section 5.3 Cosine and Sine 391
position after moving through a total angle of 4π , and so on. Thus we see
the periodic behavior of the graph of cosine.
Later in this chapter we will return to examine the properties of cosine
and its graph more deeply. For now, let’s turn to the graph of sine. Here is
the graph of sine on the interval [−6π , 6π ]:
1
11 Π
2
9Π
2
7Π
2
5Π
2
3Π
2
Π
Π
3Π
5Π
7Π
9Π
11 Π
2
2
2
2
2
2
2
Θ
1
The graph of sine on the interval [−6π , 6π ].
This graph goes through the origin, as expected because sin 0 = 0. Moving
to the right along the θ-axis from the origin, we see that the graph hits its
π
π
highest value when θ = 2 , as expected because sin 2 = 1. Continuing further
to the right, we see that the graph crosses the θ-axis at the point (π , 0),
as expected because sin π = 0. The graph then hits its lowest value when
3π
3π
θ = 2 , as expected because sin 2 = −1. Then the graph crosses the θ-axis
again at (2π , 0), as expected because sin 2π = 0.
Surely you have noticed that the graph of sine looks much like the graph
of cosine. It appears that shifting one graph somewhat to the left or right
produces the other graph. We will see that this is indeed the case when we
delve more deeply into properties of cosine and sine later in this chapter.
The word “sine”
comes from the Latin
word “sinus”, which
means curve.
exercises
Give exact values for the quantities in Exercises
1–10. Do not use a calculator for any of these
exercises—otherwise you will likely get decimal
approximations for some solutions rather than
exact answers. More importantly, good understanding will come from working these exercises
by hand.
1.
(a) cos 3π
2.
(a) cos(−
3π
2
(b) sin 3π
)
(b) sin(− 3π
)
2
3.
(a) cos
11π
4
(b) sin
11π
4
4.
(a) cos
15π
4
(b) sin
15π
4
5.
(a) cos
2π
3
(b) sin
2π
3
6.
(a) cos
4π
3
(b) sin
4π
3
7.
(a) cos 210◦
(b) sin 210◦
8.
(a) cos 300◦
(b) sin 300◦
9.
10.
(a) cos 360045◦
(b) sin 360045◦
(a) cos(−360030◦ )
(b) sin(−360030◦ )
11. Find the smallest number θ larger than 4π
such that cos θ = 0.
12. Find the smallest √
number θ larger than 6π
2
such that sin θ = 2 .
13. Find the four smallest positive numbers θ such
that cos θ = 0.
14. Find the four smallest positive numbers θ such
that sin θ = 0.
15. Find the four smallest positive numbers θ such
that sin θ = 1.
16. Find the four smallest positive numbers θ such
that cos θ = 1.
17. Find the four smallest positive numbers θ such
that cos θ = −1.
18. Find the four smallest positive numbers θ such
that sin θ = −1.
392
chapter 5 Trigonometric Functions
Suppose − π2 < θ < 0 and cos θ = 0.3. Evaluate sin θ.
19. Find the four smallest positive numbers θ such
1
that sin θ = 2 .
26.
20. Find the four smallest positive numbers θ such
1
that cos θ = 2 .
27. Find the smallest number x such that sin(ex ) =
0.
21. Suppose 0 < θ <
sin θ.
π
2
and cos θ = 5 . Evaluate
2
22. Suppose 0 < θ <
cos θ.
π
2
and sin θ = 37 . Evaluate
23. Suppose
cos θ.
π
2
< θ < π and sin θ = 29 . Evaluate
24. Suppose
cos θ.
π
2
< θ < π and sin θ = 38 . Evaluate
25.
π
Suppose − 2 < θ < 0 and cos θ = 0.1. Evaluate sin θ.
28. Find the smallest number x such that
cos(ex + 1) = 0.
29.
Find the smallest positive number x such
that
sin(x 2 + x + 4) = 0.
30.
Find the smallest positive number x such
that
cos(x 2 + 2x + 6) = 0.
problems
31.
(a) Sketch a radius of the unit circle making
an angle θ with the positive horizontal
6
axis such that cos θ = 7 .
(b) Sketch another radius, different from the
6
one in part (a), also illustrating cos θ = 7 .
32.
(a) Sketch a radius of the unit circle making
an angle θ with the positive horizontal
axis such that sin θ = −0.8.
(b) Sketch another radius, different from the
one in part (a), also illustrating sin θ =
−0.8.
33. Find angles u and ν such that cos u = cos ν but
sin u = sin ν.
34. Find angles u and ν such that sin u = sin ν but
cos u = cos ν.
35. Suppose you have borrowed two calculators
from friends, but you do not know whether
they are set to work in radians or degrees.
Thus you ask each calculator to evaluate
cos 3.14. One calculator replies with an answer
of −0.999999; the other calculator replies with
an answer of 0.998499. Without further use of
a calculator, how would you decide which calculator is using radians and which calculator is
using degrees? Explain your answer.
36. Suppose you have borrowed two calculators
from friends, but you do not know whether
they are set to work in radians or degrees.
Thus you ask each calculator to evaluate
sin 1. One calculator replies with an answer
of 0.017452; the other calculator replies with
an answer of 0.841471. Without further use of
a calculator, how would you decide which calculator is using radians and which calculator is
using degrees? Explain your answer.
37. Explain why ecos x < 3 for every real number x.
38. Explain why the equation
(sin x)2 − 4 sin x + 4 = 0
has no solutions.
39. Explain why there does not exist a real number
x such that esin x = 14 .
40. Explain why the equation
(cos x)99 + 4 cos x − 6 = 0
has no solutions.
41. Explain why there does not exist a number θ
such that log cos θ = 0.1.