Algebra 2

13-5 The Cosine Function CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

Objectives To graph and write cosine functions MF-TAFF.SB.951  2C.hFo-ToFse.2t.r5ig  oCnhomoseetrtircigfounocmtioentrsictofumncotdioenls to
To solve trigonometric equations mpeordioedlipceprihoednicompheennao.m.e.nAals.o. .FA-lIsFo.BM.4A, FF-SIF.9.C1.27.eF-,IF.2.4,
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MP 1, MP 2, MP 3, MP 4, MP 5

“cos” is short for The calculator screen shows graphs
cosine.
of the sine and cosine functions.
MATHEMATICAL
The graphs are translations of each
PRACTICES
other.

For what value of a are the
graphs of f1(x) = sin (x + a) and
f2(x) = cos x identical? For what
value of b do f1(x) = sin x and
f2(x) = cos(x + b) have identical
graphs? Can you find other values of

a and b? Explain.

Lesson The cosine function, y = cos u, matches u with the x-coordinate of the point on the
unit circle where the terminal side of angle u intersects the unit circle. The symmetry of
Vocabulary the set of points (x, y) = (cos u, sin u) on the unit circle guarantees that the graphs of
• cosine sine and cosine are congruent translations of each other.

function Essential Understanding  For each and every point along the unit circle the
radian measure of the arc has a corresponding cosine value. The colored bars represent
the cosine values of the points on the circle translated onto the cosine graph. So as the
terminal side of an angle rotates about the origin (beginning at 0°), its cosine value on
the unit circle decreases from 1 to -1, and then increases back to 1.

1 y y ‫ ؍‬cos␪

Ϫ1 1 p p 3p 2p ␪
Ϫ1 Ϫ1 2 2

Lesson 13-5  The Cosine Function 861

Problem 1 Interpreting a Graph

A What are the domain, period, range, and amplitude of the cosine function?

The domain of the function is all real numbers.

The function goes from its maximum value of 1 to its minimum value of -1 and back
again in an interval from 0 to 2p. The period is 2p. The midline is y = 0.

The range of the function is -1 … y … 1.
amplitude = 21(maximum - minimum) = 21[1 - ( -1)] = 1

B Examine the cycle of the cosine function in the interval from 0 to 2P.
How do you find a Where in the cycle does the maximum value occur? Where does the
zero of a function? minimum occur? Where do the zeros occur?

Zeros are where the

g raph of a function The maximum value occurs at 0 and 2p. The minimum value occurs at p. The zeros
crosses the x-axis. p 32p.
occur at 2 and

Got It? 1. Use the graph. What are the domain, period, range, y y ‫ ؍‬sin ␪
and amplitude of the sine function? Where do the 1
␪
maximum and minimum values occur? Where do p
Ϫ1 2 p 3p 2p
the zeros occur? 2

Concept Summary  Properties of Cosine Functions

Suppose y = a cos bu, with a ≠ 0, b 7 0, and u in radians.

• 0 a 0 is the amplitude of the function.

• b is the number of cycles in the interval from 0 to 2p.
2p
• b is the period of the function.

To graph a cosine function, locate five points equally spaced through one cycle.
For a 7 0, this five-point pattern is max-zero-min-zero-max.

Problem 2 Sketching the Graph of a Cosine Function Choose scales for axes that
p Ϸ1).
Sketch one cycle of y = 1.5 cos 2U. are about equal ( 3

What should you
f ind to graph the 0 a 0 = 1.5, so the amplitude is 1.5.
fFa uinnnddcpttheireoionad?m. plitude, cycle, Ob n=e2c,yscolethisefgroramph0 htoaspt.wo full cycles from 0 to 2p. max

2bp = p, so the period is p. 1 zero u
Divide the period into fourths. Plot the five-point pattern O

p

Ϫ1

for one cycle. Use 1.5 for the maximum and -1.5 for min

the minimum. Sketch the curve.

862 Chapter 13  Periodic Functions and Trigonometry

Got It? 2. Sketch one cycle of y = 2 cos u .
3

Problem 3 Modeling with a Cosine Function STEM

Oceanography  ​The water level varies from low tide to high tide as shown. What is a
cosine function that models the water level in inches above and below the average
water level? Express the model as a function of time in hours since 10:30 a.m..

Low Tide 60 in. High Tide
10:30 AM 4:40 PM

• The difference of the 2 water levels A cosine function • Find values of a and b.
is 60 in. that models the
water level as a • Substitute the values of a and b
• The time difference between the function of time into y = a cos bu.
low level and the high level is
6 h 10 min.

Amplitude is 1 (60) = 30. Since the tide is at - 30 inches at time zero, the curve follows
2
the min-zero-max-zero-min pattern, so a = -30.

The cycle is halfway complete after 6 h and 10 min, so the full period is 12 hours
1
and 20 minutes or 12 3 hours.

12 1 = 2p
3 b

b = 6p
37

( )So, the function f(t) = -30cos 6p t models the water level.
37

Got It? 3. a. Suppose that the water level varies 70 inches between low tide at 8:40 a.m.
and high tide at 2:55 p.m. What is a cosine function that models the
variation in inches above and below the average water level as a function
of the number of hours since 8:40 a.m.?

b. At what point in the cycle does the function cross the midline? What does
the midline represent?

You can solve an equation by graphing to find an exact location along a sine
or cosine curve.

Lesson 13-5  The Cosine Function 863

Problem 4 Solving a Cosine Equation

Suppose you want to find the time t in hours when the water level from Problem 3

is exactly 10 in. above the average level represented by f (t) = 0. What are all
( )solutions to the equation −30 cos
How can two 6P t = 10 in the interval from 0 to 25?
equations help find 37
the solutions?
Step 1 Use two equations. Graph y = 10 and
The solutions occur ( )y = -30 cos 6p
37 t on the same screen. Xmin ϭ 0
where their graphs Xmax ϭ 25
Step 2 Use the Intersect feature to find the points Xscl ϭ 5
intersect. at which the two graphs intersect. Ymin ϭ Ϫ35
Ymax ϭ 35
The graphs show four solutions in the interval. Intersection Yscl ϭ 5
They are t ≈ 3.75, 8.58, 16.08, and 20.92. Xϭ3.75 Yϭ10

The water level is 10 in. above the average level at about 3.75 h, 8.58 h, 16.08 h, and
20.92 h after 10:30 a.m.

Got It? 4. What are all solutions to each equation in the interval from 0 to 2p?
a. 3 cos 2t = -2 b. -2 cos u = 1.2
c. Reasoning ​In the interval from 0 to 2p, when is -2 cos u less than 1.2?

Greater than 1.2?

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

PRACTICES

Sketch the graph of each function in the interval from 5. Open-Ended  W​ rite a cosine function with amplitude 5
0 to 2P.
and between 2 and 3 cycles from 0 to 2p.

1. y = cos 12u 2. y = 2 cos p u 6. Assume u is in the interval from 0 to 2p.
3
a. For what values of u is y positive for y = cos u?
Write a cosine function for each description. Assume b. For what values of u is y positive for y = -sin u?
that a + 0. c. Reasoning ​What sine function has the same graph
2p
3. amplitude 3, period 2p 4. amplitude 1.5, period p as y = - 3 cos 3 u?

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Find the period and amplitude of each cosine function. Determine the values of x See Problem 1.
for 0 " x " 2P where the maximum value(s), minimum value(s), and zeros occur.
Xmin ϭ Ϫ2p
7. 8. Xmax ϭ 2p
Xscl ϭ p
Xmin ϭ Ϫ2p Ymin ϭ Ϫ2
Xmax ϭ 2p Ymax ϭ 2
Xscl ϭ p Yscl ϭ 1
Ymin ϭ Ϫ4
Ymax ϭ 4
Yscl ϭ 1

864 Chapter 13  Periodic Functions and Trigonometry

9. 10.

Xmin ϭ Ϫ2p Xmin ϭ Ϫ2p
Xmax ϭ 2p Xmax ϭ 2p
Xscl ϭ p Xscl ϭ p
Ymin ϭ Ϫ2 Ymin ϭ Ϫ4
Ymax ϭ 2 Ymax ϭ 4
Yscl ϭ 1 Yscl ϭ 1

Sketch one cycle of the graph of each cosine function. 14. y = cos p u See Problem 2.
11. y = cos 2u 12. y = -3 cos u 13. y = -cos 3t 2 15. y = -cos pu

Write a cosine function for each description. Assume that a + 0. See Problem 3.
18. amplitude p, period 2
16. amplitude 2, period p 17. amplitude p2 , period 3

Write an equation of a cosine function for each graph.

19. y 20. y x
2 u 8
4
p 2p Ϫ2

Solve each equation in the interval from 0 to 2P. Round your answer to the See Problem 4.

nearest hundredth.

21. cos 2t = 12 22. 20 cos t = -8 23. -2 cos pu = 0.3
24. 3 cos 3t = 2 26. 8 cos p3 t = 5
25. cos 1 u = 1
4

B Apply Identify the period, range, and amplitude of each function.

27. y = 3 cos u 28. y = -cos 2t 29. y = 2 cos 21t 30. y = 1 cos u
33. y = 16 cos 32pt 3 2
( ) 31. y = 3 cos -3u 1
32. y = - 2 cos 3u 34. y = 0.7 cos pt

35. Think About a Plan  ​In Buenos Aires, Argentina, the average monthly temperature
is highest in January and lowest in July, ranging from 83°F to 57°F. Write a cosine

function that models the change in temperature according to the month of the year.
• How can you find the amplitude?
• What part of the problem describes the length of the cycle?

36. Writing  E​ xplain how you can apply what you know about solving cosine equations
to solving sine equations. Use -1 = 6 sin 2t as an example.

Lesson 13-5  The Cosine Function 865

Solve each equation in the interval from 0 to 2P. Round your answers to the
nearest hundredth.

37. sin u = 0.6 38. -3 sin 2u = 1.5 39. sin pu = 1

40. a. Solve -2 sin u = 1.2 in the interval from 0 to 2p.
b. Solve -2 sin u = 1.2 in the interval 2p … u … 4p. How are these solutions

related to the solutions in part (a)?

( ) 6p
41. a. Graph the equation y = -30 cos 37 t from Problem 3.

b. The independent variable u represents time (in hours). Find four times at which

the water level is the highest.

c. For how many hours during each cycle is the water level above the line y = 0?
Below y = 0?

STEM 42. Tides  ​The table at the right shows the times for high tide Tide Table
and low tide of one day. The markings on the side of a
local pier showed a high tide of 7 ft and a low tide of 4 ft High tide 4:03 A.M.
on the previous day. Low tide 10:14 A.M.

a. What is the average depth of water at the pier? High tide 4:25 P.M.
What is the amplitude of the variation from the
average depth? Low tide 10:36 P.M.

b. How long is one cycle of the tide?
c. Write a cosine function that models the relationship

between the depth of water and the time of day.
Use y = 0 to represent the average depth of water.
Use t = 0 to represent the time 4:03 a.m.
d. Reasoning ​Suppose your boat needs at least 5 ft of water
to approach or leave the pier. Between what
times could you come and go?

C Challenge 43. Graph one cycle of y = cos u, one cycle of y = -cos u, and one cycle of
y = cos ( -u) on the same set of axes. Use the unit circle to explain any
relationships you see among these graphs.

STEM 44. Biology  ​A helix is a three-dimensional 1 x

spiral. The coiled strands of DNA and the –1 6 18
edges of twisted crepe paper are examples

of helixes. In the diagram, the y-coordinate

of each edge illustrates a cosine function.

Write an equation for the y-coordinate

of one edge.

