Hungerford T.W., Shaw D.J. Contemporary Precalculus: A Graphing Approach
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506 CHAPTER 6 Trigonometric Functions
■
Basic Identities:
sin
2
t cos
2
t 1 sin(t) sin t
1 tan
2
t sec
2
t cos(t) cos t
1 cot
2
t csc
2
t tan(t) tan t
tan t
co
1
t t
cot t
ta
1
n t
sin(t 2p) sin t csc(t 2p) csc t
cos(t 2p) cos t sec(t 2p) sec t
tan(t p) tan t cot(t p) cot t
■
If A 0 and b 0, then each of the f (t) A sin(bt c) and g(t) A cos(bt c) has
amplitude A, period 2p/b, phase shift c/b.
CATALOG OF BASIC FUNCTIONS—PART 4
2π
−2π
1
f(t) = sin tf(t) = cos tf(t) = tan t
−1
t
y
2π
−2π
1
−1
t
y
2π
−2π
t
y
1. Find a number t between 0 and 2p such that an angle of
t radians in standard position is coterminal with an angle
of 23p/3 radians in standard position.
2. Through how many radians does the second hand of a clock
move in 2 minutes and 40 seconds?
3.
9
5
p
radians degrees.
4. 36 degrees radians.
5. 220° radians.
6.
1
1
7
2
p
radians degrees.
7.
11
4
p
radians degrees.
8. 135° radians.
9. If an angle of v radians has its terminal side in the second
quadrant and sin v 8/9
, then find cos v.
10. cos
47
2
p
?
11. sin(13p) ?
12. Simplify:
s
t
i
a
n
n
(
(
t
t
2
p
p
)
)
Use the figure on the next page in Questions 13–23.
13. cos
p
5
? 14. sin
7
6
p
?
15. tan
6
5p
? 16. tan
16
6
p
?
17. sin
3
4p
? 18. cos
5
9p
?
19. csc
p
5
? 20. sec
7
6
p
?
21. cot
3
4p
? 22. csc
5
9p
?
23. sec
3
4p
?
REVIEW QUESTIONS
24.
3 sin
5
p
500
2
3 cos
5
p
500
2
?
25. Fill in the blanks (approximations not allowed):
26. Express as a single real number:
cos
3
4
p
sin
5
6
p
sin
3
4
p
cos
5
6
p
27.
sin
p
6
1
2
?
28. sin(p/2) sin 0 cos 0 ?
29. If f (x) log
10
x and g(t) cos t, then ( f g)(p) ?
30. Cos t is negative when the terminal side of an angle of
t radians in standard position lies in which quadrants?
31. If sin t 1/3
and the terminal side of an angle of
t radians in standard position lies in the second quadrant,
then cos t ?
32. Which of the following could possibly be a true statement
about a real number t?
(a) sin t 2 and cos t 1
(b) sin t 1/2 and cos t 2
/2
(c) sin t 1 and cos t 1
(d) sin t p/2 and cos t 1 (p/2)
(e) sin t 3/5 and cos t 4/5
33. If sin t 4/5 and the terminal side of an angle of
t radians in standard position lies in the third quadrant, then
cos t .
34. If sin(101p/2) 1, then sin(105p/2) ?
35. If p/2 t p and sin t 5/13, then cos t ?
36. cos
p
6
? 37. cos
2
3
p
?
36°
60°
30°
x
(1, 0)
(0, −1)
(−1, 0)
(
−−,
)
3
2
(0, 1)
(.809, .588)
y
1
2
(
− ,
)
3
2
1
2
CHAPTER 6 Review 507
38. sin
11
6
p
? 39. sin
p
3
?
40. tan (5p/3) ?
41. Which of the following is not true about the graph of
f (t) sin t?
(a) It has no sharp corners.
(b) It crosses the horizontal axis more than once.
(c) It rises higher and higher as t gets larger.
(d) It is periodic.
(e) It has no vertical asymptotes.
42. Which of the following functions has the graph in the figure
between p and p?
sin x,ifx 0
(a) f (x)
cos x,ifx 0
(b) g(x) cos x 1
sin x,ifx 0
(c) h(x)
sin(x), if x 0
(d) k(x) cos x
(e) p(x)
1 si
n
2
x
The point (3/50,7/50) lies on the terminal side of an
angle of t radians (in standard position). Find:
43. sin t 44. cos t
45. tan t 46. csc t
47. sec t 48. cot t
49. Find the equation of the straight line containing the terminal
side of an angle of 5p/3 radians (in standard position).
