15. 2

tan( ) tan 1x x

16. sec tan sec tan

seccos

17.

2 2 2sin tan cossec

sec

y y yy

y

18.

2 2

2 2

sec tan

cos sin1

w w

x x

       19.

33 cot cot

csccos

w ww

w

Warm Up: Fundamental Identities Day 2

Perform the operation without a calculator…

20. 5 3

8 4 21.

5 3

8 7 22.

7

5 3

x x 23.

14

15

x

x

 

Pre Calculus Worksheet: Fundamental Identities Day 2 Use the indicated strategy from your notes to simplify each expression. Each section may use the indicated strategy and those strategies before. Strategy V: Combining Fractions with LCD

1. 2

2 2

1 cos

sin sin

2.

sin 1 cos

1 cos sin

x x

x x

Strategy VI: Factoring with Difference of Squares

3. 2sec 1

1 sec

x

x

4. 4 4sin cos

Strategy VII: Factoring trinomials with box

5. 4 2 2 4sin sin cos 2cosx x x x   

 

 

 

 

In general we don’t want to leave a trigonometric expression with a fraction in it. Sometimes, however, we have no choice (you may have noticed we left a fraction on example 10 of the notes). If we do have to leave a fraction in our expression, we want to make it a “nice” fraction. For trigonometry, this means we prefer the fraction only have one trig. function and we prefer any addition or subtractions to be left in the numerator. To make this occur, we multiply by a special form of “1” similar to what we did with division involving “i”or division of radicals…

For example, 3 4 2 2 11

2 2 5

i i i

i i

or

1 5 5

55 5 .

Let’s try it with trigonometric expressions. Strategy VIII: Multiply by “1”

6. cos

1 sin

x

x HINT: Multiply by 1 sin x in numerator and denominator.

7. tan

sec 1

x

x

Verify (Prove) each identity. Use extra paper as necessary!!

8. 1 1

2 cot cscsec 1 sec 1

x xx x

9. sec x

sin x

sin x

cos x cot x

10. 2

22

coscsc csc sin

sin

11. 2

2

1 3cos 4cos 1 4cos

sin 1 cos

u u u

u u

12. 2 2 2

3cos csc sincot

sin sec

13. sin sin

2csc1 cos 1 cos

14. 1 tan

tan1 cot

15. 21 12 cot

sec 1 sec 1w

w w

Warm Up Lesson 5.3: Tell whether each statement is True or False. Then, given an example to justify your answer.

1. 2 2 2x y x y 2. x y x y 3.

3 3

4 4

x

x

4. log log logxy x y 5. sin 30 60 sin 30 sin 60

Pre Calculus Worksheet 5.3 Day 1 Use the sum or difference identities to prove each identity. Use extra paper as necessary!!

1. 3

cos sin2

2. cos sin2

x x

3. 3

tan cot2

u u

4. sin 2 sin 3 cos cos 3 sinx x x x x

Write the expression as the sine, cosine or tangent of a single angle. Then, evaluate if possible.

5. sin cos cos sin5 7 5 7

6. tan19 tan 41

1 tan19 tan 41

7.

2 3

2 3

tan tan

1 tan tan

8. cos 26 cos94 sin 26 sin 94

Use the sum and difference identities to find the exact value for each function.

9. cos 75 10. sin 195

11. cos12

12. 11

tan12

Now let’s get a little deeper into where these identities come from…We first have to start with the assumption that

cos cos cos sin sina b a b a b . This identity CAN be derived…just ask!

13. Using cos a b , let’s find cos a b .

a) From Algebra, subtraction is defined as adding the opposite. Use this definition to rewrite cos a b as

the difference of two angles.

b) Now apply cos cos cos sin sina b a b a b to what you wrote in part (a). Then, simplify using

Even-Odd Identities.

14. Using cos a b , let’s find sin a b .

a) From Fundamental Identities Day 1, recall we can use the Co-Function Identity: sin cos2

to

rewrite the sine of an angle as a cosine function. Apply this identity to sin a b . Let a b .

b) Next, distribute your negative to write your expression as the cosine of the difference of two angles. You will need to regroup.

c) Now apply cos cos cos sin sina b a b a b to what you wrote in part (a). Then, simplify using

sin cos2

again.

15. Using sin a b , let’s find sin a b .

a) Again, subtraction is defined as adding the opposite. Use this definition to rewrite sin a b as the sum

of two angles.

b) Now apply sin cos cos sinsin a b a b a b to what you wrote in part (a). Then, simplify using Even-

Odd Identities.  

Pre Calculus Worksheet 5.3 Day 2 1. A refresher as to why the sum/difference rules don’t work the way many people want them to:

a) Find ( )sin 30 60+ , and then find ( ) ( )sin 30 sin 60 + . Are they the same?

b) Find ( )cos 120 60+ , and then find ( ) ( )cos 120 cos 60 + . Are they the same?

c) Find ( )tan 60 30- , and then find ( ) ( )tan 60 tan 30 - . Are they the same?

