math41 notes

MATH 41

College Algebra IIand Trigonometry

Lecture Notes

MATH 41 1.1 Basic Equations College Algebra II

This section is a review of basic equation solving techniques. Each of thesetechniques will be used frequently throughout the semester. Remember that solvingan equation means isolating the variable on one side of an equation.

Solving linear equations ax + b = 0 Solving equations involving fractional expressions Solving nth-degree equations xn = a Solving equations with fractional exponents Solving for one variable in terms of another

LinearEquations

Solve each linear equation.1. 4x 6 = 14

2. 3t 7 = t + 3

3. 1t1 +t

3t2 =13

4. 13t +4

3+t +16

9t2 = 0

nth-degreeEquations

Find all real solutions to the equation.5. x3 + 8 = 0

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MATH 41 1.1 Basic Equations College Algebra II

6. x4 1 = 0

7. x2 12 = 0

8. 2(x + 1)2 4 = 0

9. x3/2 8 = 0

Solving for aSpecificVariable

Solve for the indicated variable.10. a2 + b2 = c2, for a

11. e = mc2, for c

12. a+bb =a1

b +b+1

a , for a

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MATH 41 1.2 Modeling with Equations College Algebra II

To solve a story problem, first identify what the unknown quantity is, and thenassign it a variable. Then set up equations from the information given in the prob-lem. Solve the equation, and check your answer. Dont make the problems anymore difficult than they are!

Money Problems Mixture Problems Geometric Problems Distance, Rate, and Time Problems

MoneyProblems

1. Butch earns $8 an hour at his job, but if he works more than 40 hours in a week,he is paid 112 times his regular salary for the overtime hours worked. One week, heearns $392. How many overtime hours did Butch work that week?

2. Stu invested $2,000, part at 4% interest, and the rest at 10%. During that year,Stu earned the same amount from interest as he would have if he had invested the$2000 at 8.5% interest. How much did Stu invest at each interest rate?

3. My couch has $2.50 under the cushions in nickels, dimes, and quarters. If thereare three times as many nickels as quarters, and the same number of dimes andquarters, how many coins of each type are there?

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MixtureProblems

4. Hectors favorite drink is Sprite ($2.70/gal) mixed with Eggnog ($4.00/gal). Hecant recall the proper proportions to mix, but what he does remember is that thetotal cost of the mixture is $3.00 per gallon. If Hector wants to mix up 5 gallonsfor his big holiday party, how much of each drink should he use?

GeometricProblems

5. A 6 ft tall man wants to estimate the height of a light post. He notes that hisshadow is 4 feet long, and light posts shadow is 28 feet long. How tall is the lightpost?

6. Find the length x in the figure, if the shaded area is 126 cm2.

2x

3x

x

xx

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Distance,Rate, TimeProblems

7. Harry flew his broom 550 miles from Hogwarts to Paris and back in 8 hours. A15 mi/hr tailwind assisted him on the way to Paris, but impeded him on his returntrip. What was Harrys flight speed (without the wind)?

8. My wife (Jenny) can clean our cluttered living room in 30 minutes. My son(Andrew) can clutter up the living room in 40 minutes. If Jenny starts cleaning thecluttered living room while Andrew is busy cluttering it back up, how long will ittake for the room to get cleaned?

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MATH 41 1.3 Quadratic Equations College Algebra II

A quadratic equation is an equation of the form

ax2 + bx + c = 0 a , 0.

Quadratic equations can be solved by factoring and using the property that AB = 0if and only if A = 0 or B = 0. When a quadratic equation doesnt factor, then it canbe solved by completing the square or using the quadratic formula.

Solving quadratic equations by factoring Completing the square x2 + bx + + c = 0 Quadratic formula x = b

b24ac

2a Discriminant b2 4ac

Factoring Solve the equation by factoring.1. y2 + 7y + 12 = 0

2. x2 + 8x + 12 = 0

3. 3x2 8x + 4 = 0

4. 2x(x + 1) = 7x 2

Completingthe Square

Solve the equation by completing the square.5. x2 4x 12 = 0

6. w2 + 6w 18 = 0

7. t2 + 5t 3 = 0

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MATH 41 1.3 Quadratic Equations College Algebra II

8. 4x2 8x + 8 = 0

9. 6y2 + 7y 5 = 0

QuadraticFormula

Solve the equation using the quadratic formula.10. x2 4x 5 = 0

11. 32 + 4 18 = 0

12. x2 2 = 0

Discriminant Find the number of real solutions of the equation using the discriminant.13. 5x2 + 7x + 2 = 0

Applications 14. A box with a square base and no top is to be made from a square piece ofcardboard by cutting out a 2 in 2 in squares from the corner and folding up thesides as shown. The box is to hold 72 in3. How big a piece of cardboard is needed?

2

2

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MATH 41 1.4 Complex Numbers College Algebra II

To solve equations of the form x2 = 1, we introduce a number i which satisfiesi2 = 1. Complex numbers are numbers of the form a + bi. We call a the real partand b the imaginary part. To do calculations with complex numbers, simply treat ilike a variable, and always simplify i2 to be 1. Definition of complex numbers Calculations with complex numbers Complex numbers as solutions of equations

ComplexNumbers

Decide whether each statement is true or false.1. T F 3 + 4i has imaginary part 4 and real part 3.

2. T F 6 is a complex number.

3. T F i4 = 1.

4. T F16 = 4i.

Calculationswith

ComplexNumbers

Perform the indicated operations5. (3 + 4i) + (2 5i)

6. (3 + 4i) (2 5i)

7. (3 + 4i)(2 5i)

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MATH 41 1.4 Complex Numbers College Algebra II

8.3 + 4i2 5i

9.1

2 + i 2

1 + 2i

10. i17

11. i17

12. (

5 6)(

10 +

3)

SolvingQuadratics

Solve each equation.13. x2 + 4 = 0

14. x2 3x + 3 = 0

15. 2x2 + x + 1 = 0

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MATH 41 1.5 Other Types of Equations College Algebra II

Polynomials are expressions such as x4 + 3x2 2x + 4. Polynomial equationscan often be solved by factoring. Equations involving radicals can be solved eitherby substitution or by raising both sides of the equation to a power to remove theradicals.

Solving by factoring Equations involving radicals Equations of quadratic type

Solving byFactoring

Find all real solutions of the equation.1. x4 + 3x3 + 2x2 = 0

2. x4 + 3x3 2x2 6x = 0

3. 1x+3 +x

x+4 =x24

x2+7x+12

Quadratic-Type

Equations

4. (x + 4)2 + 13(x + 4) + 36 = 0

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5.( x

x 3

)2 6x

x 3 + 8 = 0

6. 3x4 + 2x7 6 = 0

7. x + x1/2 6 = 0

8. x8 + 15x4 16 = 0

9. x3/2 + 3x1/2 10x1/2 = 0

10. x6 9x4 4x2 + 36

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RadicalEquations

11. 2x +

x + 1 = 8

12.34x2 4x = x

13.