45. a. Graphing Calculator  Graph y = cos u and y = cos 1u - p 2 in the interval
2
from 0 to 2p. What translation of the graph of y = cos u produces the graph of
b. Gy r=apchosy1=u c-op2s 21?u - p2 2 and y = sin u in the interval from 0 to 2p. What do

you notice?

c. Reasoning ​Explain how you could rewrite a sine function as a cosine function.

866 Chapter 13  Periodic Functions and Trigonometry

Standardized Test Prep

SAT/ACT 46. Which function has a period of 2p and an amplitude of 4?

f (x) = 2 cos 4u f (x) = 2 cos u

f (x) = 4 cos 2u f (x) = 4 cos u

47. Which equation corresponds to the graph shown at the right? The screen
dimensions are -4p … x … 4p and -2 … y … 2.

y = 1 cos 4x y = 1 cos 4x
2 2

y = 2 cos 4x y = 2 cos 4x

Short 48. Which equation has the same graph as y = -cos t? y = cos(t - p)
Response y = cos ( -t) y = -sin t
y = sin(t - p)

49. How many solutions does the equation 1 = -sin 2t have for 0 … t … 2p? 4
1 2 3

50. What are the amplitude and period of y = -0.2 cos p u?
3

PERFORMANCE TASK

Apply What You’ve Learned MATHEMATICAL

PRACTICES
MP 7

In the Apply What You’ve Learned in Lesson 13-1, you considered a function

d = f (t) that gives the horizontal distance d of the dragonfly from the y-axis
after t seconds, and a function h = g (t) that gives the height h of the dragonfly

above the x-axis after t seconds. Refer to the problem on page 827. Choose from

the following words, numbers, and equations to complete the sentences below

to model d = f (t) and h = g (t) using cosine and sine functions.

2 4 5p

2p 5 p p
5 2p 2 5

a. For the functions y = a cos bx and y = a sin bx to have the same amplitude as the
functions d = f (t) and h = g (t), a must be equal to ? .

b. For the functions y = a cos bx and y = a sin bx to have the same period as the
functions d = f (t) and h = g (t), b must be equal to ? .

Lesson 13-5  The Cosine Function 867

13-6 The Tangent Function MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

Objective To graph the tangent function. FM-IAFF.CS..79e1 2G.Fr-aIpF.h3.7.e.  trGigroanpohm. .e.trtircigfounoctmioentsr,icshowing
pfuenricotdio,nms,idshlinoew,ianngdpaemriopdli,tumdied.liAnels, oanFd-TaFm.Apl.i2tu, dFe-T. AF.lBs.o5
MMAP F1S,.M91P2.2F,-MTFP.13.2, ,MMPA4F,SM.9P125.F-TF.2.5
MP 1, MP 2, MP 3, MP 4, MP 5

The dot plots give clues to graphs of functions you can build from sine

and cosine. How are these dot plots made? What would a dot plot look
sin xx?
like for cos Explain.

sin x is in some y yy
cos x
ways like a rational

function.

MATHEMATICAL xx x

PRACTICES Ϫ1 Ϫ1 Ϫ1 sin x
sin x ؉ cos x sin x cos x cos x

Lesson The tangent function is closely associated with the sine and cosine functions, but it
differs from them in three dramatic ways.
Vocabulary
• tangent of u Essential Understanding  T​ he tangent function has infinitely many points of
• tangent discontinuity, with a vertical asymptote at each point. Its range is all real numbers.
Its period is p, half that of both the sine and cosine functions. Its domain is all real
function numbers except odd multiples of p2 .

Key Concept  Tangent of an Angle

Suppose the terminal side of an angle u in standard position y
intersects the unit circle at the point (x, y). Then the y 1 P (x, y)
ratio x is

the tangent of U,, denoted tan u.

In this diagram, x = cos u, y = sin u, and y = tan u. x
x 1

1O

1

868 Chapter 13  Periodic Functions and Trigonometry

Problem 1 Finding Tangents Geometrically

What is the value of each expression? Do not use a calculator.

W ill a graph help? A tan P p
(Ϫ1, 0)
Y es; use a graph of the An angle of p radians in standard position has a terminal side that
unit circle to visualize the QϪͱ23 , Ϫ21Q
problem. intersects the unit circle at the point ( -1, 0).

tan p = -01 = 0

( ) ( ) B tAthanantain−ngtl5ee6Prsoefc-ts5t6pheraudniaitncsiricnlestaatntdhaerdpopionstiti-on12h3,a-sa terminal side
.
1
2

( ) tan -56p = --12123 = 113 = 133. Ϫ5p
6

Got It? 1. What is the value of each expression? Do not use a calculator.
( ) a. tan p2 b. tan 23p p
c. tan - 4

There is another way to geometrically define tan u.

The diagram shows the unit circle and the vertical line x = 1. Q
The angle u in standard position determines a point P (x, y).

By similar triangles, the length of the vertical red segment P tan(u)
␪y
divided by the length of the horizontal red segment is equal to x1
y
x . The horizontal red segment has length 1 since it is a radius y
the unit circle, so the length of the vertical red x
of segment is

or tan u, which is also the y-coordinate of Q.

If u is an angle in standard position and not an odd multiple of p2 , then the line
containing the terminal side
of u intersects the line x = 1 (1, tan p3) y ‫ ؍‬tan U
at a point Q with y

y-coordinate tan u. 2

The graph at the right shows (1, tan p4) 1
(1, tan p6)
one cycle of the tangent ␪
(1, tan 0)
function, y = tan u, for (1, tan (Ϫ p6 )) Ϫ p p
- p 6 6 p2 . (1, tan (Ϫ p4 )) 2 2
2 u The pattern Ϫ1
(1, tan (Ϫ p3 ))
repeats periodically with
period p. At u = { p2 , the
line through P fails to Ϫ2

intersect the line x = 1, so The graph approaches
tan u is undefined. two vertical asymptotes.

Lesson 13-6  The Tangent Function 869

Concept Summary   Properties of Tangent Functions

Suppose y = a tan bu, with a ≠ 0, b 7 0, and u in radians.

• p is the period of the function.
b
• One cycle occurs in the interval from - p to 2pb.
2b
• There are vertical asymptotes at each end of the cycle.

You can use asymptotes and three points to sketch one cycle asymptote asymptote
of a tangent curve. As with sine and cosine, the five ؊1 1
elements are equally spaced through one cycle. Use the zero
pattern asymptote–( -a)–zero–(a)–asymptote. In the graph
at the right, a = b = 1. 2 y y ‫ ؍‬tan U

The next example shows how to use the period, midline Ϫp2 Ϫ2 p ␪
asymptotes, and points to graph a tangent function. 2
3p
2

Problem 2 Graphing a Tangent Function
Sketch two cycles of the graph of y = tan PU.

Use the formula for the period = P
period. Substitute p for b b
and simplify.
= P = 1
P

One cycle occurs in the −P = −P = − 1
2b 2P 2

interval from - p to 2pb. P = P = 1
2b 2b 2P 2

One cycle is from − 1 to 21 .
2

Asymptotes occur at Asymptotes are at u = − 1 , 21 , and 3 .
each end of the cycle. 2 2

Divide the period into y
fourths and locate 3
points between the a( r−e41o,n−t1h)e, (0, 0), and 1 41 , 12 1
asymptotes. Plot the graph.
points and sketch the u
curve. –1 1 2

Got It? 2. Sketch two cycles of the graph of each tangent curve.
a. y = tan 3u, 0 … u … 23p b.
y = tan p u, 0 … u … 4
2

870 Chapter 13  Periodic Functions and Trigonometry

Problem 3 Using the Tangent Function to Solve Problems STEM

How should you Design  A​ n architect is designing the front facade ␪
graph the function? of a building to include a triangle, similar to the
“Degree mode” suggests one shown. The function y = 100 tan U models 200 ft
that you use a graphing the height of the triangle, where U is the angle
calculator. Then use indicated. Graph the function using the degree
TABLE to show y values mode. What is the height of the triangle if
for different u values. U = 16°? If U = 22°?

Step 1 Graph the function.

Xmin ϭ 0 X Y1
Xmax ϭ 470
Xscl ϭ 50 16 28.675
Ymin ϭ Ϫ300 17 30.573
Ymax ϭ 300 18 32.492
Yscl ϭ 90 19 34.433
20 36.397
21 38.386
22 40.403

X=16

Step 2 Use the TABLE feature.

When u = 16°, the height of the triangle is about 28.7 ft. When u = 22°, the height of
the triangle is about 40.4 ft.

Got It? 3. a. What is the height of the triangle when u = 25°?
b. Reasoning ​The architect wants the triangle to be at least one story tall.

The average height of a story is 14 ft. What must u be for the height of the
triangle to be at least 14 ft?

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

PRACTICES

Find each value without using a calculator. 5. Vocabulary  S​ uccessive asymptotes of a tangent
p - p3 .
1. tan p4 2. tan 7p curve are x = 3 and x = What is the period?
6
6. Error Analysis  A quiz contained a question asking
( ) 3. tan- p ( )4. tan - 3p
4 3 students to solve the equation 8 = -2 tan 3u to the
nearest hundredth of a radian. One student did not

receive full credit for writing u = -1.33. Describe
and correct the student’s error.

7. Writing  Explain how you can write a tangent
function that has the same period as y = sin 4u.

Lesson 13-6  The Tangent Function 871

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Find each value without using a calculator. See Problem 1.

8. tan( -p) 9. tan p 10. tan 3p 11. tan p
4 2

( ) 12. tan -74p 13. tan 2p ( )14. tan - 3p ( )15. tan 3p
4 2

Each graphing calculator screen shows the interval 0 to 2P. What is the period See Problem 2.
of each graph?

16. 17.

Identify the period and determine where two asymptotes occur for each function.

18. y = tan 5u 19. y = tan 3u 20. y = tan 4u 21. y = tan 2 u
2 3p

Sketch the graph of each tangent curve in the interval from 0 to 2P.

22. y = tan u 23. y = tan 2u 24. y = tan 2p u 25. y = tan( -u)
3

Graphing Calculator  Graph each function on the interval 0 " x " 2P and See Problem 3.
 − 200 " y " 200. Evaluate each function at x = P4 , P2 , and 34P.
( )28. y = 125 tan 12x
26. y = 50 tan x 27. y = -100 tan x

29. Graphing Calculator  Suppose the architect in Problem 3 reduces the length of

the base of the triangle to 100 ft. The function that models the height of the triangle
becomes y = 50 tan u.
a. Graph the function on a graphing calculator.
b. What is the height of the triangle when u = 16°?
c. What is the height of the triangle when u = 22°?

B Apply Identify the period for each tangent function. Then graph each function in the

interval from −2P to 2P.

30. y = tan p6 u 31. y = tan 2.5u ( )32. y = tan-3 u
2p

Graphing Calculator  Solve each equation in the interval from 0 to 2P. Round
your answers to the nearest hundredth.

33. tan u = 2 34. tan u = -2 35. 6 tan 2u = 1

36. a. Open-Ended  W​ rite a tangent function.
b. Graph the function on the interval -2p to 2p.
c. Identify the period and the asymptotes of the function.

872 Chapter 13  Periodic Functions and Trigonometry

37. Think About a Plan  A​ quilter is making hexagonal placemats by sewing together
six quilted isosceles triangles. Each triangle has a base length of 10 in. The function
y = 5 tan u models the height of the triangular quilts, where u is the measure of
one of the base angles. Graph the function. What is the area of the placemat if the
triangles are equilateral?

• How can a graph of the function help you find the height of each triangle?
• How can you find the area of each triangle?
• What will be the last step in your solution?

38. Ceramics  ​An artist is making triangular ceramic tiles for a triangular patio. The
patio will be an equilateral triangle with base 18 ft and height 15.6 ft.

a. Find the area of the patio in square feet.
b. The artist uses tiles that are isosceles triangles with base 6 in. The function

y = 3 tan u models the height of the triangular tiles, where u is the measure
of one of the base angles. Graph the function. Find the height of the tile when
u = 30° and when u = 60°.
c. Find the area of one tile in square inches when u = 30° and when u = 60°.
d. Find the number of tiles the patio will require if u = 30° and if u = 60°.

