Section 4.3 Cosine and Sine 307
4.3 Cosine and Sine
Learning Objectives
By the end of this section you should be able to
• evaluate the cosine and sine of any multiple of 30° or 45° ( 7r6 radians or 7r4 radians);
• determine whether the cosine (or sine) of an angle is positive or negative from the location of the corresponding radius;
• sketch the radius corresponding to 0 if given either cos 0 or sin 0 and the sign of the other quantity;
• find cos 0 and sin 0 if given one of these quantities and the quadrant of the corresponding radius.
Definition of Cosine and Sine
The table here shows the endpoint of the radius of the unit circle corresponding to some special angles. This table comes from tables in Sections 4.1 and 4.2.
We might consider extending the table above to other angles. For example, suppose we want to know the endpoint of the radius corresponding to 7r18 radians. Unfortunately, the coordinates of the endpoint of that radius do not have a nice form—neither coordinate is a rational number or even the square root of a rational number. The cosine and sine functions, which we are about to introduce, were invented to help us extend this table to all angles.
Before introducing the cosine and sine functions, we state a common assumption about notation in trigonometry.
Angles without units
If no units are given for an angle, then assume the units are radians.
The figure below shows a radius of the unit circle corresponding to 0 (here 0 might be measured in either radians or degrees). The endpoint of this radius is used to define the cosine and sine, as follows.
Cosine
The cosine of 0, denoted cos 0, is the first coordinate of the endpoint of the radius of the unit circle corresponding to 0.
Sine
The sine of 0, denoted sin 0, is the second coordinate of the endpoint of the radius of the unit circle corresponding to 0.
The two definitions above can be combined into a single statement, as follows.
Cosine and sine
The endpoint of the radius of the unit circle corresponding to 0 has coordinates (cos 0, sin 0).
0 0 endpoint of radius
radians degrees
0 0° ( (1, 0)
7r 30° ^3 1
6 45°
7r 60° 90° 180° 360°
4
7r
3
7r
2
7r
27r
2 , 2
( )
^2 ^2
2 , 2 )
1 ^3
2, 2
( )
(0, 1)
(^1, 0)
(1, 0)
Coordinates of the endpoint of the
radius of the unit circle
corresponding to special angles.
This figure defines cosine and sine.
If you understand this figure well,
then you can figure out a big
chunk of trigonometry.
308 Chapter 4 Trigonometric Functions
The radius corresponding to ^ 7r2 radians has endpoint (0, ^1).
Most calculators can work in radians or degrees. When you use a calculator to compute values of cosine or sine, be sure your calculator is set to work in the appropriate units.
Evaluate cos 7r 2 and sin 7r 2 .
Example 1
Here units are not specified for the angle 7r2 . Thus we assume we are
dealing with 7r2 radians.
The radius corresponding to 7r2 radians has endpoint (0, 1).
Compare this table to the first table in this section and make sure you understand what is going on here.
0 (radians) 0 (degrees) cos 0 sin 0
0 4 10 1
7r 0^ ^3 2
6 30^ ^2
7r 45^
7r 60^
3 90^
7r 180^
2 360^
7r
27r
2
^2
2 2
1 ^3
2
0
^1
10
2
1
0
solution The radius corresponding to 7r2 radians has endpoint (0, 1). Thus
cos 7r2 = 0 and sin 7r2 = 1.
Using degrees instead of radians, we could write cos 90^ = 0 and sin 90^ = 1.
The table below gives the cosine and sine of some special angles. This table is obtained by breaking the last column of the previous table into two columns, with the first coordinate labeled as cosine and the second coordinate labeled as sine.
The next example extends the table above to another special angle.
Evaluate cos(^ 7r ) and sin(^ 7r ).
Example 2 2 2
solution The radius corresponding to ^ 7r2 radians has endpoint (0, ^1) as shown
here. Thus
cos(^ 7r) = 0 and sin(^7r) = ^1.
2 2
Using degrees instead of radians, we have cos(^90^) = 0 and sin(^90^) = ^1.