1 +

x +

2x + 1 =

5 +

x

14. A students debt D (in dollars) can be modeled by the formula

D(t) = 10t + 36

t + 196,

where t is time (in weeks) since moving to State College. How many weeks will ittake for the students debt to reach $500?

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MATH 41 1.6 Inequalities College Algebra II

Inequalities are preserved when quantities are added or subtracted from bothsides. Multiplying or dividing by a negative reverses the direction of the inequal-ity. Polynomial or rational inequalities are solved by getting 0 on one side of theinequality and then factoring the other side to break up the numberline into testintervals.

Linear inequalities Polynomial inequalities Rational inequalities

Linearinequalities

Solve each inequality. Write the solution using interval notation.1. 3x 5 7

2.3x + 4

2 2

3. 3 2x + 7 9

PolynomialInequalities

4. (x + 3)(x 4) < 0

5. x2 x + 2

6. x3 4x 0

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MATH 41 1.6 Inequalities College Algebra II

7. x5 < x3

RationalInequalities

8.x + 4x 2 0

9.x + 4x 2 1

10.3

x 1 4x 1

Applications 11. A telephone company offers two long-distance plans.Plan A: $25 per month and $.05 per minutePlan B: $5 per month and $.12 per minuteFor how many minutes of long-distance calls would plan B be financially advanta-geous?

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MATH 41 1.7 Absolute Value Equations College Algebra II

The absolute value of a number is the distance from that number to 0. Thus,

|x| = c if and only if x = c,|x| < c if and only if c < x < c,|x| > c if and only if x < c or x > c.

Absolute value equations Absolute value inequalities

AbsoluteValue

Equations

Solve each equation.1. |2x 1| = 5

2. |3x + 1| = 6

3. 2|x + 3| 4 = 6

4. |x + 2| = |2x 4|

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MATH 41 1.7 Absolute Value Equations College Algebra II

AbsoluteValue

Inequalities

Write the solution to each inequality in interval form.5. |3x + 4| 7

6. |2x + 1| > 7

7. 13 |x + 4| 3 1

8. 2 |x 1| 5

9. Write an inequality that describes the set of all numbers that are at least 3 unitsaway from 5.

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MATH 41 2.1 The Coordinate Plane College Algebra II

Algebraic relationships can be visualized as a graph of points in the plane. Thatis, a graph represents the solution set to some equation or inequality. The verticalaxis is the y-axis, and the horizontal axis is the x-axis. Each point is specified as anordered pair (x0, y0), where x0 is the x-coordinate, and y0 is the y-coordinate.

Plotting points Area of triangles and parallelograms Distance formula Midpoint formula

Areas andPlots

The area of a triangle is given by the formula 12 (base)(height). A parallelogramhas area given by (base)(height).1. Draw the parallelogram with vertices (1, 1), (4, 1), (0, 5) and (3, 5), and findits area.

2. Draw the triangle with vertices (2,3), (4, 2), and (2, 4), and find its area.

Sketch the region given by the set.3. {(x, y) | x 2}

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4. {(x, y) | 0 x 3 and y < 2}

5. {(x, y) | |x| 1 and |y| 2}

6. {(x, y) | xy > 0}

Distance The distance between points A(x1, y1) and B(x2, y2) is

d(A, B) =

(x2 x1)2 + (y2 y1)2.

7. Which of the points A(5, 6) or B(7, 4) is closer to the origin?

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8. Show that the triangle A(4, 2), B(0,4), C(3,2) is a right triangle by usingthe Pythagorean theorem.

9. Show that the points A(1, 0), B(4, 2), C(2, 7), and D(3, 5) are the vertices ofa square.

Midpoints The midpoint between points (x1, y1) and (x2, y2) is( x1 + x2

2,

y1 + y22

).

10. Find the midpoint of the points A(2, 1) and B(4,3).

11. If M(1, 4) is the midpoint of the line segment AB, and if A has coordinates(2,2), find the coordinates of B.

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MATH 41 2.2 Graphs of Equations College Algebra II

The graph of an equation in x and y is the set of all points (x, y) in the coordinateplane that satisfy the equation. The intercepts of a graph are the points where thegraph meets the coordinate axis (i.e. either the x or y coordinate is 0).

Sketching graphs by plotting points Intercepts Symmetry in equations Equation of a circle

Intercepts 1. Which of the points (2, 3), (3, 2), or (1,9) are on the graph of x3yyx = 19?

Find the intercepts of each of the following equations.2. x2 + y3 x2y = 64

3. y = x2 7x + 10

Symmetry An equation is symmetric about the x-axis if it remains unchanged when x isreplaced by x (e.g. y = x2). An equation is symmetric about the y-axis if it remainsunchanged when y is replaced by y (e.g. x = y2). An equation is symmetric aboutthe origin if it remains unchanged when x and y are replaced by x and y (e.g.x = y).4. Discuss the symmetries of y = x3 + x

y = 1x2 + 1

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Sketch the graph of each equation. Find the intercepts and test for symmetry.5. y = 3x 3

6. x + y2 = 4

7. y = 1 |x|

8. y = |1 x|

Circles A circle of radius r with center (h, k) has a standard form equation

(x h)2 + (y k)2 = r2.

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9. Find the equation of the circle with center at (2,1) and radius 9.

10. Find the equation of the circle that has a diameter with endpoints (1, 2) and(9,2).

11. Find the center and radius of the circle x2 + 2x + y2 = 3.

12. Find the center and radius of the circle x2 + 8x + y2 6y = 0.

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13. Sketch the region give by the set {(x, y) | 1 < x2 + y2 9}.

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MATH 41 2.4 Lines College Algebra II

Lines are the backbone of geometry. Here we discuss the algebra behind linesand the various forms of the equations of a line.

Slope Point-slope form Slope-intercept form General equation of a line Parallel and perpendicular

Slope A nonvertical line through the points (x1, y1) and (x2, y2) has slope

m =riserun

=y2 y1x2 x1

.

1. Find the slope of the line through the points (1, 4) and (3, 6).

Point-slopeForm

The equation of the line that passes through the point (x1, y1) and has slope m is

y y1 = m(x x1).

Find the equation of the line that satisfies the given conditions.2. Through (1, 2); slope 4

3. Through (1, 3); slope 0

4. x-intercept 3; y-intercept 7

5. Through (1,3); parallel to the x-axis

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Slope-intercept

Form

An equation of the line that has slope m and y-intercept b is

y = mx + b.

6. Find the equation of the line with slope 2 and y-intercept 4.

7. Find the y-intercept of the line 6x 2y = 4.

GeneralForm

The general form of the equation of a line is

Ax + By + C = 0.

8. Sketch the graph of 5x + 2y 10 = 0.

Perpendicularand Parallel

Lines

Two lines with slope m1 and m2 are parallel if m1 = m2. The lines are perpen-dicular if m1m2 = 1.9. Find the equation of a line through the point (3, 4) that is perpendicular to 2x +3y = 17.