Use the function y = 200 tan x on the interval 0° " x " 141°. Complete each
ordered pair. Round your answers to the nearest whole number.

39. (45°, ■) 40. (■°, 0) 41. (■°, -200) 42. (141°, ■) 43. (■°, 550)

Write an equation of a tangent function for each graph.

44. y 45. 3 y 46. y
2
2 O
x Ϫ3
O
2p 2p x Ϫp O p x
Ϫ2
Ϫ2

47. Construction  ​An architect is designing a hexagonal gazebo. The floor is a hexagon
made up of six isosceles triangles. The function y = 4 tan u models the height of
one triangle, where u is the measure of one of the base angles and the base of the

triangle is 8 ft long.
a. Graph the function. Find the height of one triangle when u = 60°.
b. Find the area of one triangle in square feet when u = 60°.
c. Find the area of the gazebo floor in square feet when the triangles forming the

hexagon are equilateral.

48. a. The graph of y = 1 - cos x suggests a tangent curve of the form y = a tan bx.
sin x
Graph the function using the window [ -3p, 3p] by [ -4, 4].
b. What is the period of the curve? What is the value of a?

c. Find the x-coordinate halfway between a removable discontinuity and the

asymptote to its right. Find the corresponding y-coordinate.

d. Find an equivalent function of the form y = a tan bx.

Lesson 13-6  The Tangent Function 873

C Challenge 49. Geometry ​Use the drawing at the right and similar triangles. Justify y Q(1, tan␪ )
the statement that tan = sin uu. P(cos␪, sin␪)
u cos

50. a. Graph y = tan x, y = a tan x (with a 7 0), and y = a tan x (with ␪
a 6 0) on the same coordinate plane.
O AB x
b. Reasoning ​Recall the pattern of five elements for graphing a tangent

function: asymptote-(-1)-zero-(1)-asymptote. How does the value of a
affect this pattern?

51. Writing  How many solutions does the equation x = tan x have for 0 … x 6 2p?
Explain.

Standardized Test Prep

SAT/ACT 52. Which value is NOT defined? ta1np4
13
tan 0 tan p tan 3p
2

53. What is the exact value of tan 7p ?
6
- 133 133
- 13

54. Which equation does NOT represent a vertical asymptote of the graph of y = tan u? 3p
2
Short u = - p2 u = 0 u = p u =
2
Response
55. Which function has a period of 4p?

y = tan 4u y = tan 2u y = tan 1 u y = tan 1 u
2 4

56. Does a tangent function have amplitude? Explain.

Mixed Review

Solve each equation in the interval from 0 to 2P. Round your answer to the See Lesson 13-5.
nearest hundredth.
60. 5 cos pt = 0.9
57. cos t = 41 58. 10 cos t = -2 59. 3 cos t = 1 See Lesson 11-5.
5

61. Find the mean, median, and mode for the set of values.
9 6 8 1 3 4 5 2 6 8 4 9 12 3 4 10 7 6

Find the 27th term of each sequence. 64. -11, -5, 1, c See Lesson 9-2.
62. 5, 8, 11, c 63. 59, 48, 37, c 65. 6, -7, -20, c

Get Ready!  To prepare for Lesson 13-7, do Exercises 66–68.

Identify each horizontal and vertical translation of the parent function y = ∣ x ∣ . See Lesson 2-6.

66. y = 0 x - 2 0 + 5 67. y = 0 x + 5 0 - 4 68. y = 0 x + 2 0 + 1

874 Chapter 13  Periodic Functions and Trigonometry

13-7 Translating Sine and CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Cosine Functions
MF-TAFF.SB.951  2C.hFo-ToFse.2t.r5ig  oCnhomoseetrtircigfounocmtioentrsictofumncotdioenls to
mpeordioedlipceprihoednicompheennao.m.e.na . . .
MF-IAFF.CS..79e1 2G.Fr-aIpF.h3.7.e.  trGigroapnhom. .e.trtricigfounnocmtioentrsi,csfhuonwctinogns,
spheoriwodin,gmpiderlinoed,, amnidlainme,palintuddaem. plitude.

MP 1, MP 2, MP 3, MP 4, MP 5, MP 7

Objectives To graph translations of trigonometric functions
To write equations of translations

When in doubt, make Think about riding a bike, and pumping the 12 in.
a sketch. pedals at a constant rate of one revolution 3 in.
each second. How does the graph of the
MATHEMATICAL height of one of your feet compare with the
graph of a sine function? How does it
PRACTICES compare with the graph of the height of
your other foot? Explain.

Lesson Recall that for any function f, you can graph f (x - h) by translating the graph of f  by
h units horizontally. You can graph f (x) + k by translating the graph of f  by k units
Vocabulary vertically.
• phase shift
Essential Understanding  You can translate periodic functions in the same way
that you translate other functions.

Each horizontal translation of certain periodic functions is a  phase shift .

y yk
h O

x x

O

g(x): horizontal translation of f(x) h(x): vertical translation of f(x)
g(x) ‫ ؍‬f(x ؊ h) h(x) ‫ ؍‬f(x) ؉ k

When g (x) = f (x - h), the value of h is the amount of the shift. If h 7 0, the shift is
to the right. If h 6 0, the shift is to the left. When h (x) = f (x) + k, the value of k is
the amount of the midline shift. If k 7 0, the midline shifts up. If k 6 0, the midline
shifts down.

Lesson 13-7  Translating Sine and Cosine Functions 875

Problem 1 Identifying Phase Shifts

Wr ehwyristheoxul+d you What is the value of h in each translation? Describe each phase shift (use a phrase
x − ( −4)? 4 as such as 3 units to the left).

W hen the function uses A g (x) = f (x − 2)
this form, you are less h = 2; the phase shift is 2 units to the right.

likely to misinterpret the B y = cos(x + 4)
= cos(x - ( -4))
value of h.
h = -4; the phase shift is 4 units to the left.

Got It? 1. What is the value of h in each translation? Describe each phase shift (use a
phrase such as 3 units to the left).

a. g (t) = f (t - 5)
b. y = sin(x + 3)

You can analyze a translation to determine how it relates to the parent function.

Problem 2 Graphing Translations y y ϭ sin x 6
2 45 x
Use the graph of the parent function y = sin x. What is the graph 1
How do translations of each translation in the interval 0 " x " 2P? O p
of trigonometric A y = sin x + 3 Ϫ1
functions relate to
translations of other 4y
functions?
Translating the graphs of Op x
trigonometric functions
is similar to translating k = 3
graphs of other functions.
Translate the graph of the parent function 3 units up. The midline is y = 3.

( )B y=sinx − P
2

y

2 x

O
p

Ϫ2

h = p
2

Translate the graph of the parent function p units to the right.
2

876 Chapter 13  Periodic Functions and Trigonometry

Got It? 2. Use the graph of y = sin x from Problem 2. What is the graph of each
translation in the interval 0 … x … 2p?

a. y = sin x - 2 b. y = sin(x - 2)
c. Which translation is a phase shift?
d. Which translation gives the graph a new midline?

You can translate both vertically and horizontally to produce combined translations.

Problem 3 Graphing a Combined Translation

Use the graph of the parent function y = sin x in Problem 2. What is the graph of the
translation y = sin (x + P) − 2 in the interval 0 " x " 2P?

What are the 2 y 2T ruannistlsadteowthne agnradpph oufntihtsetpoatrheenlteffut.nction
translations? O
Because p is added to
x, the horizontal phase p 2p x
shift is p units left.

Because - 2 is added
to the dependent value,

the vertical translation is

2 units down.

Got It? 3. Use the graph at the right of the parent function y

y = cos x. What is the graph of each translation O1 2 34 6x
in the interval 0 … x … 2p? p p 3p 2p
a. y = cos(x - 2) + 5
b. y = cos(x + 1) + 3 Ϫ1 2 2

The translations graphed in Problems 2 and 3 belong to the families of the sine and
cosine functions.

Concept Summary  Families of Sine and Cosine Functions

Parent Function Transformed Function

y = sin x y = a sin b(x - h) + k
y = cos x y = a cos b(x - h) + k

• 0 a 0 = amplitude (vertical stretch or shrink)

• 2p = period (when x is in radians and b 7 0)
b

•  h  = phase shift, or horizontal shift

•  k = vertical shift (y = k is the midline)

Lesson 13-7  Translating Sine and Cosine Functions 877

Problem 4 Graphing a Translation of y = sin 2x

Why should you What is the graph of y = sin 21x − P 2 − 3 in the interval from 0 to 2P?
graph the parent 3 2
function first?
Once you know the Since a = 1 and b = 2, the graph is a translation of y = sin 2x.
relative dimensions
of the curve, you can Step 1 Sketch one cycle of y = sin 2x. Use five points in the pattern
translate it according to zero-max-zero-min-zero.
the h- and k-values.
1y

O p px
Ϫ1
2

Step 2 Since h = p and k = - 23, translate the graph p units to the right and 3 units
3 3 2
down. Extend the periodic pattern from 0 to 2p.

Sketch the graph.

1y

O p 2p x
Ϫ1

The blue curve above is the graph of y = sin 21x - p 2 - 32 .
3

Got It? 4. What is the graph of each translation in the interval from 0 to 2p?

( ) ba.. yy == 2-c3ossinp22(xx+-1p3) --3 23

You can write an equation to describe a translation.

Problem 5 Writing Translations

What is an equation that models each translation?

kW  ihnayt are a, b, h, and A y = sin x, P units down B y = −cos x, 2 units to the left
= a sin
b(x − h) + k? p units down means k = -p. 2 units to the left means h = -2.
An equation is y = sin x - p. An equation is y = -cos (x + 2).
a = b = 1, h = 0,
and k = - p.

Got It? 5. What is an equation that models each translation?
a. y = cos x, p2 units up b. p
y = 2 sin x, 4 units to the right

878 Chapter 13  Periodic Functions and Trigonometry

You can write a trigonometric function to model a situation.

Problem 6 Writing a Trigonometric Function to Model a Situation STEM

Temperature Cycles  ​The table gives the average temperature in your town x days
after the start of the calendar year (0 … x … 365). Make a scatter plot of the data.
What cosine function models the average daily temperature as a function of x?

Day of Year 16 47 75 106 136 167 198 228 258 289 319 350

Temperature (؇F) 33 35 42 52 62 72 77 76 69 58 48 38

Make a scatter plot of the Xmin ϭ 0
data in the table. Xmax ϭ 370
Xscl ϭ 20
The cosine function Ymin ϭ 25
has the form Ymax ϭ 85
y = a cos b(x - h) + k. Yscl ϭ 5
Find the values a, b, h,
and k. amplitude: a = 21 (max − min)

One complete cycle takes = 1 (77 − 33) = 22
365 days. 2

The maximum point of period = 2P
the curve is (198, 77). b

Write the function 365 = 2bP, so b = 2P
in the form 365
y = a cos b(x - h) + k.
phase shift: h = 198 − 0 = 198
To check your answer,
graph the function with vertical shift: k = 77 − 22 = 55
the scatter plot to see if
it is a good fit. y = 22 cos 326P5(x − 198) + 55

Xmin ϭ 0
Xmax ϭ 370
Xscl ϭ 20
Ymin ϭ 25
Ymax ϭ 85
Yscl ϭ 5

Got It? 6. a. Use the model in Problem 6. What was the average temperature in 879
your town 150 days into the year?

b. What value does the midline of this model represent?
c. Reasoning ​Can you use this model to predict temperatures for next year?

Explain your answer.