In addition to adding a row for ^ 7r2 radians (which equals ^90^), we could add many more entries to the table for the cosine and sine of special angles. Possibilities
would include 27r3 radians (which equals 120^), 57r6 radians (which equals 150^), the negatives of all the angles already in the table, and so on. This would quickly become far too much information to memorize. Instead of memorizing, concentrate on understanding the definitions of cosine and sine.
Similarly, do not become dependent on a calculator for evaluating the cosine and sine of special angles. If you need numeric values for cos 2 or sin 17^, then use a calculator. But if you get in the habit of using a calculator for evaluating expressions such as cos 0 or sin(^180^), then cosine and sine will become simply buttons on your calculator and you will not be able to use these functions meaningfully.
Note that cos and sin are functions; thus cos(0) and sin(0) might be a better notation than cos 0 and sin 0. In an expression such as
cos 10
cos 5 ,
we cannot cancel cos in the numerator and denominator, just as we cannot cancel a
function f in the numerator and denominator of f (10)
f (5) . Similarly, the expression
above is not equal to cos 2, just as f (10)
f (5) is usually not equal to f (2).
Section 4.3 Cosine and Sine 309
The Signs of Cosine and Sine
The coordinate axes divide the coordinate plane into four regions, often called quadrants. The quadrant in which a radius lies determines whether the cosine and sine of the corresponding angle are positive or negative. The figure below shows the sign of the cosine and the sign of the sine in each of the four quadrants. Thus, for example, an angle corresponding to a radius lying in the region marked “cos B < 0, sin B > 0” (the upper-left quadrant) will have a cosine that is negative and a sine that is positive.
There is no need to memorize this figure, because you can always reconstruct it if you understand the definitions of cosine and sine.
Recall that the cosine of an angle is the first coordinate of the endpoint of the To remember that cos B is the first coordinate of the endpoint of the radius corresponding to B and sin B is the second coordinate, keep cosine and sine in alphabetical order.
corresponding radius. Thus the cosine is positive in the two quadrants where the first coordinate is positive, as shown in the figure above. Also, the cosine is negative in the two quadrants where the first coordinate is negative.
Similarly, the sine of an angle is the second coordinate of the endpoint of the corresponding radius. Thus the sine is positive in the two quadrants where the second coordinate is positive, as shown in the figure above. Also, the sine is negative in the two quadrants where the second coordinate is negative.
The next example should help you understand how the quadrant determines the sign of the cosine and sine.
(a) Evaluate cos 7r4 and sin 7r4 .
(b) Evaluate cos 37r4 and sin 37r4 .
(c) Evaluate cos(— 7r) and sin(— 7r).
4 4
(d) Evaluate cos(—37r ) and sin(— 37r ).
4 4
solution The four angles 7r4 , 37r4 , — 7r4 , and — 37r4 radians (or, equivalently, 45°, 135°, —45°, and —135°) are shown in the next figure. Each coordinate of the radius
^2 ^2
corresponding to each of these angles is either 2 or — 2 ; the only issue to worry
about in computing the cosine and sine of these angles is the sign.
Example 3
310 Chapter 4 Trigonometric Functions
7r 37r 37r
4 radians 4 radians ^ 7r 4 radians ^ 4 radians
Quadrants can be labeled by the descriptive terms upper/lower and right/left. For example, the radius (a) Both coordinates of the endpoint of the radius corresponding to 7r4 radians are
corresponding to 37r4 radians (which equals 135^) is in the upper-left quadrant. ^2 ^2
positive. Thus cos 7r 4 = 2 and sin 7r 4 = 2 .
(b) The first coordinate of the endpoint of the radius corresponding to 37r4 radi-
^2
ans is negative; the second coordinate is positive. Thus cos 37r 4 = ^ 2 and
^2
sin 37r 4 = 2 .
(c) The first coordinate of the endpoint of the radius corresponding to ^ 7r4 radians is positive and the second coordinate is negative. Thus cos(^7r ) = ^2 2 and
4
sin(^ 7r) = ^ ^2
2 .