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Applicationsof Lines

10. Student A made me 12 cookies and earned an A (4.0) in the course. Student Bmade me 4 cookies and earned a C (2.0) in this course. Write out a linear modelthat relates the number of cookies given to the instructor (x-variable) and the gradethat a student can expect (y-variable). What grade can a student expect who givesthe instructor no cookies? Interpret the slope of the equation in the context of thisproblem.

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MATH 41 3.1 What is a Function? College Algebra II

A functions is a special relation between variable quantities. A correct under-standing of the language and uses of functions is essential for success in this courseand all courses to come.

Definition of a Function Domain and Range

Definition ofa Function

A function f is a rule that assigns to each element x in a set A exactly oneelement, called f (x), in a set B. The set A is the domain of f (i.e. the input ofthe function), and the set { f (x) | x A} is the range of f (i.e. the output of thefunction).1. Let f (x) = |x|. Evaluate f (2) and f (1). Find the domain and range of f .

2. Let f (x) = (x 1)2. Evaluate f (2), f (1), f (2a), and f (x3). Find the domainand range of f .

3. Let g(x) =x|x| . Evaluate g(2), g(1), g(2a), and g(x

2). Find the domain and

range of g.

4. Find the domain of f (x) =

x2 5x + 6

x.

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MATH 41 3.1 What is a Function? College Algebra II

FunctionsInside

Functions

5. Use the function f (x) = x2 1 to calculate the following: f (x2)

( f (x))2

f (x + 1)

f (x) + 1

f(

x2

) f (x)2

6. Use the function f (x) = x2 to calculate the difference quotient:

f (x) =f (a + h) f (h)

h

PiecewiseDefined

Functions

7. Evaluate f (5), f (0), f (2), and f (5) for

f (x) =

3x if x < 0x + 1 if 0 x 2(x 2)2 if x > 2

8. Harvey earns 10/hour at his job. After 40 hours, he earns time-and-a-half. Writea piecewise defined function that gives Harveys pay as a function of the numberof hours he works.

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MATH 41 3.2 Graphs of functions College Algebra II

A graph is a tool that allows geometric intuition to aid in solving an algebraicproblem. The graph of a function f consists of points (x, f (x)).

Graphs from tables Domain and range Vertical line test Piecewise defined graphs

Plots fromTables

Plot the following functions after making a table of values.1. f (x) = x3 + x2 x 1

2. f (x) = |x| x

3. f (x) =

x + 5

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Piecewise-Defined

Functions

4. f (x) =

x if x < 0x2 if x 0

5. Find an equation of the function graphed below.

Vertical LineTest

A graph represents a function if any vertical line intersects the graph at mostonce (i.e. there is at most one y value for each x value).6. Which of the following represents a function?

Determine whether the equation defines y as a function of x.7. x2y + y = x

8. 2x + |y| = 0

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9. x2 = y2

10. Find the equation for the right half of the circle x2 + y2 = 16.

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MATH 41 3.3 Increasing and decreasing functions College Algebra II

The concept of an increasing or decreasing function is a fundamental tool in cal-culus. Here we introduce the important concept of the rate of change of a functionon an interval.

Increasing and decreasing functions Average rate of change

Increasingand

DecreasingFunctions

A function, f , is increasing on an interval I if f (x1) < f (x2) wheneverx1 < x2 in I. f is increasing on an interval I if f (x1) > f (x2) whenever x1 < x2in I.1. On what intervals is the function pictured increasing?

f (x) = 18 x4 x2

AverageRate of

Change

The average rate of change of a function f on an interval [a, b] is the slope ofthe line joining the points (a, f (a)) and (b, f (b)).

Average Rate of Change =change in ychange in x

=f (b) f (a)

b a

2. Find the average rate of change of the function pictured below between thefollowing x-values. x = 0 and x = 2 x = 2 and x = 0 x = 4 and x = 6 x = 2 and x = 4

f (x) = 3x2

8 x316

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MATH 41 3.3 Increasing and decreasing functions College Algebra II

Find the average rate of change of the function between the given values of thevariable.3. f (x) = x2 2x; x = 1, x = 4

4. g(t) =

t 1; t = 2, t = 5

5. The graph below shows the number of people who have told me that I am goingbald in each of the past few years.

(a) What is the average number of unwanted remarks about the status of my hairper year from the year 2000 to the end of 2007?

(b) Between which two successive years did the number of hair criticisms in-crease most quickly?

(c) Interpret the data and make a prediction for what can be expected in thecoming years.

People Year0 20001 20011 20028 2003

18 200422 200530 200626 2007

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MATH 41 3.4 Transformations of functions College Algebra II

Here we cover some extremely useful techniques for transforming the graph ofa function.

Vertical and horizontal shifts Reflections Vertical stretching/shrinking Horizontal stretching/shrinking Even and odd functions

VerticalShifts

The graph of y = f (x) + c shifts the graph of y = f (x) upward by c units.1. Plot the functions f (x) = x2 + 2 and g(x) = x2 4.

HorizontalShifts

The graph of y = f (x c) shifts the graph of y = f (x) to the right by c units.2. Plot the functions f (x) = (x 3)2 and g(x) = (x + 1)2.

ReflectingGraphs

To graph y = f (x) reflect the graph of y = f (x) across the x-axis. To graphy = f (x) reflect the graph of y = f (x) across the y-axis.3. Plot the functions f (x) = x2 and g(x) = (x 3)2.

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Vertical andHorizontalStretching

andShrinking

To graph y = c f (x) stretch the graph of y = f (x) vertically by a factor of c. Tograph y = f (cx) stretch the graph of y = f (x) horizontally by a factor of 1/c.

4. Plot the functions f (x) = 2x2, g(x) = 14 x2, h(x) = (2x)2 and k(x) =

(13 x

)2.

CombiningTransforma-

tions

A function is given. Write the equation that gives the requested transformations.5. f (x) = |x|; reflect across the x-axis, stretch vertically by a factor of 2, shift left1 unit and up 3 units

6. f (x) = 1x ; reflect across the y-axis, shrink horizontally by a factor of 3, shiftright 2 units and down 1 unit

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7. A function is pictured below. Plot each of the following functions.y = f (x) y = f (2x) y = f (x) + 1

y = f (x) + 2 y = 14 f (x + 4) 3 y = 2 f (4x + 4) 2

Even andOdd

Functions

A function f is even if f (x) = f (x) for all x. f is odd if f (x) = f (x) for allx.8. Determine whether each function is even, odd, or neither. f (x) = x3 x

g(x) = x4 x2

h(x) = 3

x

k(x) = 1|x|

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MATH 41 3.5 Quadratic Functions; Extrema College Algebra II

Quadratic functions take the form f (x) = ax2 + bx + c. The graph of a quadraticfunction is a parabola.

Standard form of a quadratic function Extrema of a quadratic function

StandardForm

The standard form of a quadratic function with vertex (h, k) is

f (x) = a(x h)2 + k.

The parabola f opens upward if a > 0 and downward if a < 0.Express each quadratic equation in standard form.

1. f (x) = 6x2 + 24x 5

2. f (x) = x2 + 5x + 8

Maxima andMinima

A quadratic equation f (x) = a(xh)2 + k will always have either a maximum ora minimum value of k when x = h (i.e. at the vertex of the graph). By completingthe square, we can see that f (x) = ax2 +bx+c has a relative maximum or minimumat f

( b2a

).