Lesson 13-7  Translating Sine and Cosine Functions

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

PRACTICES

1. Graph y = sin 1x + p 2 in the interval from 0 to 2p. 4. Vocabulary  W​ rite a sine function that has amplitude
4 4, period 3p, phase shift p, and vertical shift -5.

2. Describe any phase shift or vertical shift in the graph 5. Error Analysis  ​Two students disagree on the
of y = 4 cos (x - 2) + 9.
ititsraip6snpsu2lnautintiositntsoftotohrtehylee=fltecfootfsoy3f 1=yx=c+ocspo63s2x.3.AxI.smSebcitoehrtetlyrsassaytuysdsthethnaatttit
3. What is an equation that shifts y = cos x, 3 units up
2p correct? Describe any errors of each student.
and 3 units to the right?

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Determine the value of h in each translation. Describe each phase shift (use a See Problem 1.
phrase like 3 units to the left).

6. g (x) = f (x + 1) 7. g (t) = f (t + 2) 8. f (z) = g(z - 1.6)
9. f (x) = g (x - 3) 10. y = sin (x + p)
( )11. y = cosx-5p
7

Use the function f(x) at the right. Graph each translation. See Problem 2.

12. g (x) = f (x) + 1 13. g (x) = f (x) - 3 2y
14. g (x) = f (x + 2) 15. g (x) = f (x - 1)
x
Graph each translation of y = cos x in the interval from 0 to 2P. O2 4 6 8

Ϫ2

16. y = cos (x + 3) 17. y = cos x + 3 18. y = cos x - 4
19. y = cos (x - 4) 20. y = cos x + p 21. y = cos (x - p)

Describe any phase shift and vertical shift in the graph. See Problem 3.
See Problem 4.
22. y = 3 sin x + 1 23. y = 4 cos (x + 1) - 2
25. y = sin (x - 3) + 2
( ) 24. y = sin x + p2 + 2

Graph each function in the interval from 0 to 2P. ( )27. y = sin x + p +1
3
( ) 26. y = 2 sin x + p4 - 1 ( )29. y = 2 sin
x - p +2
28. y = cos (x - p) - 3 6

Graph each function in the interval from 0 to 2P. ( )31. y + p
30. y = 3 sin 12x = cos 2 x 2 -2
32. y = 12 sin 2x - 1
34. y = sin 2(x + 3) - 2 ( )33. y = x + p
sin 3 3

35. y = 3 sin p (x - 2)
2

880 Chapter 13  Periodic Functions and Trigonometry

Write an equation for each translation. See Problem 5.
36. y = sin x, p units to the left
38. y = sin x, 3 units up 37. y = cos x, p units down
2

39. y = cos x, 1.5 units to the right

STEM 40. Temperature  T​ he table below shows water temperatures at a buoy in the See Problem 6.
Gulf of Mexico on several days of the year.

Day of Year 16 47 75 106 136 167 198 228 258 289 319 350

Temperature (؇F) 71 69 70 73 77 82 85 86 84 82 78 74

a. Plot the data. b. Write a cosine model for the data.

B Apply Write an equation for each translation.

41. y = cos x, 3 units to the left and p units up
42. y = sin x, p2 units to the right and 3.5 units up

43. Think About a Plan  T​ he function y = 1.5 sin p (x - 6) + 2 represents the average
6
monthly rainfall for a town in central Florida, where x represents the number of

the month (January = 1, February = 2, and so on). Rewrite the function using a

cosine model.

• How does the graph of y = sin x translate to the graph of y = cos x?
• What parts of the sine function will stay the same? What must change?

Write a cosine function for each graph. Then write a sine function for each graph.

44. 1 y x 45. 10 y
x
Op 2p
10 20 30 40
Ϫ3
Ϫ10

46. The graphs of y = sin x and y = cos x are shown at the right. 1 y y ‫؍‬ sin x x
a. What phase shift will translate the cosine graph onto the sine graph?
p
Write your answer as an equation in the form sin x = cos (x - h). Ϫ1 y ‫؍‬ cos x
b. What phase shift will translate the sine graph onto the cosine graph?

Write your answer as an equation in the form cos x = sin (x - h).

47. a. Open-Ended ​Draw a periodic function. Find its amplitude and period. Then

sketch a translation of your function 3 units down and 4 units to the left.
b. Reasoning ​Suppose your original function is f (x). Describe your translation

using the form g (x) = f (x - h) + k.

48. a. Write y = 3 sin (2x - 4) + 1 in the form y = a sin b(x - h) + k. (Hint: Factor
where possible.)

b. Find the amplitude, midline, and period. Describe any translations.

Lesson 13-7  Translating Sine and Cosine Functions 881

C Challenge Use a graphing calculator to graph each function in the interval from 0 to 2P.
Then sketch each graph.

49. y = sin x + x 50. y = sin x + 2x

51. y = cos x - 2x 52. y = cos x + x

53. y = sin (x + cos x) 54. y = sin (x + 2 cos x)

Standardized Test Prep

SAT/ACT 55. Which function is a phase shift of y = sin u by 5 units to the left?

y = 5 sin u y = sin (u + 5)

y = sin u + 5 y = sin 5u

56. Which function is a translation of y = cos u by 5 units down?

y = -5 cos u y = cos (u - 5)

y = cos u - 5 y = cos ( -5u)

57. Which function is a translation of y = sin u that is p units up and p units to the left?
3 2
( ) ( ) p p p p
y = sin u + 3 + 2 y = sin u - 2 + 3

( ) + p + p ( ) y = sin - p - p
y = sin u 2 3 u 3 2

Short 58. Find values of a and b such that the function y = sin u can be expressed as
Response y = a cos (u + b).

Mixed Review

Identify the period of each function. Then tell where two asymptotes occur for See Lesson 13-6.

each function.

59. y = tan 6u 60. y = tan u 61. y = tan 1.5u 62. y = tan u
4 6

For the given probability of success P on each trial, find the probability of See Lesson 11-8.

x successes in n trials.

63. x = 4, n = 5, p = 0.2 64. x = 3, n = 5, p = 0.6
65. x = 4, n = 8, p = 0.7 66. x = 7, n = 8, p = 0.7

Get Ready!  To prepare for Lesson 13-8, do Exercises 67–71. See p. 973.

Find the reciprocal of each fraction. 69. 21p 70. 41m5 71. 14
67. 193 68. -85 -t

882 Chapter 13  Periodic Functions and Trigonometry

13-8 Reciprocal MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Trigonometric Functions
MF-IAFF.CS..79e1 2G.Fr-aIpF.h3.7.e.  trGigroapnhom. .e.trtricigfounnocmtioentrsi,csfhuonwctinogns,
spheoriwodin,gmpiderlinoed,, amnidlainme,palintuddaem. plitude.

MP 1, MP 2, MP 3, MP 4, MP 5

Objectives To evaluate reciprocal trigonometric functions
To graph reciprocal trigonometric functions

This asks only You want the extension ladder to reach the 5 ft
for the length of windowsill so you can wash the top window. 20 ft
the extension, not What expression gives the length by which 70Њ
the length of the you should extend the ladder while keeping
extension ladder. the base in place? Explain.

MATHEMATICAL

PRACTICES

Lesson To solve an equation ax = b, you multiply each side by the reciprocal of a. If a is a
trigonometric expression, you need to use its reciprocal.
Vocabulary
• cosecant Essential Understanding  C​ osine, sine, and tangent have reciprocals. Cosine
• secant and secant are reciprocals, as are sine and cosecant. Tangent and cotangent are also
• cotangent reciprocals.

Key Concept  Cosecant, Secant, and Cotangent Functions

The cosecant (csc), secant (sec), and cotangent (cot) functions are defined using
reciprocals. Their domains do not include the real numbers u that make the
denominator zero.

csc u = 1 u sec u = 1 u cot u = 1 u
sin cos tan
p
(cot u = 0 at odd multiples of 2 , where tan u is undefined.)

You can use the unit circle to evaluate the reciprocal trigonometric functions y P (x, y)
u
directly. Suppose the terminal side of an angle u in standard position intersects O1 x

the unit circle at the point (x, y).

Then csc u = 1 , sec u = 1 , cot u = x .
y x y

Lesson 13-8  Reciprocal Trigonometric Functions 883

You can use what you know about the unit circle to find exact values for reciprocal
trigonometric functions.

Problem 1 Finding Values Geometrically

( ) ( )What of cot − 5P P
are the exact values 6 and csc 6 ? Do not use a calculator.

Find the point where the ( )cot- 5p
6
unit circle intersects the

terminal side of the angle
5p
- 6 radians. y

x

( (–√3,– 1 –5p
2 2 6

Find the exact value of ( )cot− 5p = x
( )cot 5p 6 y
- 6 .

= − 13 = 13
2

− 1
2
( )cot
− 5p = 13
6

Find the point where the ( )cscp
6

unit circle intersects the y (√23 , 1 )
2
terminal side of the angle

p radians. x
6
π
6

Find the exact value of ( )csc p = 1
( )csc 6 y
p .
6

= 1 =2

1

2

( )csc p =2
6

Got It? 1. What is the exact value of each expression? Do not use a calculator. y
( ) a. csc p3 b. cot -54p n
c. sec 3p
x
d. Reasoning ​Use the unit circle at the right to find cot n, csc n,
3
and sec n. Explain how you found your answers. 5

884 Chapter 13  Periodic Functions and Trigonometry

Use the reciprocal relationships to evaluate secant, cosecant, or cotangent on a
calculator, since most calculators do not have these functions as menu options.

Problem 2 Finding Values with a Calculator

What is the decimal value of each expression? Use the radian mode on your
calculator. Round to the nearest thousandth.

A sec 2 B cot 10

Can you use the cot 10 = 1
tsc aiannlc−−u11l,akctooesry−sfo1or,natthnhede sec 2 = co1s 2 tan 10



reciprocal functions? 1/cos(2) 1/tan(10)

No; those keys are –2.402997962 1.542351045
inverse functions, not

reciprocal functions.

sec 2 ≈ -2.403 cot 10 ≈ 1.542
C csc 35° D cot P

csc 35° = sin135° cot p = 1 p
tan
To evaluate an angle in degrees
in radian mode, use the degree ERR:DIVIDE BY 0
symbol from the ANGLE menu. 1:Quit
2:Goto

1/sin(35˚) 1.743446796

csc 35° ≈ 1.743 Evaluating cot p results in an
error message, because tan p
is equal to zero.



Got It? 2. What is the decimal value of each expression? Use the radian mode on your
calculator. Round your answers to the nearest thousandth.

a. cot 13
b. csc 6.5
c. sec 15°
de.. Rseecas32opning ​How can you find the cotangent of an angle without using the tangent

key on your calculator?

Lesson 13-8  Reciprocal Trigonometric Functions 885

The graphs of reciprocal trigonometric functions have asymptotes where the
functions are undefined.

Problem 3 Sketching a Graph

For what values is What are the graphs of y = sin x and y = csc x in the interval from 0 to 2P?
csc x undefined?
Wherever sin x = 0, its Step 1 Make a table of values.
reciprocal is undefined.

x 0 p pp 2p 5p p 7p 4p 3p 5p 11p 2p
6 32 3 6 6 3 2 3 6

sin x 0 0.5 0.9 1 0.9 0.5 0 Ϫ0.5 Ϫ0.9 Ϫ1 Ϫ0.9 Ϫ0.5 0

csc x — 2 1.2 1 1.2 2 — Ϫ2 Ϫ1.2 Ϫ1 Ϫ1.2 Ϫ2 —

Step 2 Plot the points and sketch the graphs.

2 y a ys=ymcspctoxtwe willhheanveevaervietrstical

1 y ‫ ؍‬csc x denominator (sin x) is 0.