2. Use a sum or difference identity to find an exact value (NO CALCULATOR).

a) ( )12sin p- b) ( )sin 255

c) ( )712cos p d) ( )tan 105-

3. Simplify the following expressions as much as possible:

a) ( )6sin x p+ = b) ( )4cos x p- =

c) ( )4tan pq+ =

5. Prove the following identities … use a separate sheet of paper.

a) ( ) ( )sin sin 2sin cosx y x y x y+ + - = b) ( ) ( )cos cos 2cos cosx y x y x y+ + - =

c) ( ) ( )tan tan 2 tanx x xp p+ - - = d) ( ) ( ) 2 2cos cos cos sinx y x y x y+ - = -

6. Prove the following identities: Hint: 4x = 3x + x AND 2x = 3x – x … use a separate sheet of paper

a) sin 4 sin 2 2sin 3 cosx x x x+ = b) 2 2

2 2

tan 4 tantan 5 tan 3

1 tan 4 tan

u uu u

u u

-=

-

7. Use the function shown to answer the following questions. a) Write a sine function that fits the graph. b) Write a cosine function that fits the graph. c) Use identities to PROVE your answers from part a and b are the same … use a separate sheet of paper. 8. Write each trigonometric expression as an algebraic expression.

a) ( )sin arcsin arccosx x+ b) ( )1 1cos cos sinx x- --

c) ( )1 1sin tan 2 cosx x- -- d) ( )cos arcsin arctan 2x x-

9. Prove the following identities … use separate sheet of paper.

a) ( )sin 2 2sin cosx x x= b) ( ) 2 2cos 2 cos sinx x x= -

c) ( ) 2cos 2 1 2sinx x= - d) ( ) 2

2 tantan 2

1 tan

xx

x=

-

PreCalculus Worksheet 5.4 For questions 1 and 2, write as the function of one angle. Simplify, if possible, without using a calculator.

1. 21 2sin 15 2. 2sin cos6 6

For questions 3 – 5, suppose sin A = 3

5 and A is an angle in the first quadrant, find each value.

3. cos (2A) 4. tan (2A) 5. sin (2A)

For questions 6 – 8, if tan y = 5

12 and y is an angle in the third quadrant, find each value.

6. sin (2y) 7. tan (2y) 8. cos (2y)

For questions 9 – 14, prove each identity. Use a separate sheet of paper.

9. ( )( )( )2

2 tansin 2

1 tan

AA

A=

+ 10. ( ) ( ) ( )2sin 2 2 cot sinx x x=

11. ( )( )( )

sin 2cot

1 cos 2

xx

x=

- 12. ( ) ( ) ( )sin 2 cot tan 2x x xé ù+ =ë û

13. ( ) ( ) ( )csc sec 2csc 2x x x= 14. ( ) ( ) ( )2 2cos 4 1 8sin cosx x x= -

15. ( ) 2 3sin 3 3cos sin sinu u u u= - 16. ( ) ( )cos 3 cos 2 cos 2 cosx x x x+ =

PreCalculus

PreRequisites for Solving Trig Equations

Factor the following polynomials using any and all factoring techniques from Algebra 1 and Algebra 2.

1. 24 4 1x x 2. 22 3 2x x

3. 23 6x y xy 4. 29 4x

Factor the following trigonometric expressions using the same techniques from above.

5. 2cos 2cos 1x x 6. 21 2sin sinx x

7. sin sin cosx x x 8. 24 cos x

9. 2sec secx x 10. 22cos 5cos 7x x

Solve each of the following equations for x.

11. 22 4 0x x 12. (x + 2)(x – 1) = 0

13. 2 2 3x x 14. 3 2x x

Simplify each of the following expressions:

15. 1 2

2cos 16. sin

-1 (–1) 17. arctan 3 18. arctan(–1)

19. arccos 0 20. 1 3

2cos 21. 1 1

2sin 22. 1 1

2cos

23. Graph the equation y = sin x and 12y on your calculator.

a) How many times do these two graphs intersect?

b) On the interval 0, 2 , how many solutions does the equation 12

sin x have? Find them in terms of .

c) Find the solutions to the following equations on the interval 0, 2 … use your calculator to help you

determine how many solutions each equation has.

i) 2

2cos x ii) sin 1x iii) tan 1x

iv) 3

2sin x v) 1

2cos x vi) cos 0x

vii) tan 0x viii) 2

2sin x ix) tan 3x

Pre Calculus Worksheet: Solving Trigonometric Equations

1. Solve: 1 1sin

2x

2. Solve for x on the domain [0, 2 ) :

1sin

2x

3. Solve the system with your graphing calculator:

sin

1

2

y x

y

4. Explain the difference in your solutions for questions 1 - 3.

For questions 5 – 10, solve each equation on the interval [0, 2 ) .

5. 2 cos 5 4x 6. 2sin tan 2sin 0x x x

7. cos 2 sin 1x x 8. 22sin 5sin 2 0x x

9. 22cos cos 0x x 10. 22cos cos 1x x

For questions 11 – 18, find all solutions to each equation.

11. 4 cos 3 2 cosx x 12. 2sin 2sin 3x x

13. 2sin 3sin 2 0x x 14. 2cos 1x

15. 22cos 4 7cosx x 16. sin sin cos 0x x x

17. 2 23sin cos 0x x 18. 3

sin 12

x