A quadratic function is given. Express the function in standard form, sketch itsgraph, and find its maximum or minimum value.3. f (x) = x2 8x + 18

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MATH 41 3.5 Quadratic Functions; Extrema College Algebra II

4. f (x) = 12 x2 + 2x 1

5. g(x) = 3 2x x2

Find the maximum or minimum value of each function.6. f (x) = 2x2 + 8x 9

7. g(x) = 100x2 1500x

8. Stus enjoyment of his week depends on how many dates d he goes on. On ascale of 0 to 100 his enjoyment E is given by E(d) = d(48 6d) + 4. How manydates should Stu go on each week to ensure the maximum enjoyment?

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MATH 41 3.6 Combining Functions College Algebra II

Here we discuss different ways to make new functions by combining functions.

Adding and subtracting functions Multiplying and dividing functions Composing functions Domain of combined functions

Adding,Subtracting,Multiplying,

and DividingFunctions

Two functions f (x) and g(x) can be combined to form new functions in thefollowing elementary ways: ( f + g)(x) = f (x) + g(x) ( f g)(x) = f (x) g(x) ( f g)(x) = f (x)g(x)

(fg

)(x) = f (x)g(x)

1. For f (x) = x2 and g(x) = x2 2 find f + g, f g, f g, and f /g and evaluate eachof these new functions at x = 3

Compositionof Functions

Given two functions f and g, the composite functions f g is defined by

( f g)(x) = f (g(x)).

For each of the following pairs of functions, find f g, g f , f f , and g g,and then evaluate each of the composite functions at x = 1.2. f (x) = x2 1, g(x) = x + 1

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3. f (x) = 1x , g(x) = 2x

4. Find f g f where f (x) = x2 and g(x) = x 1.

5. Find functions f and g such that F = f g, where F(x) = 1x2 .

6. A circular ripple in a pond is expanding outward at a rate of 6 in/sec.(a) Find a function g that models the radius as a function of time.(b) Find a function f that models the area of the circle as a function of the radius.(c) Find f g. What does this function represent?

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Domain ofCombinedFunctions

For the purposes of this course, the domain of a combined function will be theset of all values on which the new function is defined.7. Find the domain of each of the following functions. f + g; f (x) = x + 2, g(x) =

x

f g; f (x) = x2, g(x) =x

f g; f (x) =

x + 1, g(x) =

1 x

f /g; f (x) = x, g(x) = x 3

f g, g f ; f (x) = x + 2, g(x) =

4 2x

f g, g f ; f (x) = x2, g(x) =

x

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MATH 41 3.7 1-1 Functions and Inverses College Algebra II

The inverse of a function undoes or reverses what the function has done.

One-to-one functions Horizontal line test Inverse function definition Finding an inverse function Plotting an inverse function

One-to-oneFunctions

A function with domain A is called a one-to-one function if no two elements ofA have the same image, that is

f (x1) , f (x2) whenever x1 , x2.

Decide whether each function is one-to-one.1. f (x) = x4

2. g(x) = 2x 1

3. h(x) = 1x

HorizontalLine Test

A function is one-to-one if and only if no horizontal line intersects its graphmore than once.

Decide whether each function is one-to-one using the horizontal line test.4. f (x) = x4

5. g(x) = 2x 1

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6. h(x) = 1x

InverseFunctions

Let f be a one-to-one function with domain A and range B. Then its inversefunction f 1 has domain B and range A defined by

f 1(y) = x f (x) = y

for any y in B. Alternatively, we can define the inverse function of f to be anyfunction f 1 that satisfies each of the following conditions.

f 1( f (x)) = x for every x in Af ( f 1(x)) = x for every x in B

Determine if the following pairs of functions are inverses.7. f (x) = 3x + 1, g(x) = x13

8. f (x) =

x, g(x) = x2

FindingInverse

Functions

To find the inverse of a function y = f (x), simply interchange x and y and thensolve for y in terms of x.

Find the inverse function.9. f (x) = 1x

10. f (x) = x2x+2

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11. f (x) = (3 + x7)5

Plotting anInverse

Function

To plot f 1(x), simply reflect the graph of f (x) across the line y = x (i.e. inter-change x and y values of each point).12. Plot f (x) = x3 and f 1(x).

13. Going to the movie costs $8 for admission plus $3 per bag of popcorn. Thusif you go to the movie and buy x bags of popcorn the total cost is given by thefunction f (x) = 8 + 3x. Find and interpret the meaning of f 1(x).

Page 45 of 112

MATH 41 Modeling with Functions College Algebra II

The ability to set up an equation that models a relationship is an essential skillthat is required in calculus.

Setting up models Optimization problems

The basic steps in modeling with functions are1. Express the model in words.2. Choose the variable(s).3. Set up the model.4. Use the model to answer the question.

Setting upModels

1. A poster is 10 inches longer than it is wide. Find a function that models its areaA in terms of its width w.

2. The height of a cylinder is four times its radius. Find a function that models thevolume V of the cylinder in terms of its radius r.

3. Find a function that models the surface area S of a cube in terms of its volumeV .

4. The volume of a cone is 100 in3. Find a function that models the height h of thecone in terms of its radius r.

Page 46 of 112

MATH 41 Modeling with Functions College Algebra II

5. The bases on a baseball diamond are 90 feet apart. A runner from 2nd is stealing3rd. Find a function that models the distance from the runner to 1st base when therunner is x feet from 2nd base.

x

1st

2nd

3rd

Home

OptimizationProblems

6. Find two positive numbers whose sum is 30 and the sum of whose squares is aminimum.

7. Bessie the cow is building her dream home. She has 400 feet of fencing andwill make her pasture by dividing up a rectangular pen into four pens as pictured.

(a) Find a function that models the total area of the four pens.(b) Find the largest possible total area of the four pens.

Page 47 of 112

MATH 41 4.1 Polynomials and Graphs College Algebra II

A polynomial function of degree n is a function of the form

P(x) = anxn + an1xn1 + + a1x + a0,

where n is a nonnegative integer and an , 0.

Graphing polynomials End behavior of a polynomial Zeros and graphs of polynomials Intermediate value theorem Multiplicity of a zero

A polynomial function will always have a smooth graph without any corners orholes.

GraphingPolynomials

1. Plot f (x) = x5, g(x) = (x 2)5, and h(x) = x5 2.

EndBehavior of

Polynomials

If f (x) has odd degree, then the graph of f (x) looks like . . . or . . .depending on the sign of the lead coefficient. Think of f (x) = x3.

If f (x) has even degree, then the graph of f (x) looks like . . . or . . .depending on the sign of the lead coefficient. Think of f (x) = x2.

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2. Determine the end behavior of the following polynomials. f (x) = x4 + 2x2 + x 7

f (x) = 4x7 + x6

f (x) = 3x3 2x2 + x 78

Real Zeros ofPolynomials

If P(x) is a polynomials and c is a real number, then the following are equivalent.1. c is a zero of P(x).2. x = c is a solution of P(x) = 0.3. (x c) is a factor of P(x).4. x = c is an x-intercept of the graph of P.