O y ‫ ؍‬sin x p x
Ϫ1

Ϫ2

Got It? 3. What are the graphs of y = tan x and y = cot x in the interval from 0 to 2p?

You can use a graphing calculator to graph trigonometric functions quickly.

Problem 4 Using Technology to Graph a Reciprocal Function

How can you find the Graph y = sec x. What is the value of sec 20°?
value?
Use the table feature of Step 1 Use degree mode. Step 2 Use the TABLE feature.
your calculator. 1 sec 20° ≈ 1.0642
Graph y = cos x.

Xmin ϭ –360 X Y1
Xmax ϭ 360
Xscl ϭ 30 20 1.0642
Ymin ϭ –5 21 1.0711
Ymax ϭ 5 22 1.0785
Yscl ϭ 1 23 1.0864
24 1.0946
25 1.1034
26 1.1126

X = 20

Got It? 4. What is the value of csc 45°? Use the graph of the reciprocal
trigonometric function.

886 Chapter 13  Periodic Functions and Trigonometry

You can use a reciprocal trigonometric function to solve a real-world problem.

Problem 5 Using Reciprocal Functions to Solve a Problem

A restaurant is near the top of a tower. A diner looks down U
601 ft
at an object along a line of sight that makes an angle of U

with the tower. The distance in feet of an object from the

observer is modeled by the function d = 601 sec U.

How far away are objects sighted at angles  d
of 40° and 70°?

Set your calculator to degree mode.
Enter the function and construct
a table that gives values of d for
various angles of u.



Plot1 Plot2 Plot3 TABLE SETUP X Y1
\Y1 = 601/cos(X) TblStart = 20
How can you check \Y2 = 20 639.57
that your answers \Y3 = Tbl = 10 30 693.98
are correct? \Y4 = Indpnt: Auto Ask 40 784.55
Multiply the answers by \Y5 = Depend: Auto Ask 50 934.99
\Y6 = 60 1202
cos u. If the answers are \Y7 = 70 1757.2
correct, then the product 80 3461
is 601.
X = 20

From the table, the objects are about 785 feet away and 1757 feet away, respectively.

Got It? 5. The 601 in the function for Problem 5 is the diner’s height above the ground

in feet. If the diner is 553 feet above the ground, how far away are objects
sighted at angles of 50° and 80°?

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

PRACTICES

Find each value without using a calculator. 6. Reasoning  E​ xplain why the graph of y = 5 sec u has
no zeros.
1. csc p2 2. sec 1- p 2
6
7. Error Analysis  ​On a quiz, a student wrote
U se a calculator to find each value. Round your answers
to the nearest thousandth. sec 20° + 1 = 0.5155. The teacher marked it wrong.
What error did the student make?

3. csc 1.5 4. sec 42° 8. Compare and Contrast  H​ ow are the graphs of
y = sec x and y = csc x alike? How are they
5. An extension ladder leans against a building forming different? Could the graph of y = csc x be
a transformation of the graph of y = sec x?
a 50° angle with the ground. Use the function
y = 21 csc x + 2 to find y, the length of the ladder.
Round to the nearest tenth of a foot.

Lesson 13-8  Reciprocal Trigonometric Functions 887

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Find each value without using a calculator. If the expression is undefined, See Problem 1.

write undefined.

9. sec( -p) 10. csc 54p ( )11. cot-p 12. sec p
14. csc 76p 3 2
( ) 13. cot -32p ( )15. sec3p
- 4 16. cot ( -p)

Graphing Calculator  Use a calculator to find each value. Round your answers See Problem 2.

to the nearest thousandth.

( ) 17. sec 2.5 18. csc(-0.2) 19. cot 56° 20. sec -32p

21. cot( -32°) 22. sec 195° 23. csc 0 24. cot( -0.6)

Graph each function in the interval from 0 to 2P. 27. y = csc 2u - 1 See Problem 3.
25. y = sec 2u 26. y = cot u 28. y = csc 2u

Graphing Calculator  Use the graph of the appropriate reciprocal trigonometric See Problem 4.

function to find each value. Round to four decimal places.

29. sec 30° 30. sec 80° 31. sec 110° 32. csc 30°

33. csc 70° 34. csc 130° 35. cot 30° 36. cot 60°

37. Distance  A​ woman looks out a window of a building. She is 94 feet above the See Problem 5.

ground. Her line of sight makes an angle of u with the building. The distance in
feet of an object from the woman is modeled by the function d = 94 sec u. How far
away are objects sighted at angles of 25° and 55°?

B Apply 38. Think About a Plan  A​ communications tower has wires anchoring it to the ground.
Each wire is attached to the tower at a height 20 ft above the ground. The length y
of the wire is modeled with the function y = 20 csc u, where u is the measure
of the angle formed by the wire and the ground. Find the length of wire needed to
form an angle of 45°.
• Do you need to graph the function?
• How can you rewrite the function so you can use a calculator?

39. Multiple Representations  Write a cosecant model that has the same graph as y = sec u.

Match each function with its graph.

40. y = sin1 x 41. y = 1 x 42. y = - 1 x
cos sin

a. b. c.

888 Chapter 13  Periodic Functions and Trigonometry

Graph each function in the interval from 0 to 2P.

43. y = csc u - p2 44. y = sec 1 u 45. y = -sec pu 46. y = cot u
4 3

47. a. What are the domain, range, and period of y = csc x?
b. What is the relative minimum in the interval 0 … x … p?
c. What is the relative maximum in the interval p … x … 2p?

48. Reasoning  ​Use the relationship csc x = 1 x to explain why each statement is true.
sin
a. When the graph of y = sin x is above the x-axis, so is the graph of y = csc x.
b. When the graph of y = sin x is near a y-value of -1, so is the graph of y = csc x.

Writing ​Explain why each expression is undefined.

49. csc 180° 50. sec 90° 51. cot 0°

52. Indirect Measurement  T​ he fire ladder forms an angle

of measure u with the horizontal. The hinge of the ladder y
is 35 ft from the building. The function y = 35 sec u
θ
models the length y in feet that the fire ladder must be 35 ft
8 ft
to reach the building.
a. Graph the function.

b. In the photo, u = 13°. What is the ladder’s length?
c. How far is the ladder extended when it forms an angle

of 30°?

d. Suppose the ladder is extended to its full length of 80 ft. What

angle does it form with the horizontal? How far up a building can the ladder

reach when fully extended? (Hint: Use the information in the photo.)

53. a. Graph y = tan x and y = cot x on the same axes.
b. State the domain, range, and asymptotes of each function.
c. Compare and Contrast ​Compare the two graphs. How are they alike? How are

they different?
d. Geometry ​The graph of the tangent function is a reflection image of the

graph of the cotangent function. Name at least two reflection lines for such a

transformation.

Graphing Calculator  Graph each function in the interval from 0 to 2P. Describe
any phase shift and vertical shift in the graph.

54. y = sec 2u + 3 55. y = sec 21u + p2 2 56. y = -2 sec (x - 4)

57. f (x) = 3 csc (x + 2) - 1 58. y = cot 2(x + p) + 3 59. g (x) = 2 sec 131x - p 2 2 - 2
6

60. a. Graph y = -cos x and y = -sec x on the same axes.
b. State the domain, range, and period of each function.

c. For which values of x does -cos x = -sec x? Justify your answer.
d. Compare and Contrast ​Compare the two graphs. How are they alike? How are

they different?

e. Reasoning ​Is the value of -sec x positive when -cos x is positive and negative
when -cos x is negative? Justify your answer.

Lesson 13-8  Reciprocal Trigonometric Functions 889

61. a. Reasoning  W​ hich expression gives the correct value of csc 60°?

I. sin 1 160-12°2 II. (sin 60°)-1 III. (cos 60°)-1

b. Which expression in part (a) represents sin 16102°?

C Challenge 62. Reasoning  E​ ach branch of y = sec x and y = csc x is a curve.
Explain why these curves cannot be parabolas. (Hint: Do parabolas
have asymptotes?)

63. Reasoning  ​Consider the relationship between the graphs of
y = cos x and y = cos 3x. Use the relationship to explain the distance
between successive branches of the graphs of y = sec x and y = sec 3x.

64. a. Graph y = cot x, y = cot 2x, y = cot ( -2x), and y = cot 12x on the same axes.
b. Make a Conjecture ​Describe how the graph of y = cot bx changes as the value

of b changes.

PERFORMANCE TASK

Apply What You’ve Learned MATHEMATICAL

PRACTICES

MP 4, MP 5

Refer to the problem on page 827 and look back at the tables and graphs you

made for the functions d = f (t) and h = g (t) in the Apply What You’ve Learned

in Lesson 13-1.

a. What is a phase shift and vertical translation you can use to transform the function

( )d = 2cos 25pt that results in the function d = f (t)?

b. Write a function d = f (t) that models the horizontal distance d of the dragonfly
from the y-axis after t seconds. Check your function by graphing it on your
calculator and comparing the graph to the one you made in Lesson 13-1.

c. What is a phase shift and vertical translation you can use to transform the function

( )h = 2sin 25pt that results in the function h = g(t)?

d. Write a function h = g (t) that models the height h of the dragonfly above the
x-axis after t seconds. Check your function by graphing it on your calculator and
comparing the graph to the one you made in Lesson 13-1.

890 Chapter 13  Periodic Functions and Trigonometry

13 Pull It All Together

RMANCPERFOE TASKCompleting the Performance Task

To solve these Look back at your results from the Apply What You’ve Learned sections in Lessons 13-1,
problems, 13-5, and 13-8. Use the work you did to complete the following.
you will pull
together many 1. Solve the problem in the Task Description on page 827 to determine the
concepts and coordinates of the dragonfly when the frog eats it. Round each coordinate to the
skills related to nearest hundredth. Show all your work and explain each step of your solution.
trigonometric
functions. 2. Reflect  Choose one of the Mathematical Practices below and explain how you
applied it in your work on the Performance Task.
MP 1: Make sense of problems and persevere in solving them.
MP 4: Model with mathematics.
MP 5: Use appropriate tools strategically.
MP 7: Look for and make use of structure.
MP 8: Look for and express regularity in repeated reasoning.

On Your Own

For another part of Suzanne’s computer game, a fish swims at different depths and
occasionally jumps out of the water. The circular path of the fish is shown in the graph,
where the line y = 6 represents the surface of the water. The fish starts at (4, 1), moves
counterclockwise around the circle, and completes one cycle in 3 seconds. Suzanne
wants to know when the fish will be out of the water.

8y

6

4

2
x

2468

Determine a range of times when the fish will be above the surface of the water during
its first cycle. Round times to the nearest hundredth of a second.

Chapter 13  Pull It All Together 891

13 Chapter Review

Connecting and Answering the Essential Questions

1 Modeling Angles, the Unit Circle, and Radian Translating Sine and Cosine
You can use combinations Functions (Lesson 13-7)
of circular functions (sine Measure (Lessons 13-2 and 13-3)
and cosine) to model natural
periodic behavior. One radian is For the functions
y = a sin b(x - h) + k,
the measure of 1 radian R and
a central angle y = a cos b(x - h) + k,

of a circle that

2 Function intercepts an arc R •  0 a 0 is the amplitude
of length equal
The graph of •  2p is the period
a4msipnli2tu1dxe-4,p4p2er+iod4 to the radius of b
y= •  h is the phase shift, or horizontal
has the circle.
shift
p, midline of y = 4, and
a minimum at the origin. •  k is the vertical shift ( y = k is the
Sine, Cosine, and Tangent Functions midline)

(Lessons 13-4, 13-5, and 13-6) 1
2
3 Function y sin u = Reciprocal Trigonometric
If you know the value of
1 1 cos u = { 13 Functions (Lesson 13-8)
sin u, find an angle with 2 1
measure u in standard 2u 2 csc u = sin = 2
u sin u u
position on the unit circle x tan u = cos u
ϪV23 V3 sec u = 1 = { 2
to find values of the other 2 1 cos u 13
= { 13
trigonometric functions. cot u = 1 = { 13
tan u

Chapter Vocabulary • initial side (p. 836) • sine curve (p. 852)
• intercepted arc (p. 844) • sine function (p. 851)
• amplitude (p. 830) • midline (p. 830) • sine of u (p. 838)
• central angle (p. 844) • period (p. 828) • standard position (p. 836)
• cosecant (p. 883) • periodic function (p. 828) • tangent function (p. 869)
• cosine function (p. 861) • phase shift (p. 875) • tangent of u (p. 868)
• cosine of u (p. 838) • radian (p. 844) • terminal side (p. 836)
• cotangent (p. 883) • secant (p. 883) • unit circle (p. 838)
• coterminal angle (p. 837)
• cycle (p. 828)

Choose the correct term to complete each sentence.