Suppose that an angle of w radians has its terminal side in the
fourth quadrant and cos w 2/13. Find:
50. sin w 51. tan w
52. cot w 53. cos(w)
54. sec (w) 55. csc(w)
56. Fill in the blanks (approximations not allowed):
57. Sketch the graphs of f(t) sin t and h(t) csc t on the same
set of coordinate axes (2p t 2p).
π
2
π
1
−1
−
−π
t
y
π
2
t 0
p
6
p
4
p
3
p
2
sin t
cos t
t sin t tan t sec t
p/4
2p/3
5p/6
58. Let u be the angle shown in the figure. Which of the follow-
ing statements is true?
(a) sin u
2
2
(b) cos u
2
2
(c) tan u 1
(d) cos u 2
(e) tan u 1
59. Let u be as indicated in the figure below. Which of the state-
ments (i)–(iii) are true?
(i) cos u
1
3
(ii) tan u
2
9
2
(iii) sin u
2
3
2
(a) only ii
(b) only ii and iii
(c) all of them
(d) only i and iii
(e) none of them
60. Between (and including) 0 and 2p, the function h(t)
tan t has
(a) 3 roots and is undefined at 2 places.
(b) 2 roots and is undefined at 3 places.
(
−−
θ
,
)
1
3
22
3
(0, 1)
(1, 0)
x
y
(
−−
θ
,
)
2
2
2
2
1
x
y
508 CHAPTER 6 Trigonometric Functions
(c) 2 roots and is undefined at 2 places.
(d) 3 roots and is defined everywhere.
(e) no roots and is undefined at 3 places.
61. If the terminal side of an angle of u radians in standard
position passes through the point (2, 3), then tan u .
62. Which of the statements (i)–(iii) are true?
(i) sin(x) sin x
(ii) cos(x) cos x
(iii) tan(x) tan x
(a) (i) and (ii) only
(b) (ii) only
(c) (i) and (iii) only
(d) all of them
(e) none of them
63. If sec x 1 and p/2 x p/2, then x ?
64. If tan t 4/3 and 0 t p, what is cos t?
65. Which of the following is true about sec t?
(a) sec(0) 0
(b) sec t 1/sin t
(c) Its graph has no asymptotes.
(d) It is a periodic function.
(e) It is never negative.
66. If cot t 0 and 0 t p, then t .
67. What is cot
2
3
p
?
68. Which of the following functions has the graph in the
figure?
(a) f(t) tan t
(b) g(t) tan
t
p
2
(c) h(t) 1 tan t
(d) k(t) 3 tan t
(e) p(t) tan t
69. Let f(t)
3
2
sin 5t.
(a) What is the largest possible value of f(x)?
(b) Find the smallest positive number t such that f(t) 0.
π
1
2
3
−π
t
y
−1
−2
−3
2π−2π
π
2
−
π
2
70.Sketch the graph of g(t)2 cos t.
71.Sketch the graph of f(t)
1
2
sin 2t(2pt2p).
72.Sketch the graph of f(t)sin 4t(0t2p).
In Questions 73–78, determine graphically whether the given
equation could possibly be an identity.
73.costsin
t
p
2
74.tan
2
t
1
sin
co
t
st
75.
c
s
o
in
s
t
t
s
c
i
o
n
s
3
3
t
t
tant
76.cos 2t
12
1
sin
2
t
77.sec
2
xcsc
2
x(sec
2
x)(csc
2
x)
78.tanxsec
2
x
1
79.What is the period of the function g(t)sin 4pt?
80.What are the amplitude, period, and phase shift of the
function
h(t)13 cos(14t15)?
81.The number of hours of daylight in Boston on day tof the
year is approximated by
d(t)3.1 sin(.0172t1.377)12.
(a)On the day with the most daylight, how many hours of
daylight are there? On the day with the least daylight,
how many hours are there?
(b)On what days are there less than 11 hours of daylight?
82.Acertain person’s blood pressure P(t) at time tseconds is
approximately
P(t)22 cos(2.5pt)98.
(a)The period of this function is time between heartbeats.
