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SECTION 4.3 Trigonometry Extended: The Circular Functions
381
SECTION 4.3 EXERCISES
In Exercises 1 and 2, identify the one angle that is not coterminal with
all the others.
1. 150, 510, 210, 450, 870 450°
In Exercises 3–6, evaluate the six trigonometric functions of the
angle ␪.
4.
y
θ
P(–1, 2)
y
θ
x
P(4, –3)
6.
y
y
θ
θ
x
x
P(–1, –1)
P(3, –5)
In Exercises 7–12, point P is on the terminal side of angle ␪. Evaluate
the six trigonometric functions for ␪. If the function is undefined,
write “undefined.”
7. P共3, 4兲
9. P共0, 5兲
11. P共5, 2兲
8. P共4, 6兲
10. P共3, 0兲
12. P共22, 22兲
In Exercises 13–16, state the sign 共 or 兲 of (a) sin t, (b) cos t, and
(c) tan t for values of t in the interval given.
( )
( )
(c) 共兹3
苶, 1兲 (b)
(a) 共兹3苶, 1兲 (b) 共1, 兹3苶兲
(c) 共兹3
苶, 1兲 (a)
24. ␪ 60
(a) 共1, 1兲
(b) 共1, 兹3苶兲
(c) 共兹3
苶, 1兲 (b)
In Exercises 25– 36, evaluate without using a calculator by using ratios
in a reference triangle.
x
5.
(b) 共1, 兹3苶兲
7␲
23. ␪ 6
5␲
5␲ 11␲
7␲ 365␲
5␲
2. , , , , 3
3
3
3
3
3
3.
2␲
22. ␪ 3
(a) 共1, 1兲
( )
( )
25. cos 120 12
26. tan 300 兹3苶
␲
27. sec 2
3
3␲
28. csc 兹2苶
4
13␲ 1
29. sin 6 2
7␲ 1
30. cos 3 2
15␲
31. tan 1
4
13␲
32. cot 1
4
23␲ 兹苶3
33. cos 2
6
17␲ 兹苶2
34. cos 2
4
11␲ 兹3苶
35. sin 2
3
19␲
36. cot 兹3苶
6
In Exercises 37–42, find (a) sin ␪, (b) cos ␪, and (c) tan ␪ for
the given quadrantal angle. If the value is undefined, write “undefined.”
37. 450
38. 270
39. 7␲
11␲
40. 2
7␲
41. 2
42. 4␲
In Exercises 43– 48, evaluate without using a calculator.
␲
13. 0, , , 2
␲
14. , ␲ , , 2
2
43. Find sin ␪ and tan ␪ if cos ␪ and cot ␪ 0.
3
3␲
15. ␲, , , 2
3␲
16. , 2␲ , , 2
1
44. Find cos ␪ and cot ␪ if sin ␪ and tan ␪ 0.
4
In Exercises 17–20, determine the sign 共 or 兲 of the given value
without the use of a calculator.
2
45. Find tan ␪ and sec ␪ if sin ␪ and cos ␪ 0.
5
17. cos 143 7␲
19. cos 8
3
46. Find sin ␪ and cos ␪ if cot ␪ and sec ␪ 0.
7
18. tan 192 4␲
20. tan 5
In Exercises 21–24, choose the point on the terminal side of ␪.
21. ␪ 45
(a) 共2, 2兲
(b) 共1, 兹3
苶兲
(c) 共兹3
苶, 1兲 (a)
4
47. Find sec ␪ and csc ␪ if cot ␪ and cos ␪ 0.
3
4
48. Find csc ␪ and cot ␪ if tan ␪ and sin ␪ 0.
3
382
CHAPTER 4 Trigonometric Functions
In Exercises 49–52, evaluate by using the period of the function.
(
(
(
59. Too Close for Comfort An F-15 aircraft flying at an altitude of
8000 ft passes directly over a group of vacationers hiking at 7400 ft.
If ␪ is the angle of elevation from the hikers to the F-15, find the
distance d from the group to the jet for the given angle.
)
␲
49. sin 49,000␲ 12
6
50. tan (1,234,567␲兲 tan 共7,654,321␲兲 0
(a) ␪ 45
)
5,555,555␲
51. cos 0
2
3␲ 70,000␲
52. tan 2
)
53. Group Activity Use a calculator to evaluate the expressions in
Exercises 49–52. Does your calculator give the correct answers?
Many calculators miss all four. Give a brief explanation of what
probably goes wrong.
Standardized Test Questions
61. True or False If ␪ is an angle of a triangle such that cos ␪ 0,
then ␪ is obtuse. Justify your answer.
62. True or False If ␪ is an angle in standard position determined
by the point 共8,6兲, then sin ␪ 0.6. Justify your answer.
␪1
You should answer these questions without using a calculator.
␪2
63. Multiple Choice If sin ␪ 0.4, then sin (␪) csc ␪ E
sin ␪1 ␮ sin ␪2.
If ␪1 83 and ␪2 36 for a certain
piece of flint glass, find the index of
refraction.
(c) ␪ 140
where t 1 represents January, t 2 February, and so on.
Estimate the number of Get Wet swimsuits sold in January,
April, June, October, and December. For which two of these
months are sales the same? Explain why this might be so.
54. Writing to Learn Give a convincing argument that the
period of sin t is 2␲. That is, show that there is no smaller positive real number p such that sin 共t p兲 sin t for all real numbers t.