3. Sketch the graph of f (x) = (x + 4)(x 2)(x 5).

IntermediateValue

Theorem

If a < b and f (a) and f (b) have opposite signs, then there is a zero of f (x)between a and b.4. Show that f (x) = x3 x2 + 1 has a zero between x = 1 and x = 0.

Multiplicityof Zeros

If c is a zero of a polynomial f (x) and the corresponding factor x c occurs ex-actly m times in the factorization of f (x), then we say that c is a zero of multiplicitym. Near the point x = c the graph of f (x) looks a lot like the graph of y = xm.

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5. Graph f (x) = x3(x + 4)(x 1)2(x 3)2

6. Graph f (x) = x5 9x3

7. Graph f (x) = x4 2x3 + 8x 16

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8. Find a polynomial f such that f (0) = 0, f (1) = 1, f (2) = 2, f (3) = 3, f (4) = 4,but f (5) , 5.

Page 51 of 112

MATH 41 4.2 Dividing Polynomials College Algebra II

Polynomial division is a useful technique for simplifying problems and findingroots of polynomials.

Long division of polynomials Division algorithm Synthetic division Remainder Theorem Factor Theorem

LongDivision of

Polynomials

To divide polynomials, just apply the same techniques as dividing real numbers.The key is to only focus on the lead terms.

Perform the division in the following problems.

1.278013

2.4x3 + 3x2 + x 1

2x + 1

Divisionalgorithm

If P(x) and D(x) are polynomials, with D(x) , 0, then there exist unique poly-nomials Q(x) and R(x), where R(x) is either 0 or of degree less than the degree ofD(x), such that

P(x) = D(x) Q(x) + R(x) or P(x)D(x)

= Q(x) +R(x)D(x)

.

We call Q(x) the quotient and R(x) the remainder.For each P(x) and D(x), divide P(x) by Q(x) and express the result in both forms

of the division algorithm.3. P(x) = x3 + 6x + 5, D(x) = x 4

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4. P(x) = 2x5 + 4x4 4x3 x 3, D(x) = x2 2

Syntheticdivision

Synthetic division is a quick method of dividing by a polynomial of the formx c. It is basically a streamlined version long division. Use synthetic division todivide the polynomial below.5. P(x) = x3 + 6x + 5, D(x) = x 4

RemainderTheorem

If the polynomial P(x) is divided by x c, then the remainder is the value P(c).6. Use synthetic division and the Remainder Theorem to evaluate P(11) for P(x) =2x3 21x2 + 9x 200.

7. Find P(7) for P(x) = (x14 + x8 98x)(x 7) + 19.

FactorTheorem

The value c is a zero of P(x) if and only if x c is a factor of P(x).8. Show that c = 4 are zeros of P(x) = x3 + 3x2 36x + 32, and find all other zerosof P(x).

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9. Is x 1 a factor of P(x) = 17x3 18x2 + 5x 4?

10. Find a polynomial of degree 4 with zeros 1, 3, 5, 7.

Page 54 of 112

MATH 41 4.5 Rational Functions College Algebra II

A rational function is a fraction of polynomial functions.

Vertical and horizontal asymptotes Slant asymptotes

Vertical andHorizontal

Asymptotes

An asymptote of a function is a line that the graph of a function gets closerand closer to as one travels along that line. Vertical asymptotes often occur wheredivision by 0 would take place. Horizontal asymptotes occur when the denominatorgets large at least as fast as the numerator as x .1. Plot f (x) = 1x

2. Use a transformation of f (x) = 1x to plot g(x) =2x+1

If the numerator and denominator have the same degree, then the y-value of thehorizontal asymptote can be found by taking the ratio of the lead coefficients.

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3. Plot f (x) = 4x2+1

x2x6

SlantAsymptotes

When the degree of the numerator exceeds the degree of the denominator, thenthere is no horizontal asymptote. Instead, the behavior of the graph as x canbe found by using long division.

4. Plot f (x) = x2+2xx1

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5. Plot f (x) = x3xx+4

6. Plot f (x) = x2+xx+1

7. After a certain drug is injected into a patient, the concentration c (in mg/L) ofthe drug t minutes since the injection is given by c(t) = 30tt2+2 . Draw a graph of thedrug concentration and describe what eventually happens to the drug concentra-tion.c

t

Page 57 of 112

MATH 41 5.1 Exponential Functions College Algebra II

Exponential functions are an important class of functions that describe naturallyoccuring phenomena.

Exponential functions Natural exponential functions Compound interest Exponential decay

ExponentialFunctions

The exponential function with base a (with a > 0) is defined for all real numbersto be f (x) = ax.1. f (x) = 2x. Find f (2), f (1), f (1/2), f (0), and f (1). Find the domain, range,and any asymptotes.

2. Plot f (x) = 2x, g(x) = 2x, and h(x) = 2x+1 + 6 on the same coordinate axis.

3. Plot f (x) = 3x, g(x) = 4x, and h(x) = 5x on the same coordinate axis.

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NaturalExponential

Function

The number e 2.718281828 is an important constant that we will use repeat-edly in this course and in courses to come.4. Plot f (x) = ex.

5. Find the domain, range, and asymptote(s) of the following functions. f (x) = ex

g(x) = 3ex

h(x) = e

x

k(x) =

1 ex

CompoundInterest

The amount A in an account after t years that had a principal investment P thatis compounded n times per year at an interest rate of r is given by the formula

A(t) = P(1 +

rn

)nt.

If the interest is compounded continuously (i.e. n ) the formula is

A(t) = Pert.

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6. Suppose you invest $100 in the bank at a rate of 10% interest. How much moneywill you have at end of the year if interest is compounded annually? biannually?10 times per year? continuously?

Page 60 of 112

MATH 41 5.2 Logarithmic Functions College Algebra II

Exponential functions are an important class of functions that describe naturallyoccuring phenomena.

Log base a Properties of logarithms Plots of logarithms Common log and natural log

Log base a For any positive number a , 1, the logarithmic function with base a is definedby

loga x = y ay = x.So, loga x is the exponent to which the base a must be raised to give x.1. Convert the following from exponential form to logarithmic form or vice versa. 104 = 10, 000

41/2 = 2

20 = 1

log4 16 = 2

log3 81 = 4

2. Evaluate each of the following expressions. log2 16

log16 4

log10 .01

log4 1/2

log3 1

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Properties ofLogarithms

3. Simplify each of the following expressions. loga 1

loga a

loga ax

aloga x

PlottingLogarithmic

Functions

The logarithmic function base a and the exponential function base a are in-verses, so their graphs are obtained by reflecting across the line y = x. The domainof loga is (0,) and the range is (,).4. Plot f (x) = log2 x.

5. Plot f (x) = log3 x and g(x) = log5 x.

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6. Plot f (x) = log2(x) and g(x) = 12 log2(x + 4) + 1.