4
(d) Both coordinates of the endpoint of the radius corresponding to ^ 37r4 radians
are negative. Thus cos(^37r) = ^^2
2 and sin(^ 37r) = ^^22 .
4 4
The next example shows how to use information about the signs of the cosine and sine to locate the corresponding radius.
Sketch the radius of the unit circle corresponding to an angle 0 such that
Example 4
cos 0 = 0.4 and sin 0 < 0.
solution Because cos 0 is positive and sin 0 is negative, the radius corresponding to 0 lies in the lower-right quadrant. To find the endpoint of this radius, which has first coordinate 0.4, start with the point 0.4 on the horizontal axis and then move vertically down to reach a point on the unit circle. Then draw the radius from the origin to that point, as shown here.
The radius corresponding to angle 0 with cos 0 = 0.4 and sin 0 < 0.
Relationship between cosine and sine
The Key Equation Connecting Cosine and Sine
By definition of cosine and sine, the point (cos 0, sin 0) is on the unit circle, which is the set of points in the coordinate plane such that the sum of the squares of the coordinates equals 1. In the xy-plane, the unit circle is described by the equation
x2 + y2 = 1.
Thus the following crucial equation holds.
The point (cos 0, sin 0) is on the
unit circle.
(cos 0)2 + (sin 0)2 = 1
for every angle 0.
Section 4.3 Cosine and Sine 311
Given either cos 0 or sin 0, the last equation can be used to solve for the other quantity, provided that we have enough additional information to determine the sign. The following example illustrates this procedure.
Suppose 0 is an angle such that sin 0 = 0.6, and suppose also that rc2 < 0 < rc. Evaluate cos 0.
solution The equation above implies that (cos 0)2 + (0.6)2 = 1. Because (0.6)2 =
0.36, this implies that
(cos 0)2 = 0.64.
Thus cos 0 = 0.8 or cos 0 = ^0.8. The additional information that rc2 < 0 < rc implies that cos 0 is negative, as can been seen in the figure. Thus
cos 0 = ^0.8.
The Graphs of Cosine and Sine
Before graphing the cosine and sine functions, we should think carefully about the domain and range of these functions. Recall that for each real number 0, there is a radius of the unit circle corresponding to 0.
Recall also that the coordinates of the endpoints of the radius corresponding to the angle 0 are labeled (cos 0, sin 0), thus defining the cosine and sine functions. These functions are defined for every real number 0. Thus the domain of both cosine and sine is the set of real numbers.
As we have already noted, a consequence of (cos 0, sin 0) lying on the unit circle is the equation
(cos 0)2 + (sin 0)2 = 1.
Because (cos 0)2 and (sin 0)2 are both nonnegative, the equation above implies that
(cos 0)2 ^ 1 and (sin 0)2 ^ 1.
Thus cos 0 and sin 0 must both be between ^1 and 1.
Cosine and sine are between ^1 and 1
^1 ^ cos 0 ^ 1 and ^ 1 ^ sin 0 ^ 1
for every angle 0.
These inequalities could also be written in the following form:| cos 0| ^ 1 and | sin 0| ^ 1.
The first coordinates of the points of the unit circle are precisely the values of the cosine function. Every number in the interval [^1, 1] is the first coordinate of some point on the unit circle. Thus we can conclude that the range of the cosine function is the interval [^1, 1]. A similar conclusion holds for the sine function (use second coordinates instead of first coordinates).
In summary, we have the following results.
Domain and range of cosine and sine
• The domain of both cosine and sine is the set of real numbers.
• The range of both cosine and sine is the interval [^1, 1].
Example 5
These inequalities can be used as a crude test of the plausibility of a result. For example, suppose you do a calculation and determine that cos 0 = 2, which is impossible. Thus either there is a mistake in your calculation or solutions corresponding to cos 0 = 2 should be discarded (see the solution to Exercise 33 for an example).