55. Refracted Light Light is refracted
(bent) as it passes through glass. In the
figure below ␪1 is the angle of incidence and ␪2 is the angle of refraction.
The index of refraction is a constant ␮
that satisfies the equation
(b) ␪ 90
60. Manufacturing Swimwear Get Wet, Inc. manufactures
swimwear, a seasonal product. The monthly sales x (in thousands)
for Get Wet swimsuits are modeled by the equation
␲t
x 72.4 61.7 sin ,
6
(A) 0.15
(A) 0.6
56. Refracted Light A certain piece of crown glass has an index of
refraction of 1.52. If a light ray enters the glass at an angle ␪1 42,
what is sin ␪2?
57. Damped Harmonic Motion A weight
suspended from a spring is set into motion.
Its dispacement d from equilibrium is
modeled by the equation
(E) 2.1
(B) 0.4
(C) 0.4
(D) 0.6
(E) 3.54
65. Multiple Choice The range of the function
f(t) (sin t)2 (cos t)2 is A
(A) {1}
(B) [ 1, 1]
(D) [0, 2]
(E) [0, ∞)
12
(A) 13
where d is the displacement in inches and
t is the time in seconds. Find the displacement at
the given time. Use radian mode.
(C) [0, 1]
5
(B) 12
5
(C) 13
5
(D) 12
12
(E) 13
Explorations
(a) t 0 0.4 in.
(b) t 3 ⬇ 0.1852 in.
␪ 0.25 cos t.
(D) 0.65
5
66. Multiple Choice If cos ␪ and tan ␪ 0, then sin ␪ A
13
d 0.4e0.2t cos 4t.
Find the measure of angle ␪ when t 0 and t 2.5.
(C) 0.15
64. Multiple Choice If cos ␪ 0.4, then cos (␪ ␲) B
Glass
58. Swinging Pendulum The Columbus
Museum of Science and Industry exhibits
a Foucault pendulum 32 ft long that swings back and forth
on a cable once in approximately 6 sec. The angle ␪ (in
radians) between the cable and an imaginary vertical line θ
is modeled by the equation
(B) 0
d
In Exercises 67– 70, find the value of the unique real number ␪
between 0 and 2␲ that satisfies the two given conditions.
1
67. sin ␪ and tan ␪ 0. 5␲/6
2
兹3苶
68. cos ␪ and sin ␪ 0. 11␲/6
2
69. tan ␪ 1 and sin ␪ 0. 7␲/4
兹2苶
70. sin ␪ and tan ␪ 0. 5␲/4
2
383
SECTION 4.3 Trigonometry Extended: The Circular Functions
Exercises 71–74 refer to the unit circle in this figure. Point P is on the
terminal side of an angle t and point Q is on the terminal side of an
angle t ␲2.
y
Q(–b, a)
t+ π
2
t
Extending the Ideas
77. Approximation and Error Analysis Use your grapher to
complete the table to show that sin ␪ ⬇ ␪ (in radians) when
冷 ␪ 冷 is small. Physicists often use the approximation sin ␪ ⬇ ␪ for
small values of ␪. For what values of ␪ is the magnitude of the
error in approximating sin ␪ by ␪ less than 1% of sin ␪? That is,
solve the relation
冷 sin ␪ ␪ 冷 0.01 冷 sin ␪冷.
P(a, b)
t
(1, 0)
(Hint: Extend the table to include a column for values of
冷 sin ␪ ␪ 冷
.)
冷 sin ␪ 冷
x
␪
( )
( )
␲
73. Explain why sin t cos t.
2
78. Proving a Theorem If t is any real number, prove that
1 共tan t兲2 共sec t兲2.
Taylor Polynomials Radian measure allows the trigonometric
functions to be approximated by simple polynomial functions. For
example, in Exercises 79 and 80, sine and cosine are approximated by Taylor polynomials, named after the English mathematician
Brook Taylor (1685–1731). Complete each table showing a Taylor
polynomial in the third column. Describe the patterns in the table.
␲
74. Explain why cos t sin t.
2
75. Writing to Learn In the figure for Exercises 71–74, t is an
angle with radian measure 0 t ␲2. Draw a similar figure for
an angle with radian measure ␲2 t ␲ and use it to explain
why sin 共t ␲2兲 cos t.
79.
76. Writing to Learn Use the accompanying figure to explain each
of the following.
y
Q(–a, b)
π –t
t
P(a, b)
t
(1, 0)
x
␪
sin ␪
0.3
0.2
0.1
0
0.1
0.2
0.3
0.295…
0.198…
0.099…
0
0.099…
0.198…
0.295…
80.
(a) sin 共␲ t兲 sin t
(b) cos 共␲ t兲 cos t
sin ␪ ␪
0.03
0.02
0.01
0
0.01
0.02
0.03
71. Using Geometry in Trigonometry Drop perpendiculars
from points P and Q to the x-axis to form two right triangles.
Explain how the right triangles are related.
72. Using Geometry in Trigonometry If the coordinates of
point P are 共a, b兲, explain why the coordinates of point Q are
共b, a兲.
sin ␪
␪
cos ␪
0.3
0.2
0.1
0
0.1
0.2
0.3
0.955…
0.980…
0.995…
1
0.995…
0.980…
0.955…
␪3
␪ 6
(
␪3
sin ␪ ␪ 6
(
)
␪2
␪2
␪4
␪4
1 cos ␪ 1 2
2
24
24
)