Commonand Natural

Log

When the base is 10, we often omit it from the notation log x = log10 x. We calllog base 10 common log. When the base is e, we write ln x = loge x and call thisnatural log.

Page 63 of 112

MATH 41 5.3 Laws of Logarithms College Algebra II

Logarithms allow us to do math with exponents, so the laws of exponents con-vert to laws of exponents

Laws of logarithms Expanding and combining logarithmic expressions Change of base formula

Laws ofLogarithms

For A > 0, B > 0, and any C the following properties hold: loga(AB) = loga A + loga B loga

(AB

)= loga A loga B

loga ((Ac)) = C loga A1. Evaluate each expression. log2 160 log2 5

log 8 + log 125

log

1000

log3 100 log3 18 log3 50

ln 7 + ln 17

ln(ln ee

4)

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Combiningand

ExpandingLogarithms

2. Expand each expression using the Laws of Logarithms. log2(7/2)

log73ab

log4(

4x2y

)

ln

x

y

z

ln x+43e x

3. Combine each expression using the Laws of Logarithms. log(a + b) + log(a b) log(a2 b2)

log 8 + log 50 log 4

3 log2 x 2 log2 x2

ln(x2 5x) + ln 4 ln x

ln 7 + ln 17

12 [ln a ln(a + b)

Change ofBase

Formulalogb x =

loga xloga b

The change of base formula is especially important when using a calculator forcomputations.

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4. Simplify (log2 3)(log3 5)

5. Rewrite using natural log. log2 20

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MATH 41 5.4 Exponential and Logarithmic Equations College Algebra II

Exponential and logarithmic equations will be important in calculus.

Exponential equations Logarithmic equations

Exponentialequations

Exponential equations have variables in the exponent. To solve an exponentialequation do the following steps.

1. Isolate the exponential expression.2. Take the logarithm of both sides.3. Solve.Solve each equation.

1. 2x+4 = 7

2. 3x+1 = 4x

3. 3 + 52x = 95

4. e3x2 = 2x4

5. 1001+ex = 10

Page 67 of 112

MATH 41 5.4 Exponential and Logarithmic Equations College Algebra II

6. e2x 4ex 12 = 0

Exponentialequations

Logarithmic equations involve logarithms of variables. To solve an logarithmicequation do the following steps.

1. Isolate the logarithmic expression.2. Exponentiate both sides.3. Solve.Solve each equation.

7. ln(3 x) = 6

8. log2(x2 12) = 2

9. log5 x + log5(x 1) = log5 20

10. log(x 2) = log x log 2

11. $1000 is invested in an account for 4 years, and the interest was compoundedsemiannually. If the total after 4 years was $1400.00, find the interest rate. (RecallA(t) = P

(1 + rn

)nt.)

Page 68 of 112

MATH 41 6.1 Angle Measure Trigonometry

We introduce a new unit of measure for angles called a radian. It will havevarious advantages over using degrees to measure an angle.

1

Measure of = 1 rad Measure of 57.296 Convert Radians Degrees,

multiply by

180 Convert Degrees Radians,

multiply by180

Radians and degrees Coterminal angles Arc Length Area of a circular sector Linear speed and angular speed

Radians andDegrees

An angle AOB consists of two rays R1 and R2 with a common vertex O. Wethink of R1 as stationary and R2 rotating. If a circle has radius 1, then the measureof an angle in radians is the length of the arc that subtends the angle.1. Draw and label the following angles.

= rad = 1 rad = /2 rad = 2 rad

To convert radians to degrees, multiply by 180 . To convert degrees to radians,multiply by 180

.

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2. Convert the following angle measures to radians. 60

135

90

750

3. Convert the following angle measures to degrees. 3 rad

76 rad

9 rad

4 rad

CoterminalAngles

Two angles are coterminal if their sides coincide when graphed (e.g. 360 =0).4. Determine whether the following angles are coterminal 30, 330

125, 845

3 ,73

2, 2 + 6

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Arc Length In a circle of radius r, the length s of an arc that subtends a central angle of radis

s = r.

5. Find the length of an arc that subtends a central angle of 30 in a circle of radius4 cm.

6. A central angle in a circle of radius 5 is subtended by an arc of length 6 m.Find the measure of in radians.

Area of aCircular

Sector

In a circle of radius r, the area A of a sector with central angle of rad is

A =12

r2.

7. Find the area of a sector with central angle 23 rad in a circle of radius 4 mi.

Linear Speedand Angular

Speed

If a point moves along a circle of radius r with angular speed , then its linearspeed is given by

= r.

8. The earth rotates about its axis once every 24 hours. The radius of the earthis 4000 mi. Find the linear speed of a point on the equator in mi/hr.

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MATH 41 6.2 Trigonometry of Right Angles Trigonometry

We introduce the 6 trigonometric ratios.

adjacent

hypotenuseopposite

sin =opposite

hypotenusecos =

adjacenthypotenuse

tan =oppositeadjacent

csc =hypotenuse

oppositecsc =

hypotenuseadjacent

cot =adjacentopposite

Trigonometric ratios Special Triangles Applications

TrigonometricRatios

Find all six trigonometric ratios for each triangle.1.

12

5

2.

810

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3. Calculate cos 2 + sin

2

cos 3

(cos 3 )2 + (sin3 )

2

SpecialTriangles

4. 45-45-90 triangle

5. 30-60-90 triangle

Applications Find all sides and angles of the triangle.

Page 73 of 112


6.

4

6

7. tan = 4

9

Page 74 of 112

MATH 41 6.3 Trigonometric Functions of Angles Trigonometry

We introduce the 6 trigonometric ratios.

x

yr

sin =yr

cos =xr

tan =yx

csc =ry

csc =rx

cot =xy

Trigonometric Functions Reference angles Trig identities Area of a triangle

TrigonometricFunctions

1. Describe the relationship between sin and csc . Do they share the same do-main?

ReferenceAngles

Let be an angle in standard position. The reference angle associated with is the acute angle formed by the terminal side of and the x-axis.2. Find the reference angle for each of the following angles. = 105

= 225

= 76

= 173

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3. Find the exact value of the trigonometric function. sin 135

tan 45

sec 300

cos 76

cot173

csc 54

4. In which quadrant does lie if sin > 0 and tan < 0?

5. If lies in quadrant II and cos = 45 , find tan .

6. If tan > 0 and csc = 3, find cos .

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TrigIdentities

Pythagorean Theorem tells us that

sin2 + cos2 = tan2 , 1 + tan2 = sec2 , cot2 + 1 = csc2 .

7. Write tan in terms of cos .

Area of aTriangle

The area A of a triangle with sides a and b with included angle is

A = 12ab sin .

8. Find the area of the triangle with sides 3 and 12 with included angle = 30.

9. An isosceles triangle has an area of 24 cm2, and the angle between the sides is5/6. What is the length of the two sides?

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MATH 41 6.4 The Law of Sines Trigonometry

A triangle with sides a, b, and c and angles A, B, and C can usually be deter-mined if we know 3 of the 6 parts, as long as at least one of these three is a side.