312 Chapter 4 Trigonometric Functions
The Greek mathematician
Hipparchus, depicted here in a
19th-century illustration,
developed trigonometry over 2100
years ago as a tool for calculations
in astronomy.
The word sine comes from the
Latin word sinus, which means
curve.
Because the domain of the cosine and the sine is the set of real numbers, we cannot show the graph of these functions on their entire domain. To understand what the graphs of these functions look like, we start by looking at the graph of cosine on the interval [^67c, 67c].
The graph of cosine on the interval [^67c, 67c].
Let’s begin examining the graph above by noting that the point (0, 1) is on the graph, as expected from the equation cos 0 = 1. Note that the horizontal axis has been called the B-axis.
Moving to the right along the B-axis from the origin, we see that the graph crosses the B-axis at the point (7c2 , 0), as expected from the equation cos 7c2 = 0. Continuing further to the right, we see that the graph hits its lowest value when B = 7c, as expected from the equation cos 7c = ^1. The graph then crosses the B-axis again at the point (37c2 , 0), as expected from the equation cos 37c2 = 0. Then the graph hits its highest value again when B = 27c, as expected from the equation cos(27c) = 1.
The most striking feature of the graph above is its periodic nature—the graph repeats itself. To understand why the graph of cosine exhibits this periodic behavior, consider a radius of the unit circle starting along the positive horizontal axis and moving counterclockwise. As the radius moves, the first coordinate of its endpoint gives the value of the cosine of the corresponding angle. After the radius moves through an angle of 27c radians, it returns to its original position. Then it begins the cycle again, returning to its original position after moving through a total angle of 47c, and so on. Thus we see the periodic behavior of the graph of cosine.
In Section 4.6 we will 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 [^67c, 67c].
The graph of sine on the interval [^67c, 67c].
This graph goes through the origin, as expected because sin 0 = 0. Moving to the right along the B-axis from the origin, we see that the graph hits its highest value when B = 7c 2 , as expected because sin 7c2 = 1. Continuing further to the right, we see that the graph crosses the B-axis at the point (7c, 0), as expected because sin 7c = 0. The graph then hits its lowest value when B = 37c2 , as expected because sin 37c2 =^1. Then the graph crosses the B-axis again at (27c, 0), as expected because sin(27c) = 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 in Section 4.6.
Section 4.3 Cosine and Sine 313
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) (a) (a) (a) (a) (a) (a) (a) (a) (a) cos(37r) (b) (b) (b) (b) (b) (b) (b) (b) (b) (b) sin(37r)
2 cos(^37r ) 44 sin(^37r )
3 2
4 117r
5 cos
6
7
8
9
10
2
117r
sin
157r 157r
cos sin
4 4
cos 27r 27r
3
cos 47r
3
cos 210^
cos 300^
cos 360045^
cos(^360030^)
sin 3
47r
sin 3
sin 210^
sin 300^
sin 360045^
sin(^360030^)
11 Find the smallest number 0 larger than 47r such that cos 0 = 0.
12 Find the smallest number 0 larger than 67r such that
^2
sin 0 = 2 .
13 Find the four smallest positive numbers 0 such that cos 0 = 0.
14 Find the four smallest positive numbers 0 such that sin 0 = 0.
15 Find the four smallest positive numbers 0 such that sin 0 = 1.
16 Find the four smallest positive numbers 0 such that cos 0 = 1.
17 Find the four smallest positive numbers 0 such that cos 0 = ^1.
18 Find the four smallest positive numbers 0 such that sin 0 = ^1.
19 Find the four smallest positive numbers 0 such that sin 0 = 12.
20 Find the four smallest positive numbers 0 such that cos 0 = 12.
21 Suppose 0 G 0 G 7r2 and cos 0 = 25. Evaluate sin 0.
22 Suppose 0 G 0 G 7r2 and sin 0 = 37. Evaluate cos 0.
23 Suppose 7r2 G 0 G 7r and sin 0 = 29. Evaluate cos 0.
24 Suppose 7r2 G 0 G 7r and sin 0 = 38. Evaluate cos 0.
25 Suppose ^ 7r2 G 0 G 0 and cos 0 = 0.1. Evaluate sin 0.
26 Suppose ^ 7r2 G 0 G 0 and cos 0 = 0.3. Evaluate sin 0.
27 Find the smallest number x such that
sin(ex) = 0.