What is this person’s pulse rate (heartbeats per minute)?
(b)What is this person’s blood pressure [systolic pressure
(maximum) over diastolic pressure (minimum)]?
83.State the rule of a periodic function whose graph from t0
tot2pclosely resembles the one in the figure.
π
1
2
t
y
−1
−2
2π
2π
5
4π
5
6π
5
8π
5
84.State the rule of a periodic function with amplitude 3,
periodp, and phase shift p/3.
85.State the rule of a periodic function with amplitude 8,
period 5, and phase shift 14.
86.Ifg(t)20 sin(200t), for how many values of t, with
0t2p, is it true that g(t)1?
In Exercises 87 and 88, estimate constants A, b, c such that
f(t)Asin(btc).
87.f(t)6 sin(4t7)5 cos(4t8)
88.f(t)5 sin(5t3)2 cos(5t2)
In Exercises 89 and 90, find a viewing window that shows a
complete graph of the function.
89.f(t)3 sin(300t5)2 cos(500t8)
90.g(t)5 sin(400pt1)2 cos(150pt6)
91.The average monthly temperatures in St. Louis, Missouri,
are shown in the table.*
(a)Let x1 correspond to January. Plot 24 data points
(two years).
(b)Use regression to find a sine function that models this
data.
(c)What is the period of the function in part (b)? Does it fit
the data well?
CHAPTER 6Review509
*Based on data from the National Climatic Data Center.
MonthTemperature (F)
Jan.30
Feb.35
March46
April57
May67
June77
July80
Aug.78
Sept.70
Oct.58
Nov.45
Dec.34
510 CHAPTER 6 Trigonometric Functions
Chapter
6
Test
Sections 6.1–6.3; Special Topics 6.1.A
1. Find the radian measure of an angle in standard position
formed by rotating the terminal side
7
1
2
of a circle.
2. Find the exact values:
(a) sin
13
3
p
(b) cos
13
3
p
(c) tan
13
3
p
3. Factor: sin
3
t sin t
4. Assume that a wheel of a car has radius 36 cm. Find the
angle (in radians) through which the wheel turns when the
car travels 5 kilometers.
5. Match each angle in column 1 with an angle in column II
that is coterminal with it.
III
(a)
8
7
p
(i)
25
7
p
(b)
3
7
p
(ii)
4
7
p
(c)
9
7
p
(iii)
6
7
p
(iv)
23
7
p
6. The terminal side of an angle of t radians in standard posi-
tion passes through the point (3, 2). Find:
(a) sin t (b) cos t
(c) tan t
7. Find the exact value of sin t if cos t
1
17
and
3
2
p
t 2p.
8. If the radius of the circle is 4 cm and angle u measures 2
radians, find the area of the shaded sector.
9. Do not use a calculator and show your work.
(a) 480________ radians
(b)
4
p
0
radians ________.
10. The terminal side of an angle of t radians in standard posi-
tion lies in quadrant I on the straight line that passes through
the origin and is parallel to the line 7y 2x 4. Find
(a) sin t (b) cos t (c) tan t
θ
11. If sin t 35 and 0 t
p
2
, find tan (5p t).
12. The second hand on a clock is 5 inches long. Find the linear
speed (in inches per minute) and the angular speed (in radi-
ans per minute) of the tip of the second hand as it moves on
the face of the clock.
13. Simplify the expression (assume all denominators are
nonzero).
8
si
c
n
o
2
s
t
t
sin
s
2
in
t
2
t
sin
co
t
s
c
2
o
t
s t
14. Find the exact average rate of change of f(t) tan t from
t
p
6
to t
p
4
. Do not give decimal approximations.
15. Find the exact value of sin
17
4
p
.
16. Simplify the expression (assume all denominators are
nonzero).
Sections 6.4–6.6; Special Topics 6.5.A
17. If the terminal side of an angle of t radians lies on the line
y 6.2t, then tan t ________.
18. Let f(t) cos(2pt). Find the values of t (with 0 t 2p)
for which f(t) 1.
19. Find the exact values:
(a) sin
11
6
p
(b) cos
11
6
p
(c) tan
11
6
p
(d) cot
11
6
p
(e) csc
11
6
p
(f) sec
11
6
p
20. Sketch the graph of a periodic function with period 2 that is
not a trigonometric function.