A B

C

ab

c

sin Aa

=sin B

b=

sin Cc

Law of Sines The ambiguous case Applications

Law of Sines The Law of Sinessin A

a=

sin Bb

=sin C

ccan be used to solve triangles in the cases of ASA, or SAA.

Find the indicated quantity in the following cases.1. A = /6, B = /4, a = 4; Find b.

2. A = /6, B = /4, a = 4; Find c.

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MATH 41 6.4 The Law of Sines Trigonometry

3. A = 70, C = 25, c = 10; Find a.

TheAmbiguous

Case

There will often be two solutions in the SSA case.ind the indicated quantity in the following cases.

4. A = /6, c = 5, a = 4; Find b.

5. B = 45, b = 10, a = 9; Find c.

Applications 6. A tree on a hillside casts a shadow of length 100 feet down a hill that has slope14. If the angle of elevation of the sun (above horizontal) is 44, find the heightof the tree.

Page 79 of 112

MATH 41 6.5 The Law of Cosines Trigonometry

A triangle with sides a, b, and c and angles A, B, and C can usually be deter-mined if we know 3 of the 6 parts, as long as at least one of these three is a side.

A B

C

ab

c

a2 = b2 + c2 2bc cos Ab2 = a2 + c2 2ac cos Bc2 = a2 + b2 2ab cos C

Law of cosines Solving triangles Herons Formula Applications

Law ofCosines

The Law of Cosines

a2 = b2 + c2 2bc cos Ab2 = a2 + c2 2ac cos Bc2 = a2 + b2 2ab cos C

can be used to solve triangles in the cases of SSS or SAS.Find the indicated quantity in the following cases.

1. a = 2, b = 3, c = 4; Find A.

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2. A = 5/6, b = 18, c = 4; Find a.

3. A = 70, b = 8, c = 10; Find B.

SolvingTriangles

Find the missing pieces using either the Law of Sines or Law of Cosines.4. C = /3, c = 6, a = 5

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5. B = 45, c = 12, a = 8

HeronsFormula

The semiperimeter s of a triangle ABC is half the perimeter (i.e. s = 12 (a+b+c)).Herons formula gives the area of this triangle:

Area(ABC) =

s(s a)(s b)(s c).

6. Butch wants to buy a triangular field for $100,000 per square kilometer. Hedrives his car around the field and notes that the three sides measure 5 km, 6 km,and 7 km respectively. How much will the field cost Butch?

Applications 7. two straight roads diverge an an angle of 30. Two cars leave the intersection at12:00 noon, one traveling 40 mi/h, the other at 60 mi/h. How far apart are the carsat 1:30 p.m.?

Page 82 of 112

MATH 41 7.1 The Unit Circle Trigonometry

All circles are just scale models of a circle with radius 1.

(1, 0)02

(

32 ,

12 )

/6

(

22 ,

2

2 )

/4

( 12 ,

32 )

/3

(0, 1)

/2

(

32 ,

12 )

5/6

(

22 ,

2

2 )

3/4

(12 ,

32 )

2/3

(0,1)3/2

(

32 ,

12 )

11/6

(

22 ,

2

2 )

7/4

(12 ,

32 )

5/3

(1, 0)

(

32 ,

12 )

7/6

(

22 ,

2

2 )

5/4

(12 ,

32 )

4/3

The unit circle Terminal points Reference number

The UnitCircle

The unit circle is the circle of radius 1 centered at the origin:

x2 + y2 = 1 (1)

1. Show that the point(

15 ,

2

65

)is on the unit circle.

2. The point P lies on the unit circle. The x-coordinate of P is 35 and the y-coordinate is negative. Find the y coordinate.

Page 83 of 112

MATH 41 7.1 The Unit Circle Trigonometry

TerminalPoints

A point P that is found by tracing the unit circle from (1, 0) in the counterclock-wise direction for t units is called a terminal point for the angle t.3. Find the terminal point P(x, y) on the unit circle determined by t. t = 23

t = 32

t =

ReferenceNumber

The reference number t is the shortest distance from the terminal point for t tothe x-axis along the unit circle.4. Find the terminal point P(x, y) and the reference number determined by t. t = 74

t = 76

t = 34

t = 313

Page 84 of 112

MATH 41 7.2 Trig Functions of Real Numbers Trigonometry

We compute exact values of trigonometric functions using the geometry of theunit circle.

Exact values of trig functions Signs and quadrants Trig functions in terms of trig functions

Exact Valuesof Trig

Functions

1. Find the exact values of each of the trigonometric functions. cos 6

tan 23

csc(3

)

cot 76

sec 114

sin 15

2. Find sin t, cos t, and tan t for the terminal point P(x, y) determined by t.

(5

13 ,1213

)

(35 ,

45

)

Signs andQuadrants

3. Find the quadrant where each of the following is satisfied. cos t > 0 and sin t < 0

tan t > 0 and sin t > 0

csc t < 0 and cot t > 0

Page 85 of 112

MATH 41 7.2 Trig Functions of Real Numbers Trigonometry

4. Determine whether the function is even, odd, or neither. f (x) = cos x

g(x) = sin x

h(x) = x2 tan x

k(x) = sin(cos x)

TrigFunctions in

Terms of TrigFunctions

5. If sin t < 0, write sin t in terms of cos t.

Page 86 of 112

MATH 41 7.3 Trigonometric Graphs Trigonometry

y = sin xy = cos x

Periodic properties Transformations Amplitude and period Mixing functions

Both y = sin x and y = cos x have period 2. Thus their graphs repeat every 2units. Plot each of the following:1. f (x) = sin x, g(x) = cos x

2. f (x) = 1 + 2 sin x

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3. f (x) = | cos x|

Amplitudeand Period

The functions y = a sin kx and y = a cos kx have amplitude |a| and period 2/k.Find the amplitude and period for each function and the plot the function.4. f (x) = sin 2x

5. f (x) = cos x

6. f (x) = 3 cos 12 x

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7. f (x) = 2 sin(x /2)

8. f (x) = cos(2x )

MixingFunctions

9. f (x) = x sin x

Page 89 of 112

MATH 41 7.4 More Trigonometric Graphs Trigonometry

Tangent and cotangent Secant and cosecant

Tangent andCotangent

Both y = tan x and y = cot x have period . That is, tan( + x) = tan x andcot( + x) + cot x. The period of y = a tan(kx) is 2/k.

Plot each of the following:1. f (x) = tan x

3 2 1 1 2 3

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2. f (x) = cot x

3 2 1 1 2 3

3. f (x) = cot(x + /2)

3 2 1 1 2 3

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4. f (x) = 2 tan(x)

Secant andCosecant

Both y = sec x and y = csc x have period 2. That is, sec(2 + x) = sec x andcsc(2 + x) + csc x. The period of y = a sec(kx) is /k.5. f (x) = sec x and g(x) = cos x

3 2 1 1 2 3

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6. f (x) = sec 2x

3 2 1 1 2 3

7. f (x) = 12 csc(12 x)

3 2 1 1 2 3

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8. f (x) = 12 csc(4x + 2)

3 2 1 1 2 3

Page 94 of 112

MATH 41 8.1 Trigonometric Identities Trigonometry

Trigonometric identities are formulas that are always true. They can be used tosimplify complicated expressions into forms that are equivalent but more friendly.