1. The ? of a periodic function is the length of one cycle.

2. Centered at the origin of the coordinate plane, the ? has a radius of 1 unit.

3. An asymptote of the ? occurs at u = p , and repeats every p units.
2

4. A horizontal translation of a periodic function is a(n) ? .

5. The ? is the reciprocal of the cosine function.

892 Chapter 13  Chapter Review

13-1  Exploring Periodic Data

Quick Review Exercises

A periodic function repeats a pattern of y-values at regular 6. Determine whether the function below is or is not
intervals. One complete pattern is called a cycle. A cycle may periodic. If it is, identify one cycle in two different
begin at any point on the graph. The period of a function ways and find the period and amplitude.
is the length of one cycle. The midline is the line located
2 y
midway between the maximum and the minimum values x
of the function. The amplitude of a periodic function is half
1 35
the difference between its maximum and minimum values. Ϫ2

Example 2y 7. Sketch the graph of a wave with a period of 2,
an amplitude of 4, and a midline of y = 1.
What is the period of the
periodic function? 8. Sketch the graph of a wave with a period of 4,
an amplitude of 3, and a midline of y = 0.
Ϫ4 Ϫ2 24

One cycle is 5 units long, so the period of the function is 5.

13-2  Angles and the Unit Circle

Quick Review Exercises

An angle is in standard position if the vertex is at the 9. Find the measurement of the angle in standard
origin and one ray, the initial side, is on the positive x-axis. position below.
The other ray is the terminal side of the angle. Two angles
in standard position are coterminal if they have the same y
terminal side.
45؇ O
The unit circle has radius of 1 unit and its center at the x
origin. The cosine of U (cos u) is the x-coordinate of the
point where the terminal side of the angle intersects the unit 10. Sketch a -30° angle in standard position.
circle. The sine of U (sin u) is the y-coordinate.
11. Find the measure of an angle between 0° and 360°
Example y coterminal with a -120° angle.

What are the cosine and sine of −210°? 12. Find the exact values of the sine and cosine of 315°
and -315°. Then find the decimal equivalents.
Sketch an angle of -210° in standard Ϫ210؇ x Round your answers to the nearest hundredth.
position with a unit circle. The terminal

side forms a 30°-60°-90° triangle with 13
2
hypotenuse = 1, shorter leg = 21, longer leg =

Since the terminal side lies in Quadrant II, cos ( -210°) is
negative and sin ( -210°) is positive.

cos ( -210°) = - 13 and sin ( -210°) = 1
2 2

Chapter 13  Chapter Review 893

13-3  Radian Measure

Quick Review Exercises

A central angle of a circle is an angle whose vertex is at the The measure U of an angle in standard position is given.
center of a circle and whose sides are radii of the circle. An a. Write each degree measure in radians and each
intercepted arc is the portion of the circle whose endpoints
are on the sides of the angle and whose remaining points radian measure in degrees rounded to the nearest
lie in the interior of the angle. A radian is the measure of
a central angle that intercepts an arc equal in length to a degree.
radius of the circle.
b. Find the exact values of cos U and sin U for each

angle measure.

13. 60° 14. -45°

Example 15. 180° 16. 2p radians

#What is the radian measure of an angle of −210°?p7p 17. 5p radians 18. - 3p radians
180° 6 6 4
-210° = -210° radians = - radians

19. Use the circle to find the length of m
the indicated arc. Round your 5p
answer to the nearest tenth. 3 5 ft

13-4  The Sine Function

Quick Review Exercises

The sine function y = sin u matches the measure u of an Sketch the graph of each function in the interval
angle in standard position with the y-coordinate of a point from 0 to 2P.
on the unit circle. This point is where the terminal side
of the angle intersects the unit circle. The graph of a sine 20. y = 3 sin u
function is called a sine curve.
21. y = sin 4u
For the sine function y = a sin bu, the amplitude equals
22. Write an equation of a sine function with a 7 0,
0 a 0 , there are b cycles from 0 to 2p, and the period is 2bp. amplitude 4, and period 0.5p.

Example

Determine the number of cycles the sine function
y = −7 sin 3U has in the interval from 0 to 2P. Find the

amplitude and period of each function.

For y = -7 sin 3u, a = -7 and b = 3. Therefore there are
3 cycles from 0 to 2p. The amplitude is 0 a 0 = 0 -7 0 = 7.
2p = 23p.
The period is b

894 Chapter 13  Chapter Review

13-5  The Cosine Function

Quick Review Exercises

The cosine function y = cos u matches the measure u of an Sketch the graph of each function in the interval from
angle in standard position with the x-coordinate of a point
0 to 2P.
on the unit circle. This point is where the terminal side of
( ) 23. y = 2 cos p u
the angle intersects the unit circle. 2

For the cosine function y = a cos bu, the amplitude equals 24. y = -cos 2u
2p
0 a 0 , there are b cycles from 0 to 2p, and the period is b . 25. Write an equation of a cosine function with a 7 0,
amplitude 3, and period p.

Example Solve each equation in the interval from 0 to 2P. Round
your answer to the nearest hundredth.
Find all solutions to 5 cos U = −4 in the interval from
0 to 2P. Round each answer to the nearest hundredth. 26. 3 cos 4u = -2

On a graphing calculator graph the equations y = -4 and 27. cos (pu) = -0.6
y = 5 cos u.

Use the Intersect feature to find the points at which the
two graphs intersect. The graph shows two solutions in the
interval. They are u ≈ 2.50 and 3.79.

13-6  The Tangent Function

Quick Review Exercises

The tangent of an angle u in standard position is the Graph each function in the interval from 0 to 2P. Then
y-coordinate of the point where the terminal side of P P2 .
the angle intersects the tangent line x = 1. A tangent evaluate the function at t = 4 and t = If the tangent is
function in the form y = a tan bu has a period of pb.
Unlike the graphs of the sine and the cosine, the tangent undefined at that point, write undefined.
is periodically undefined. At these points, the graph has
vertical asymptotes. 28. y = tan 1 t
2

29. y = tan 3t

30. y = 2 tan t

Example 31. y = 4 tan 2t

What is the period of y = tan P U? Tell where two
4
asymptotes occur.

period = p = p =4
b
p
4
One cycle occurs in the interval from -2 to 2, so there are
asymptotes at u = -2 and u = 2.

Chapter 13  Chapter Review 895

13-7  Translating Sine and Cosine Functions

Quick Review Exercises

Each horizontal translation of certain periodic functions is Graph each function in the interval from 0 to 2P.
a phase shift. When g(x) = f (x - h) + k, the value of h is
the amount of the horizontal shift and the value of k is the ( ) 32. y = cos x + p 33. y = 2 sin x - 4
amount of the vertical shift. y = k is the midline of the graph. 2

Example 34. y = sin (x - p) + 3 35. y = cos (x + p) - 1

What is an equation for the translation of y = sin x, 2 units Write an equation for each translation.
to the right and 1 unit up?
2 units to the right means h = 2, and 1 unit up means 36. y = sin x, p units to the right
k = 1. 4
An equation is y = sin (x - 2) + 1.
37. y = cos x, 2 units down

13-8  Reciprocal Trigonometric Functions

Quick Review Exercises

The cosecant (csc), secant (sec), and cotangent (cot) Evaluate each expression. Write your answer in
functions are defined as reciprocals for all real numbers u exact form.
(except those that make a denominator zero).
38. sec ( -45°) 39. cot 120°
1 1 1
csc u = sin u sec u = cos u cot u = tan u 40. csc 150° 41. cot ( -150°)

Example Graph each function in the interval from 0 to 4P.

Suppose sin U = − 35. Find csc U. 42. y = 2 csc u 43. y = sec u - 1

csc u = 1 = 1 = - 5 44. y = cot 1 u 45. y = csc 1 u + 2
sin 3 4 2
u -3
5

896 Chapter 13  Chapter Review

13 Chapter Test M athX

OLMathXL® for School
R SCHO Go to PowerAlgebra.com
L®

FO

Do you know HOW? Find the amplitude and period of each function. Then
sketch one cycle of the graph of each function.
Determine whether each function is or is not periodic.
If it is periodic, find the period and amplitude. 13. y = 4 sin (2x) 14. y = 2 sin (4x)

1. 2y Solve each equation in the interval from 0 to 2P.
Give an exact answer and an answer rounded to the
O
Ϫ3 1 3 x nearest hundredth.

15. cos t = 12 16. 3 tan 2t = 13

2. 2y Graph each function in the interval from 0 to 2P.

x 17. y = 2 cos x 18. y = cos (x + p)
20. y = tan p3 u
Ϫ5 Ϫ3 Ϫ1 1 3 5 19. y = - cos u 22. y = sec u + 1
p
24. y = csc (u + 1)
Ϫ2 21. y = cot x

23. y = csc u
2
Find the measure of an angle between 0° and 360°
coterminal with the given angle. Write an equation for each translation.

3. - 32° 4. -229° 5. 375° 25. y = cos x, 7.5 units to the right
26. y = sin x, 3 units to the left, 1.5 units down
Write each measure in radians. Express your answer in
terms of P and also as a decimal rounded to the nearest Evaluate each expression. Write your answer in exact
hundredth. form. If the expression is undefined, write undefined.

6. - 225° 7. 120° 8. 600° 27. sin 30° 28. sin ( -330°)

Write each radian measure in degrees. If necessary, 29. sec 270° 30. tan ( -60°)
round your answer to the nearest degree.

9. 56p 10. -2.5p 11. 0.8 Do you UNDERSTAND?

12. Using the graph below, determine how many cycles 31. Open-Ended  S​ ketch a function with period 4,
the sine function has in the interval from 0 to 2p. amplitude 7, and midline y = 3.
Find the amplitude, period, and midline.
32. Writing  ​Explain how to convert an angle measure
2y in radians to an angle measure in degrees. Include
an example.

STEM 33. Physics  ​On each swing, a pendulum 18 inches long
u 3p
O p 2p 3p travels through an angle of 4 radians. How far does
Ϫ2
the tip of the pendulum travel in one swing? Round

your answer to the nearest inch.

34. Reasoning  W​ hat are the steps you take to find the
asymptotes of the function y = tan (ax + b)?

Chapter 13  Chapter Test 897

13 Common Core Cumulative ASSESSMENT
Standards Review

Some problems ask you to The table below shows the results of TIP 2
use given results to find the spinning a spinner.
probability of independent Find the number of times the
events. Outcome Frequency events “landing on orange”
Orange 8 and “landing on blue” occur.
TIP 1 Green 7
Blue 5 Think It Through
Find the number of trials
for the experiment. The number of trials for the

experiment is 8 + 7 + 5 = 20.
Of those, 8 are orange.

What is the probability of landing on orange 8 = 2
and then blue? 20 5

The probability of spinning blue
5 14.
1 2 is 20 = The probability of
10 5
spinning orange, then blue is
#2
7 1230 1 = 2 = 110.
20 5 4 20

The correct answer is A.