21. Find a viewing window that accurately shows exactly six
complete waves of the graph of f(t) sin(1100t).
22. Multiply and simplify your answer:
(sin t csc t)(sin
2
t csc
2
t 1).
23. Use graphs to determine the number of solutions of the
equation sin t
2
9
between 0 and 4p. You need not solve
the equation.
24. The table on the next page shows the average monthly tem-
perature in a western city.
(a) Use 12 data points (with x 1 corresponding to January)
and sine regression to find a function that models this data.
4 tan t sin t 2 sin t
6 sin
2
t 2 sin t
(b) Use 24 data points (two years of data) to find another
model for this data.
(c) Which model is likely to be more accurate? [Hint: what
is the period of each model?]
CHAPTER 6 Test 511
Month Temperature (°F)
Jan 22.2
Feb 27.2
Mar 37.5
Apr 48.0
May 58.9
Jun 68.4
Jul 73.5
Aug 71.9
Sep 64.0
Oct 52.3
Nov 39.5
Dec 27.6
26. Use graphs to determine whether this equation could possi-
bly be an identity or is definitely not an identity:
(cos
2
t 1)(tan
2
t 1) tan
2
t.
27. (a) State the amplitude, period, and phase shift of the func-
tion f(t) 3 cos(4t 7).
(b) Write the rule of a periodic function that has amplitude 6,
period 2, and phase shift 1.
28. Assume that t is a number such that
sin t
6
7
and sec t 0.
Find the exact value of:
(a) cos t (b) tan t
(c) cot t (d) csc t
(e) sec t.
29. Let f(t) 2 sin(6t). Use algebra (not a calculator) to find all
the local maxima of this function in the interval [0, p] (that
is, the coordinates of the highest points on the graph over
this interval).
30. Find constants A, b, and c (rounded to four decimal places)
such that
A cos(bt c) 2 sin(3t 5) 2 cos(3t 2).
25. Prove the identity: 1 tan
2
x sec
2
x.
Andre Jenny/Focus Group/PictureQuest
512
DISCOVERY PROJECT 6 Pistons and Flywheels
A common and well-proven piece of technology is the piston and flywheel combi-
nation. It is clearly visible in photographs of steam locomotives from the middle of
the nineteenth century. The structure involves a wheel (or crankshaft) connected to
a sliding plug in a cylinder by a rigid arm. The axis of rotation of the wheel is per-
pendicular to the central axis of the cylinder. The sliding plug, the piston, moves in
one dimension, in and out of the cylinder. The motion of the piston is periodic, like
the basic trigonometric functions, but doesn’t have the same elegant symmetry.
It is quite easy to superimpose a coordinate plane on the flywheel-piston system
so that the center of the flywheel is the origin, the flywheel rotates counterclockwise,
and the piston moves along the x-axis. As you can see in the diagram, the flywheel
typically has a larger radius than the radial distance from the center to the point where
the arm attaches. In this particular figure, the radius of the flywheel is 50 centimeters,
and the radial distance to the attachment point Q is 45 centimeters. The arm is
150 centimeters long measured from the base of the piston to the attachment point.
1. What are the coordinates of A and B in the figure? How close does the base
of the piston come to the flywheel? What is the length of the piston stroke?
2. Show that Q has coordinates (45 cos t,45sint), where t is the radian mea-
sure of angle BOQ. [Hint: If Q has coordinates (x, y), use the point-in-the-
plane description to compute cos t and solve for x; find y similarly.]
3. Let p be the x-coordinate of the center point of the base of the piston.
Express p as a function of t. [Hint: Use the distance formula to express
the distance from Q to (p, 0) in terms of p and t. Set this expression
equal to 150 (why?) and solve for p.]
4. Let p(t) be the function found in Question 3 (that is, p(t) the x-coordinate
of the base of the piston when angle BOQ measures t radians). What is
the range of this function? How does this relate to Question 1?
5. Approximate the values of p(0), p(p/2), and p(p). Note that p(p/2) is
not halfway between p(0) and p(p). Which side of the halfway point is it
on? Does this mean that the piston moves faster on the average when it is
to the right or left of the halfway point? Find the value of t that places the
base of the piston at the halfway point of its motion.