28 Find the smallest number x such that
cos(ex + 1) = 0.
29 Find the smallest positive number x such that
sin(x2 + x + 4) = 0.
30 Find the smallest positive number x such that
cos(x2 + 2x + 6) = 0.
31 Let 0 be the acute angle between the positive horizontal axis and the line with slope 3 through the origin. Evaluate cos 0 and sin 0.
32 Let 0 be the acute angle between the positive horizontal axis and the line with slope 4 through the origin. Evaluate cos 0 and sin 0.
33 Suppose 5(cos 0)2 ^ 7 cos 0 ^ 6 = 0. Evaluate | sin 0|.
34 Suppose 13(sin x)2 ^ 14 sin x ^ 24 = 0. Evaluate | cos x|.
Problems
35 (a) Sketch a radius of the unit circle corresponding to an angle 0 such that cos 0 = 67.
(b) Sketch another radius, different from the one in part (a), also illustrating cos 0 = 67.
36 (a) Sketch a radius of the unit circle corresponding to an angle 0 such that sin 0 = ^0.8.
(b) Sketch another radius, different from the one in part (a), also illustrating sin 0 = ^0.8.
37 Find angles u and v such that cos u = cos v but sin u =6 sin v.
38 Find angles u and v such that sin u = sin v but cos u =6 cos v.
39 Show that ln(cos 0) is the average of ln(1 ^ sin 0) and
ln(1 + sin 0) for every 0 in the interval (^ 7r2 , 7r).
2
40 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 gives an answer of ^0.999999; the other calculator gives 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.
41 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 gives an answer of 0.017452; the other calculator gives 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.
^ 100^ Chapter 4 Trigonometric Functions
42 A good scientific calculator will show that
cos 710 ^ 0.999999998,
where of course the left side means the cosine of 710 radians. Thus cos 710 is remarkably close to 1. Use the approximation 7r ^ 355
113 (which has an error of less than 3 × 10^7) to explain why cos 710 ^ 1.
43 Suppose m is a real number. Let 0 be the acute angle between the positive horizontal axis and the line with slope m through the origin. Evaluate cos 0 and sin 0.
44 Explain why there does not exist a real number x such that 2sin x = 3 7 .
45 Explain why 7rcos x < 4 for every real number x. 46 Explain why 13 < esinx for every number real number x.
47 Explain why the equation
(sin x)2 ^ 4 sin x + 4 = 0
has no solutions.
48 Explain why the equation
(cos x)99 + 4 cos x ^ 6 = 0
has no solutions.
49 Explain why there does not exist a number 0 such that log cos 0 = 0.1.
Worked-Out Solutions to Odd-Numbered 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(37r) (b) sin(37r)
solution Because 37r = 27r + 7r, an angle of 37r radians (as measured counterclockwise from the positive horizontal axis) consists of a complete revolution around the circle (27r radians) followed by another 7r radians (180^), as shown below. The endpoint of the corresponding radius is (^1, 0). Thus cos(37r) = ^1 and sin(37r) = 0.
3 (a) cos 117r (b) sin 117r
4 4
solution Because 114 7r = 27r + 7r2 + 7r4 , an angle of 117r4 radians (as measured counterclockwise from the positive horizontal axis) consists of a complete revolution around the circle (27r radians) followed by another 7r2 radians
(90^), followed by another 7r4 radians (45^), as shown below. Hence the endpoint of the corresponding radius is
(^ ^2^2). Thus cos 117r ^2 ^2
2 , 4 = ^ 2 and sin 117r 4 = 2 .