LVVeooscsacoabnubluarlayry Review Selected Response

As you solve test items, you must understand Read each question. Then write the letter of the correct
the meanings of mathematical terms. Match each answer on your paper.
term with its mathematical meaning.
1. What is the sum of the geometric series

A . amplitude I. the measure of a central 2 + 4 + 8 + c + 64?
B. initial side angle that intercepts an arc
C . radian equal in length to a radius of 30 126
the circle
62 252

D. terminal side II. the ray not on the x-axis of an 2. Solve the following equation for x.
E. phase shift angle in standard position
y = 1 6
I II. a horizontal translation of 1x +
certain periodic functions 1 1
x = 1y - 6 x = y2 - 6

I V. the ray on the x-axis of an x = 1 + 6 x = 1 - 6
angle in standard position y2 y

V. half the difference between
the minimum and maximum
values of a periodic function

898 Chapter 13  Common Core Cumulative Standards Review

3. Which point is not a solution of the inequality 1 0. What are the roots of 5x4 - 12x 3 - 11x 2 + 6x = 0?

y 7 2 0 x + 3 0 - 7? - 1, 52, 3 0, 1, 52, 3

 (0, 0) (1, 1)

 ( -2, -1) ( -1, 1) - 1, 0, 25, 3 - 1, - 25, 0, 3

4. Which of the following equations is an equivalent 11. Which equation represents a circle with center ( -3, 8)
and radius 12?
form of 3x - 4y = 36 that makes it easy to identify the
y-intercept? (x - 8)2 + ( y + 3)2 = 144

 y = - 3 x - 9 y - 6 = 3 (x + 4) (x - 8)2 - ( y + 3)2 = 144
4 4

 y + 6 = 3 (x - 4) y = 3 x - 9 (x + 3)2 + ( y - 8)2 = 144
4 4
(x - 3)2 - ( y - 8)2 = 144

5. Which point in the feasible region below maximizes 12. The table below shows the results of spinning a
the objective functionC = 2x + 3y ? spinner. What is the probability of the spinner landing
on white, then red?
x Ú 0, y Ú 0
c x + y … 8
Outcome Frequency
3x + 2y … 18

(0, 0) (2, 6) (0, 8) (18, 0) Red 7

6. Which best describes the transformations used to obtain Blue 3

the graph of y = 31-x + 323 from the graph of y = x3? White 5

 r eflect across the x-axis, shift left 3 units, stretch by  91 7
a factor of 3 18

 r eflect across the y-axis, shift right 3 units, stretch
by a factor of 3
 475 4
 r eflect across the x-axis, shift right 3 units, stretch 5
by a factor of 3
1 3. During a promotional event, two customers at a
 r eflect across the y-axis, shift left 3 units, stretch by clothing store between 9 and 10 a.m. will be randomly
a factor of 3 selected to win a gift certificate. Suppose you and a
friend visit the store and there are 19 other customers
7. A sixth degree polynomial equation with rational between 9 and 10 a.m. What is the probability you and
coefficients has roots -1, 2i, and 1 - 15. Which of your friend will NOT both win the gift certificate?

the following cannot also be a root of the polynomial?  2110 2
21
I.  1 II. -2i III. 15 IV. 1 - i

 I only III and IV only  129 209
210
 II and III only II, III, and IV only
1 4. What is the sixth term in the expansion of (2x - 3y)7?
8. Which equation shows an inverse variation?  21 x 2y 5
x
 y = 5x 6 = y - 126 x 2y 5
- 20,412 x2y 5
xy - 4 = 0 y = -4  20,412 x 2y 5

9. Which is equivalent to 29 = 512?

 log5122 = 9 log29 = 512

 log2512 = 9 log9512 = 2

Chapter 13  Common Core Cumulative Standards Review 899

Constructed Response 2 5. Find all asymptotes of the graph of y = 3x 2 - 5x - 82.
x2 + 2x -

15. What is the integer equivalent of i6? 2 6. Using Pascal’s Triangle, what is the expansion of

16. What is the distance of 4 - 5i from the origin? (r - 3)4?

1 7. To the nearest hundredth, what is the theoretical 1
probability of rolling a 3 on a standard number cube?
11

18. What is the determinant of 5 - 3 d ? 121
c4 1
1331
19. A periodic function has an amplitude of 14 and a
minimum value of -3. What is its maximum value? 14641

2 0. Suppose y varies directly with the square of x. If y = 20 2 7. What is the directrix of the graph of 6x = x2 + 8y + 9?
when x = 3, what is |x| when y = 80?
2 1
21. What is the solution to the equation 72x = 75 to the 28. What is the inverse matrix of c -3 - 2 d ?
nearest hundredth?
29. Solve 8 + 15x + 2 = 10.

22. The table below shows a family’s daily water usage for 2 -1 3 4 07
2 weeks. Find the standard deviation of the data to
the nearest hundredth. 3 0. If B = £ 4 -8 0 § and C = £ -3 1 2 § ,

Daily Water Usage (gal) 5 69 9 -5 6

what is B + C?

92.3 81.3 Extended Response
85.3 81.6
89.7 76.9 31. The table shows the values of an investment after
101.2 94.0 the given number of years of continuously
80.3 89.6 compounded interest.
91.4 96.3
88.8 102.1

Years 0 1 2 3

Value $750.00 $780.61 $812.47 $845.62

23. If A = 3 -1 0 R, what is the entry a12 of the a. What is the rate of interest?
J -5 4 2 b. Write an equation to model the growth of the

matrix 3A? investment.
c. To the nearest year, when will the investment be
24. Find the sum of the rational expressions below. What
is the coefficient of the x2 term in the numerator of worth more than $1000?
x+1 x+1
the sum? x2 + 2 + 3x + 6 32. In a geometric sequence, a1 = 8 and a4 = 27. Explain
how to find a7.

900 Chapter 13  Common Core Cumulative Standards Review

Get Ready! CHAPTER

Lesson 4-5 Solving Quadratic Equations 14

Solve each equation. 22. 56p

1. 4x 2 = 25 2. x 2 - 23 = 0 3. 3x 2 = 80
6. 6x 2 - 13 = 11
4. 8x 2 - 44 = 0 5. 0.5x 2 = 15

Lesson 6-7 Finding the Inverse of a Function

For each function f, find f −1 and the domain and range of f and f −1.
Determine whether f −1 is a function.

7. f (x) = 5x + 2 8. f (x) = 1x + 3

9. f (x) = 13x - 4 10. f (x) = 5
11. f (x) = x 1-0 1 x

12. f (x) = 10 - 1
x

Lesson 7-5 Solving Exponential and Logarithmic Equations

Solve each equation.

13. 4x = 81 14. log 5x + 1 = -1
15. 73x = 500 16. log 3x + log x = 9

17. log (4x + 3) - log x = 5 18. 3x = 243

Lessons 13-4, Evaluating Trigonometric Functions

13-5, and 13-6 For each value of U, find the values of cos U, sin U, and tan U. Round your
answers to the nearest hundredth.

19. 48° 20. -105° 21. 16°

Looking Ahead Vocabulary

23. The equation 1 + tan2 u = sec2 u is a trigonometric identity. Use what you know
about identities to make a conjecture about this equation.

24. The Pythagorean Theorem is a special case of the Law of Cosines. What do you
suppose you will use the Law of Cosines to find?

Chapter 14  Trigonometric Identities and Equations 901

CHAPTER Trigonometric Identities

14 and Equations

Download videos VIDEO Chapter Preview 1 Equivalence
connecting math Essential Question  How do you verify
to your world.. 14-1 Trigonometric Identities that an equation involving the variable x
14-2 Solving Trigonometric Equations is an identity?
Interactive! ICYNAM
Vary numbers, ACT I V I TI Using Inverses 2 Function
graphs, and figures D 14-3 Right Triangles and Trigonometric Ratios Essential Question  A trigonometric
to explore math ES 14-4 Area and the Law of Sines function corresponds one number to many,
concepts.. 14-5 The Law of Cosines so how can its inverse be a function?
14-6 Angle Identities
14-7 Double-Angle and Half-Angle Identities 3 Equivalence
Essential Question  How do the
The online trigonometric functions relate to the
Solve It will get trigonometric ratios for a right triangle?
you in gear for
each lesson.

Math definitions VOC ABUL ARY Vocabulary DOMAINS
in English and • Trigonometric Functions
Spanish English/Spanish Vocabulary Audio Online: • Similarity, Right Triangles, and Trigonometry

English Spanish

Law of Cosines, p. 936 Ley de cosenos

Online access Law of Sines, p. 929 Ley de senos
to stepped-out
problems aligned trigonometric identity, identidad
to Common Core p. 904 trigonométrica
Get and view
your assignments trigonometric ratios razones trigonométricas
online. for a right triangle, para un triángulo
NLINE p. 922 rectángulo
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Extra practice
and review
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tutorials with
built-in support

PERFORMANCE TASK

Common Core Performance Task

Determining the Length of a Zip Line

A crew is setting up a zip line between two towers with support wires, as shown in
the figure. They know the heights of the towers and the angles that two of the
support wires make with the ground, which is level between the towers.

Angle of Depression

B

A Zip-Line

Tower 1 Tower 2
65 ft 85 ft

41° 62°
D CE

The crew’s leader is concerned that the angle of depression of the zip line may be
too great, resulting in a ride that is too fast. She wants to know how the length of
the zip line will change if the angle of depression is halved. (In this case, the crew
would set up a taller Tower 1.)

Task Description

Describe how the length of the zip line will change if its angle of depression is
halved. Include the length of each zipline in your answer.

Connecting the Task to the Math Practices MATHEMATICAL

As you complete the task, you’ll apply several Standards for Mathematical PRACTICES
Practice.

• You’ll analyze the given information and identify relationships among
important quantities in a real-world situation. (MP 1, MP 4)

• You’ll calculate accurately and efficiently to determine lengths and angle
measures. (MP 6)

• You’ll look for structure in a geometric figure and apply appropriate properties.
(MP 7)

Chapter 14  Trigonometric Identities and Equations 903

14-1 Trigonometric Identities CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

Objective To verify trigonometric identities FM-TAFF.SC.981  2P.rFo-vTeF.t3h.e8P  yPtrhoavgeotrheeaPnyitdheangtoitryean identity
sin2(u) + cos2(u) = 1 and use it ttooffiinnddssiinn((uu)),,ccooss((uu)),,oorr
tan(u), given sin(u), cos(u),or tan(u), and tthheeqquuaaddrraanntt
of the anglee..

MP 1, MP 2, MP 3, MP 4

y y

Ϫp O pu Ϫp O p u
Ϫ1 Ϫ1
Graphs of rational
functions had holes What could be the function for each graph? Explain your reasoning.
like these.

MATHEMATICAL

PRACTICES

You may recognize x 2 = 5x - 6 as an equation that you are to solve to find the few,

if any, values of x that make the equation true. On the other hand, you may recognize
exxx35 p=rexs2siaosnasninidtehnetietqy,uaantioeqnuaarteiodneftihnaetdi.s(tHrueeref,oxxr53ailsl
values of x for which the
not defined for x = 0.)

Lesson A trigonometric identity in one variable is a trigonometric equation that is true for all
values of the variable for which all expressions in the equation are defined.
Vocabulary
• trigonometric Essential Understanding  The interrelationships among the six basic

identity

trigonometric functions make it possible to write trigonometric expressions in various

equivalent forms, some of which can be significantly easier to work with than others in

mathematical applications.

Some trigonometric identities are definitions or follow immediately from definitions.

Key Concept  Basic Identities

Reciprocal Identities csc u = 1 u sec u = 1 u tan u = 1 u
sin cos cot

sin u = 1 u cos u = 1 u cot u = 1 u
csc sec tan

Tangent Identity tan u = sin uu Cotangent Identity cot u = cos u
cos sin u

The domain of validity of an identity is the set of values of the variable for which all
expressions in the equation are defined.

