Download Problems

454
chapter 5 Trigonometric Functions
chapter summary
To check that you have mastered the most important concepts and skills covered in this chapter,
make sure that you can do each item in the following list:
Explain what it means for an angle to be
negative.
Compute the cosine, sine, and tangent of any
angle of a right triangle if given the lengths of
two sides of the triangle.
Explain how an angle can be larger than 360◦ .
Compute the lengths of all three sides of a
right triangle if given any angle (in addition
to the right angle) and the length of any side.
Convert angles from radians to degrees.
Convert angles from degrees to radians.
Compute the length of a circular arc.
Use the basic trigonometric identities involving
π
−θ, 2 − θ, and θ + π .
Compute the cosine, sine, and tangent of any
π
π
multiple of 30◦ or 45◦ ( 6 radians or 4 radians).
Give the domain and range of the cosine, sine,
and tangent functions.
2
Explain why cos θ + sin θ = 1 for every angle
θ.
2
Give the domain and range of cos−1 , sin−1 , and
tan−1 .
Compute cos θ, sin θ, and tan θ if given just
one of these quantities and the location of the
corresponding radius.
Compute the composition of a trigonometric
function and an inverse trigonometric function.
To review a chapter, go through the list above to find items that you do not know how to do,
then reread the material in the chapter about those items. Then try to answer the chapter
review questions below without looking back at the chapter.
chapter review questions
1. Find all points where the line through the origin with slope 5 intersects the unit circle.
2. Sketch a unit circle and the radius of that circle
that makes an angle of −70◦ with the positive
horizontal axis.
3. Explain how to convert an angle from degrees
to radians.
9. Find three distinct angles, expressed in radians,
whose sine equals − 12 .
10. Find three distinct angles, expressed in radians,
whose tangent equals 1.
11. Explain why cos2 θ + sin2 θ = 1 for every angle
θ.
4. Convert 27◦ to radians.
12. Explain why cos(θ + 2π ) = cos θ for every angle
θ.
5. Explain how to convert an angle from radians
to degrees.
π
13. Suppose 2 < x < π and tan x = −4. Evaluate
cos x and sin x.
6. Convert
7π
9
radians to degrees.
7. Give the domain and range of each of the following functions: cos, sin, and tan.
8. Find three distinct angles, expressed in de1
grees, whose cosine equals 2 .
14.
Find the lengths of both circular arcs of the
3 4
unit circle connecting the points ( 5 , 5 ) and
5 12
( 13 , 13 ).
Chapter Summary and Chapter Review Questions 455
Use the right triangle below for Questions 15–35.
This triangle is not drawn to scale corresponding
to the data in the questions.
c
Ν
b
u
a
34.
Suppose a = 4 and c = 7. Evaluate u and ν
in radians.
35.
Suppose a = 6 and b = 7. Evaluate u and ν
in degrees.
3
36. Suppose θ is an angle such that cos θ = 8 . Evaluate cos(−θ).
4
37. Suppose x is a number such that sin x = 7 .
Evaluate sin(−x).
2
15. Suppose a = 4 and b = 9. Evaluate c.
16. Suppose a = 4 and b = 9. Evaluate cos u.
17. Suppose a = 4 and b = 9. Evaluate sin u.
18. Suppose a = 4 and b = 9. Evaluate tan u.
19. Suppose a = 4 and b = 9. Evaluate cos ν.
20. Suppose a = 4 and b = 9. Evaluate sin ν.
21. Suppose a = 4 and b = 9. Evaluate tan ν.
22. Suppose a = 3 and c = 8. Evaluate b.
23. Suppose a = 3 and c = 8. Evaluate cos u.
24. Suppose a = 3 and c = 8. Evaluate sin u.
25. Suppose a = 3 and c = 8. Evaluate tan u.
26. Suppose a = 3 and c = 8. Evaluate cos ν.
27. Suppose a = 3 and c = 8. Evaluate sin ν.
28. Suppose a = 3 and c = 8. Evaluate tan ν.
38. Suppose y is a number such that tan y = − 9 .
Evaluate tan(−y).
2
39. Suppose u is a number such that cos u = − 5 .
Evaluate cos(u + π ).
5
40. Suppose θ is an angle such that tan θ = 6 . Evalπ
uate tan( 2 − θ).
41. Find a formula for tan θ solely in terms of
cos θ.
42. Give the domain and range of each of the following functions: cos−1 , sin−1 , and tan−1 .
43. Evaluate cos−1
√
3
.
2
44. Evaluate sin−1
√
3
.
2
2
45. Evaluate cos(cos−1 5 ).
46. Without using a calculator, sketch the unit
circle and the radius that makes an angle of
cos−1 (−0.8) with the positive horizontal axis.
47. Explain why your calculator is likely to be unhappy if you ask it to evaluate cos−1 3.
29.
Suppose b = 4 and u = 28◦ . Evaluate a.
30.
Suppose b = 4 and u = 28◦ . Evaluate c.
31.
Suppose u = 28◦ . Evaluate cos ν.
32.
Suppose u = 28◦ . Evaluate sin ν.
49. Evaluate sin−1 (sin
33.
Suppose u = 28◦ . Evaluate tan ν.
50. Evaluate cos(tan−1 5).
48.
Find the smallest positive number x such
that
3 sin2 x − 4 sin x + 1 = 0.
19π
8
).