2
5 (a) cos 27r (b) sin 27r
3 3
solution Because 27r3 = 7r2 + 7r6 , an angle of 27r3 radians (as measured counterclockwise from the positive horizontal axis) consists of 7r2 radians (90^) followed by another
7r6 radians (30^), as shown below. The endpoint of the
corresponding radius is (^ 1 ^3). Thus cos 27r
2, 3 = ^1
2 2
^3
and sin 27r3 = 2 .
7 (a) cos 210^ (b) sin 210^
solution Because 210 = 180 + 30, an angle of 210^ (as measured counterclockwise from the positive horizontal axis) consists of 180^ followed by another 30^, as shown below. The endpoint of the corresponding radius
is (^ 2^3,^1 ). Thus cos 210^ = ^ 2 ^3 and sin 210^ = ^1 2 . 2
Section 4.3 Cosine and Sine 315
9 (a) cos 360045^ (b) sin 360045^
solution Because 360045 = 360 × 1000 + 45, an angle of 360045^ (as measured counterclockwise from the positive horizontal axis) consists of 1000 complete revolutions around the circle followed by another 45^. The endpoint
of the corresponding radius is ~ ^2 ^2 ~. Thus
2 , 2
^2 ^2
cos 360045^ = 2 and sin 360045^ = 2 .
11 Find the smallest number 0 larger than 47r such that cos 0 = 0.
solution Note that
0 = cos 7r2 = cos 37r2 =cos 57r2 = . . .
and that the only numbers whose cosine equals 0 are of
the form (2n+1)7r 2 , where n is an integer. The smallest
number of this form larger than 47r is 97r2 . Thus 97r2 is the smallest number larger than 47r whose cosine equals 0.
13 Find the four smallest positive numbers 0 such that cos 0 = 0.
solution Think of a radius of the unit circle whose endpoint is (1, 0). If this radius moves counterclockwise, forming an angle of 0 radians with the positive horizontal axis, the first coordinate of its endpoint first becomes 0 when 0 equals 7r2 (which equals 90^), then again when 0
equals 37r2 (which equals 270^), then again when 0 equals 57r2 (which equals 360^ + 90^, or 450^), then again when
0 equals 77r 2 (which equals 360^ + 270^, or 630^), and so on. Thus the four smallest positive numbers 0 such that
cos 0 = 0 are 7r2 , 37r2 , 57r2 , and 77r2 .
15 Find the four smallest positive numbers 0 such that sin 0 = 1.
solution Think of a radius of the unit circle whose endpoint is (1, 0). If this radius moves counterclockwise, forming an angle of 0 radians with the positive horizontal axis, then the second coordinate of its endpoint first becomes 1 when 0 equals 7r2 (which equals 90^),
then again when 0 equals 57r2 (which equals 360^ + 90^, or 450^), then again when 0 equals 97r2 (which equals
2 × 360^ + 90^, or 810^), then again when 0 equals 137r 2 (which equals 3 × 360^ + 90^, or 1170^), and so on. Thus the four smallest positive numbers 0 such that sin 0 = 1
7r 57r 97r 2 , and 137r
are 2 , 2 , 2 .
17 Find the four smallest positive numbers 0 such that cos 0 = ^1.
solution Think of a radius of the unit circle whose endpoint is (1, 0). If this radius moves counterclockwise, forming an angle of 0 radians with the positive horizontal axis, the first coordinate of its endpoint first becomes ^1 when 0 equals 7r (which equals 180^), then again when 0 equals 37r (which equals 360^ + 180^, or 540^), then again when 0 equals 57r (which equals 2 × 360^ + 180^, or 900^), then again when 0 equals 77r (which equals
3 × 360^ + 180^, or 1260^), and so on. Thus the four smallest positive numbers 0 such that cos 0 = ^1 are 7r, 37r, 57r, and 77r.