6. If the flywheel spins at a constant speed, does the piston move back and
forth at a constant speed? How do you know?
x
45
150
(p, 0)
O
y
t
B
Q
A
TRIGONOMETRIC IDENTITIES AND EQUATIONS
Beam me up, Scotty!
When a light beam passes from one medium to
another (for instance, from air to water), it changes
both its speed and its direction. If you know what
some of these numbers are (say, the speed of light in
air or the angle at which a light beam hits the water),
then you can determine the unknown ones by solving a
trigonometric equation. See Exercises 39–42 on
page 564.
513
Chapter
© Stefano Torrione/Getty Images
2
−4
−6
6
514
514
Chapter Outline
Interdependence of
Sections
7.1 Basic Identities and Proofs
7.2 Addition and Subtraction Identities
7.2.A Special Topics: Lines and Angles
7.3 Other Identities
7.4 Inverse Trigonometric Functions
7.5 Trigonometric Equations
Until now, the variable t has been used for trigonometric functions, to
avoid confusion with the x’s and y ’s that appear in their definitions. Now
that you are comfortable with these functions, we shall usually use the let-
ter x (or occasionally y) for the variable. Unless stated, all trigonometric
functions in this chapter are considered to be functions of real numbers,
rather than functions of angles in degree measure.
Two kinds of trigonometric equations are considered here. Identities
(Sections 7.1–7.3) are equations that are valid for all values of the vari-
able for which the equation is defined, such as
sin
2
x cos
2
x 1 and cot x
ta
1
n x
.
Identities can be used for simplifying expressions, rewriting the rule of a
trigonometric function, performing numerical computations, and in other
ways. Conditional equations (Section 7.5) are valid only for certain values
of the variable, such as
sin x 0 (true only when x is an integer multiple of p).
Inverse trigonometric functions, which have a number of uses, are
considered in Section 7.4.
7.1 Basic Identities and Proofs
■ Examine possible identities graphically.
■ Use basic identities to simplify trigonometric expressions.
■ Learn strategies for proving identities algebraically.
When you suspect that an equation might be an identity, it’s a good idea to see
whether there is any graphical evidence to support this conclusion.
EXAMPLE 1
Is either of the following equations an identity?
(a) 2 sin
2
x cos x 2 cos
2
x sin x
(b)
1 si
c
n
o
x
s
x
sin
2
x
cos x tan x
Section Objectives
7.1 7.2 7.3
7.4
7.5
Sections 7.1, 7.4, and 7.5 are
independent of one another and
may be read in any order.
SOLUTION
(a) Test the equation by graphing these two equations on the same screen:
y
1
2 sin
2
x cos x [left side of equation being tested]
y
2
2 cos
2
x sin x [right side of equation being tested]
If the given equation is an identity (meaning that y
1
y
2
), then the two
graphs will be identical. As Figure 7–1 shows, however, the graphs are quite
different. Hence, this equation is not an identity.
(b) Test the second equation graphically.
The exploration suggests that the equation may be an identity because the two
graphs appear to be identical. However, this graphical evidence is not a
proof—that must be done algebraically. ■
Any equation can be graphically tested, as in Example 1, to see whether it
might be an identity. If the left-side and right-side graphs are different, then it
definitely is not an identity. If the graphs appear to be the same, then it is possible,
but not certain, that the equation is an identity.
The fact that two graphs appear identical on a calculator
screen does not prove that they are actually the same,
as the following Graphing Exploration shows.
PROVING IDENTITIES
The phrases “prove the identity” and “verify the identity” mean “prove that the
given equation is an identity.” We shall assume the elementary identities that were
proved in Chapter 6 and are summarized here.
SECTION 7.1 Basic Identities and Proofs 515
3
y
1
y
2
−2
2π
−2π
Figure 7–1
Graph the functions
f(x)
1 si
c
n
o
x
s
x
sin
2
x
and g(x) cos x tan x
on the same screen, using a viewing window with 2p x 2p. Do the graphs
appear to be identical?
GRAPHING EXPLORATION
In the viewing window with p x p and 2 y 2, graph both sides of the
equation
cos x 1
x
2
2
2
x
4
4
7
x
2
6
0
40
x
,3
8
20
.
Do the graphs appear to be identical? Now change the viewing window so that
2p x 2p. Is the equation an identity?
GRAPHING EXPLORATION