904 Chapter 14  Trigonometric Identities and Equations

Problem 1 Finding the Domain of Validity

What is the domain of validity of each trigonometric identity?

How can an A cos U = 1 U.
expression be sec
undefined?
An expression could The domain of cos u is all real numbers. The y sec u
contain a denominator 1 1 cos u
that could be zero or domain of sec u excludes all zeros of sec u (of
it could contain an
expression that is itself which there are none) and all values u for p
undefined for some 2
values. which sec u is undefined (odd multiples of ). Ϫp2 O p p u
Ϫ1
Therefore the domain of validity of 2

cos u = 1 u is the set of real numbers
sec multiples
except for the odd of p .
2

B sec U = 1 U.
cos

The domain of validity is the same as part (a), because sec u is not defined for odd
p2 , p
multiples of and the odd multiples of 2 are the zeros of cos u.

Got It? 1. What is the domain of validity of the trigonometric identity sin u = 1 u ?
csc

You can use known identities to verify other identities. To verify an identity, you can
use previously known identities to transform one side of the equation to look like the
other side.

Problem 2 Verifying an Identity Using Basic Identities

Verify the identity. What is the domain of validity?

What identity do # A (sin U)(sec U) = tan U 1
cos
you know that you (sin u)(sec u) = sin u u Reciprocal Identity

can use? sin u
cos u
Look for a way to write = Simplify.

the expression on the = tan u Tangent Identity

left in terms of sin u

and cos u. The identity The domain of sin u is all real numbers. The domains of sec u and tan u exclude all
1
sec u = cos u does the zeros of cos u. These are the odd multiples of p . The domain of validity is the set of
job. 2
real numbers except for the odd multiples of p2 .

B co1t U = tan U
1
1 u = 1 Definition of cotangent
cot
tan u

         = tan u Simplify.

The domain of cot u excludes multiples of p. Also, cot u = 0 at the odd multiples of
p2 . The domain of validity is the set of real numbers except all multiples of p2 .

Got It? 2. Verify the identity csc u = cot u. What is the domain of validity?
sec u

Lesson 14-1  Trigonometric Identities 905

You can use the unit circle and the Pythagorean Theorem to verify (cos u, sin u)
another identity. The circle with its center at the origin with a 1
radius of 1 is called the unit circle, and has an equation
x2 + y2 = 1. y = sin u

x = cos u

Every angle u determines a unique point on the unit circle
with x- and y-coordinates (x, y) = (cos u, sin u).

Therefore, for every angle u, This form allows you to
(cos u)2 + (sin u)2 = 1  or  cos2 u + sin2 u = 1. hsmwr1ite2_thae2i_desnet_itcy1w4i_thLo0u1t _t0001.ai

using parentheses.

This is a Pythagorean identity. You will verify two others in Problem 3.

You can use the basic and Pythagorean identities to verify other identities. To prove
identities, transform the expression on one side of the equation to the expression on the
other side. It often helps to write everything in terms of sines and cosines.

With which side Problem 3 Verifying a Pythagorean Identity
should you work? Verify the Pythagorean identity 1 + tan2 U = sec2 U.
It usually is easier to
begin with the more ( )1 + tan2u = 1 + sin u 2 Tangent Identity
complicated-looking side. cos u

= 1 + sin2 u Simplify.
cos2 u

= cos2 u + sin2 u Find a common denominator.
cos2 u cos2 u

= cos2 u + sin2 u Add.
cos2 u

= 1 Pythagorean identity
cos2 u

= sec2 u Reciprocal identity

You have transformed the expression on the left side of the equation to become the
expression on the right side. The equation is an identity.

Got It? 3. a. Verify the third Pythagorean identity, 1 + cot2 u = csc2 u.
b. Reasoning  Explain why the domain of validity is not the same for all

three Pythagorean identities.

You have now seen all three Pythagorean identities.

Key Concept  Pythagorean Identities

cos2 u + sin2 u = 1 1 + tan2 u = sec2 u 1 + cot2 u = csc2 u

906 Chapter 14  Trigonometric Identities and Equations

There are many trigonometric identities. Most do not have specific names.

Problem 4 Verifying an Identity

Verify the identity tan2 U − sin2 U = tan2 U sin2 U.

How do you begin tan2 u - sin2 u = sin2 u - sin2 u Tangent identity
when both sides look cos2 u
complicated?
It often is easier to = sin2 u - sin2 u cos2 u Use a common denominator.
collapse a difference (or cos2 u cos2 u
sum) into a product than
to expand a product into = sin2 u - sin2 u cos2 u Simplify.
a difference. cos2 u

= sin2 u(1 - cos2 u) Factor.
cos2 u

= sin2 u(sin2 u) Pythagorean identity
cos2 u

= sin2 u sin2 u Rewrite the fraction.
cos2 u

= tan2 u sin2 u Tangent Identity

Got It? 4. Verify the identity sec2 u - sec2 u cos2 u = tan2 u.

You can use trigonometric identities to simplify trigonometric expressions.

Problem 5 Simplifying an Expression
What is a simplified trigonometric expression for csc U tan U?

Write the expression. 1 #csc = 1
Then replace csc u with sin u. u tan u sin u tan u

Replace tan u with sin uu. #= 1 u sin u
cos sin cos u

= sin u u
sin u cos

Simplify. = 1
cos
u

1 = sec u. = sec u
cos u

Got It? 5. What is a simplified trigonometric expression for sec u cot u?

Lesson 14-1  Trigonometric Identities 907

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
Verify each identity.
5. Vocabulary  H​ ow does the identity
1. tan u csc u = sec u cos2 u + sin2 u = 1 relate to the Pythagorean
2. csc2 u - cot2 u = 1 Theorem?
3. sin u tan u = sec u - cos u
4. Simplify tan u cot u - sin2 u. 6. Error Analysis  ​A student simplified the expression
2 - cos2 u to 1 - sin2 u. What error did the student
make? What is the correct simplified expression?

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Verify each identity. Give the domain of validity for each identity. See Problems 1–4.
9. cos u tan u = sin u
7. cos u cot u = sin1 u - sin u 8. sin u cot u = cos u 12. tan u cot u = 1
15. csc u - sin u = cot u cos u
10. sin u sec u = tan u 11. cos u sec u = 1

13. sin u csc u = 1 14. cot u = csc u cos u

Simplify each trigonometric expression. See Problem 5.

16. tan u cot u 17. 1 - cos2 u 18. sec2 u - 1

19. 1 - csc2 u 20. sec2 u cot2 u 21. cos u tan u

22. sin u cot u 23. sin u csc u 24. sec u cos u sin u

25. sin u sec u cot u 26. sec2 u - tan2 u 27. sin u u
cos u tan

B Apply 28. Think About a Plan  ​Simplify the expression sec tan u u.
u- cos

• Can you write everything in terms of sin u, cos u, or both?

• A re there any trigonometric identities that can help you simplify the expression?

Simplify each trigonometric expression. 30. csc u cos u tan u

29. cos u + sin u tan u 32. sin2 u + cos2 u + tan2 u
31. tan u(cot u + tan u)
33. sin u(1 + cot2 u) 34. sin2 u csc u sec u
35. sec u cos u - cos2 u
37. csc2 u(1 - cos2 u) 36. csc u - cos u cot u

39. coscuotcusc u 38. sin u csc u cot u
+ cos u

40. sin2 u csc u sec u
tan u

908 Chapter 14  Trigonometric Identities and Equations

Express the first trigonometric function in terms of the second.

41. sin u, cos u 42. tan u, cos u 43. cot u, sin u
46. sec u, tan u
44. csc u, cot u 45. cot u, csc u

Verify each identity. 48. sec u - sin u tan u = cos u 49. sin u cos u (tan u + cot u) = 1
47. sin2 u tan2 u = tan2 u - sin2 u
50. 1 c-ossiun u = 1 c+ossiun u 51. cot sec u u = sin u 52. 1cot u + 122 = csc2 u + 2 cot u
u+ tan

53. Express cos u csc u cot u in terms of sin u.

54. Express sec uco+s utan u in terms of sin u.

Use the identity sin2U + cos2U = 1 and the basic identities to answer the
following questions. Show all your work.

55. Given that sin u = 0.5 and u is in the first quadrant, what are cos u and tan u?
56. Given that sin u = 0.5 and u is in the second quadrant, what are cos u and tan u?
57. Given that cos u = -0.6 and u is in the third quadrant, what are sin u and tan u?
58. Given that sin u = 0.48 and u is in the second quadrant, what are cos u and tan u?
59. Given that tan u = 1.2 and u is in the first quadrant, what are sin u and cos u?
60. Given that tan u = 3.6 and u is in the third quadrant, what are sin u and cos u?
61. Given that sin u = 0.2 and tan u 6 0, what is cos u?

C Challenge 62. The unit circle is a useful tool for verifying identities. Use the diagram
at the right to verify the identity sin(u + p) = -sin u.
a. Explain why the y-coordinate of point P is 1y

sin(u + p). p (cos U, sin U)
b. Prove that the two triangles shown are congruent. u x

c. Use part (b) to show that the two blue segments are 1

congruent.

d. Use part (c) to show that the y-coordinate of Ϫ1

P is -sin u.
e. Use parts (a) and (d) to conclude that
sin(u + p) = -sin u. P

Use the diagram in Exercise 62 to verify each identity. Ϫ1

63. cos(u + p) = -cos u 64. tan(u + p) = tan u

Simplify each trigonometric expression. 66. (1 - sin u)(1 + sin u)csc2 u + 1
65. ctaont22 uu -- scescc22 uu

Lesson 14-1  Trigonometric Identities 909

STEM 67. Physics  W​ hen a ray of light passes from one medium into a second, the angle of u1
incidence u1 and the angle of refraction u2 are related by Snell’s law: u2
n1 sin u1 = n2 sin u2, where n1 is the index of refraction of the first medium and
n2 is the index of refraction of the second medium. How are u1 and u2 related if
n2 7 n1? If n2 6 n1? If n2 = n1?

Standardized Test Prep

SAT/ACT 68. Which expression is equivalent to 2 cot u? sin u

2 1 u co2t u 2 cos u 1 cos u
tan sin u 2

69. Which equation is NOT an identity?

cos2 u = 1 - sin2 u sin2 u = cos2 u - 1
cot2 u = csc2 u - 1 tan2 u = sec2 u - 1

70. Which expressions are equivalent?
I. (sin u)(csc u - sin u) II. sin2 u - 1 III. cos2 u

I and II only II and III only I and III only I, II, and III

71. How can you express csc2 u - 2 cot2 u in terms of sin u and cos u?

1 - 2 cos2 u 1 - 2 sin2 u sin2 u - 2 cos2 u sin12 u - 2 u
sin2 u sin2 u tan2

72. Which expression is equivalent to cos tan u u ?
u- sec
tan2 u
csc u sec u -csc u

Short 73. Show that (sec u + 1)(sec u - 1) = tan2 u is an identity.
Response

Mixed Review

Graph each function in the interval from 0 to 2P. See Lesson 13-8.
74. y = csc( -u) 75. y = -sec 0.5 u 76. y = -sec(0.5 u + 2) 77. y = p sec u

Find the measure of an angle between 0° and 360° that is coterminal with the See Lesson 13-2.

given angle.

78. 395° 79. 405° 80. -225° 81. - 149°

Get Ready!  To prepare for Lesson 14-2, do Exercises 82–84.

For each function f, find f −1. See Lesson 6-7.

82. f (x) = x + 1 83. f (x) = 2x - 3 84. f (x) = x 2 + 4

910 Chapter 14  Trigonometric Identities and Equations