19 Find the four smallest positive numbers 0 such that sin 0 = 12.
solution Think of a radius of the unit circle whose endpoint is (1, 0). If this radius moves counterclockwise, forming an angle of 0 radians with the positive horizontal axis, the second coordinate of its endpoint first becomes 12
when 0 equals 7r6 (which equals 30^), then again when 0 equals 57r6 (which equals 150^), then again when 0 equals 137r 6 (which equals 360^ + 30^, or 390^), then again when
0 equals 177r 6 (which equals 360^ + 150^, or 510^), and so on. Thus the four smallest positive numbers 0 such that
sin 0 = 12 are 67r, 657r, 6137r, and 6177r.
21 Suppose 0 G 0 G 7r2 and cos 0 = 25. Evaluate sin 0.
solution We know that
(cos 0)2 + (sin 0)2 = 1.
Thus
(sin 0)2 = 1 ^ (cos 0)2
(2)2
= 1 ^ 5
21
25.
Because 0 G 0 G 7r2 , we know that sin 0 > 0. Thus taking square roots of both sides of the equation above gives
^
21
sin 0 = 5 .
23 Suppose 7r2 G 0 G 7r and sin 0 = 29. Evaluate cos 0.
solution We know that
(cos 0)2 + (sin 0)2 = 1.
Thus
=
316 Chapter 4 Trigonometric Functions
(cos o)2 = 1 ^ (sin o)2
~ 2 ~2 = 1 ^ 9
77
81.
Because 7r2 G o G 7r, we know that cos o G 0. Thus taking square roots of both sides of the equation above gives
^77
cos o = ^ 9 .
25 Suppose ^ 7r2 G o G 0 and cos o = 0.1. Evaluate sin o.
solution We know that
(cos o)2 + (sin o)2 = 1.
Thus
(sin o)2 = 1 ^ (cos o)2
= 1 ^ (0.1)2
= 0.99.
Because ^ 7r2 G o G 0, we know that sin o G 0. Thus taking square roots of both sides of the equation above gives
^
sin o = ^ 0.99 ^ ^0.995.
27 Find the smallest number x such that
sin(ex) = 0.
solution Note that ex is an increasing function. Because ex is positive for every real number x, and because 7r is the smallest positive number whose sine equals 0, we want to choose x so that ex = 7r. Thus x = ln 7r.
29 Find the smallest positive number x such that
sin(x2 + x + 4) = 0.
solution Note that x2 + x + 4 is an increasing function on the interval [0, ^). If x is positive, then x2 + x + 4 > 4. Because 4 is larger than 7r but less than 27r, the smallest number bigger than 4 whose sine equals 0 is 27r. Thus we want to choose x so that x2 + x + 4 = 27r. In other words, we need to solve the equation
x2 + x + (4 ^ 27r) = 0.
Using the quadratic formula, we see that the solutions to this equation are
^1 ± ^87r ^ 15
.
2
A calculator shows that choosing the plus sign in the equation above gives x ^ 1.0916 and choosing the minus sign gives x ^ ^2.0916. We seek only positive values of x, and thus we choose the plus sign in the equation above, getting x ^ 1.0916.
31 Let o be the acute angle between the positive horizontal axis and the line with slope 3 through the origin. Evaluate cos o and sin o.
solution From the solution to Exercise 5 in Section 4.1,
we see that the endpoint of the relevant radius on the
unit circle has coordinates ( ^10
10 , 3^10 . Thus
10
^10 3^10
cos o = 10 and sin o = 10 .
33 Suppose 5(cos o)2 ^ 7 cos o ^ 6 = 0. Evaluate | sin o|.
solution Let y = cos o. The equation above can be rewritten as
5y2 ^ 7y ^ 6 = 0.
Solving for y (using either the quadratic formula or by factoring the left side of the equation above), we have
y = ^3 5 or y = 2.
However, y = cos o, and there does not exist a real number o such that cos o = 2. Hence y = ^ 35, and hence cos o = ^3 5. Now
(sin o)2 = 1 ^ (cos o)2
=1^259
16
25.
~16
Thus sin o = ± 25 = ± 45. Thus | sin o| = 45.
=
x =
=