Reciprocal identities Pythagorean identities Even-Odd identities Cofunction identities

ReciprocalIdentities csc x =

1sin x

sec x =1

cos xcot x =

1tan x

tan x =sin xcos x

cot x =cos xsin x

PythagoreanIdentities sin2 x + cos2 x = 1 tan2 x + 1 = sec2 x 1 + cot2 x = csc2 x

Even-OddIdentities sin(x) = sin x cos(x) = cos x tan(x) = tan x

CofunctionIdentities sin

(

2 x

)= cos x tan

(

2 x

)= cot x sec

(

2 x

)= csc x

cos(

2 x

)= sin x cot

(

2 x

)= tan x csc

(

2 x

)= sec x

Write the trigonometric expression in terms of sine and cosine, then simplify.1. cos2 (1 + cot2 )

2. tan csc

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Simplify the trigonometric expression.3. cos3 x + sin2 x cos x

4.sin csc

+ cos sec

5.cos x

1 sin x +cos x

1 + sin x

Verify the identity. (Often things are easiest if you write everything in terms ofsines and cosines.)

6.cos

sec sin = csc sin

Page 96 of 112


7.sec t cos t

sec t= sin2 t

8.sin x

sin x + cos x=

tan x1 + tan x

9. sec4 x tan4 x = sec2 x + tan2 x

Page 97 of 112


10.tan x + tan ycot x + cot y

= tan x tan y

11. Show that the equation1

sin x + cos x= csc x + sec x is not an identity.

Page 98 of 112

MATH 41 8.2 Addition and Subtraction Formulas Trigonometry

(1, 0)

(cos t, sin t)

(cos(s + t), sin(s + t))

(cos s, sin s)

t

s

s

Formulas for sine Formulas for cosine Formulas for tangent

Formulas forSine sin(s + t) = sin s cos t + cos s sin t

sin(s t) = sin s cos t cos s sin tFormulas for

Cosine cos(s + t) = cos s cos t sin s sin tcos(s t) = cos s cos t + sin s sin t

Formulas forTangent tan(s + t) =

tan s + tan t1 tan s tan t

tan(s t) = tan s tan t1 + tan s tan t

The proof of the addition formula for cosine can be proved by finding the lengthof the line segments pictured above using Pythagorean theorem. The other formulascan be derived through a similar procedure or from the addition formula for cosine.

Use an addition or subtraction formula to find the exact value of each expres-sion.1. sin 15

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2. cos 512

3. Write the expression as a single value.tan 18 + tan

9

1 tan 18 tan9

Prove each identity.

4. sin(2 x

)= sin

(2 + x

)

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5. tan x tan y = sin(xy)cos x cos y

6. cos(x + y) cos(x y) = cos2 x sin2 y

Skip problems 45-47,54,55.

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MATH 41 8.3 Double-Angle/Half-Angle/Product-Sum Formulas Trigonometry

We provide some formulas that allows us to compute the values of trig functionsfor nonstandard angles.

Double-angle formulas Formulas for lowering powers Half-angle formulas Product-sum formulas

Double-Angle

Formulassin 2x = 2 sin x cos x

cos 2x = cos2 x sin2 x= 1 2 sin2 x= 2 cos2 x 1

tan 2x =2 tan x

1 tan2 xTo prove these formulas, just use the addition formula for angles.1. Find sin 2x, cos 2x, and tan 2x if sin x = 45 and x is in quadrant I.

2. Calculate sin 15 cos 15.

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FormulasFor Lowering

Powerssin2 x =

1 cos 2x2

cos2 x =1 + cos 2x

2

tan2 x =1 cos 2x1 + cos 2x

To prove these formulas, just take square roots in the appropriate double angleformula for cos 2x3. Write cos4 x in terms of the first power of cosine.

Half-AngleFormulas sin

x2

=

1 cos x2

cosx2

=

1 cos x2

tanx2

=1 cos x

sin x=

sin x1 + cos x

To prove these formulas, just replace x with x/2 in the formulas for lowering pow-ers.4. Find the exact value of cos 75.

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5. Simplify 1cos 4sin 4

Product-to-Sum

Formulassin x cos y =

12

[sin(x + y) + sin(x y)]

cos x sin y =12

[sin(x + y) sin(x y)]

cos x cos y =12

[cos(x + y) + cos(x y)]

sin x sin y =12

[cos(x y) cos(x + y)]To prove these formulas, just combine the addition and subtraction formulas for thesine function.6. Write the product as a sum.cos x sin 5x

Sum-to-Product

Formulassin x + sin y = 2 sin

x + y2

cosx y

2

sin x sin y = 2 cos x + y2

sinx y

2

cos x + cos y = 2 cosx + y

2cos

x y2

cos x cos y = 2 sin x + y2

sinx y

2To prove these formulas, just reverse engineer the product-to-sum formulas

(substituting x+y2 for x andxy2 for y).

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7. Write the sum as a product.sin 75 + sin 15

8. Prove the identity(sin x + cos x)2 = 1 + sin 2x

9. Show that sin 130 sin 110 = sin 10

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MATH 41 8.4 Inverse Trigonometric Functions Trigonometry

We define the inverse trigonometric functions.

Inverse sine and cosine Inverse tangent Other inverses

Inverse Sineand Cosine

The inverse sine function is the function sin1 with domain [1, 1] and range[/2, /2] defined by

sin1 y = sin = y.The inverse cosine function is the function cos1 with domain [1, 1] and range

[0, ] defined bycos1 x = cos = x.

Evaluate each of the following.

1. sin1

32

2. sin1 12

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3. sin(sin1 12 )

4. cos1

22

5. cos1 1

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6. cos1(cos 53 )

InverseTangent

The inverse tangent function is the function tan1 with domain (,) andrange (/2, /2) defined by

tan1 x = tan = x.7. tan1 1

8. tan1

3

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9. Plot f (x) = tan1 x.

OtherInverses

The inverses of sec, csc, and cot also exist and are defined analogously.

10. Show that sin(tan1 x) =

1x2+1 . (Hint: Draw a triangle in the unit circle with

angle tan1 x and set x = u1u2

and then solve for u.)

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MATH 41 8.5 Trigonometric Equations Trigonometry

We solve trigonometric functions.

Intersection Points Solving by Factoring Using a trig identity Extra solutions

IntersectionPoints

1. Find the all points of intersection of y = sin x and y = tan x.

Solve each equation.

Solving byFactoring

2. tan x sin x + sin x = 0.

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3. sin2 x cos2 x = 0.

4. 2 cos3 x + 2 cos2 x 1 cos x 1 = 0

Using a TrigIdentity

5. sin x + 1 = cos x (Hint: Square both sides.)

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6. sin 2x + cos x = 0 (Hint: Use the double angle formula)

7. tan x 3 cot x = 0

ExtraSolutions

8. 2 sin 4x cos x cos x = 0

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math41 notes

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