19.4 Angles and Arc Lengths 621
1, 0; the line drawn from the origin to 0, 1makes an angle of radians with the positive horizontal axis.
v v v
2
2 u
5
2 u u
—3 4
Figure 19.22
Radians are unitless. When we write sin1.1 and want to think of 1.1 as an angle, then it is an angle in radians. In other words, sin1.1sin1.1 radians).
Converting Degrees to Radians and Radians to Degrees
You may be accustomed to measuring angles in degrees, but radians are much more convenient for calculus. We need a way to go back and forth between the two measures. The fact that one full revolution is 360, or 2 radians, allows us to do this. Equivalently, half a revolution is 180, or radians.
180 radians
1 180 radians
radians 180 degrees
1 radian 180 degrees 57
EXERCISE 19.12 Convert from degrees to radians or radians to degrees.
(a) 45 (b) 30 (c) 3 radians (d) 2 radians
Answers
(a) (b) (c) 270 (d) 360 114.59
Arc Length
Arc lengths are usually described in terms of the angle that subtends the arc and the radius of the circle. We know that on a unit circle, an angle of radians subtends an arc length of units. What about on a circle of radius 3? The circumference of a circle of radius 3 is three times that of the unit circle, so the arc length subtended by an angle of radians on the circle of radius 3 should be 3.
622 CHAPTER 19 Trigonometry: Introducing Periodic Functions
v
3
3x
x 1.5x
1 1.5 3 u
angle of x radians
Figure 19.23
An arc subtended by an angle of # radians on a circle of radius $ will have
arc length $#.
Noticethattheformulaforarclengthissimplewhenthesubtendingangleisgiveninradian measure. The corresponding arc length formula for an angle of # degrees is
$ # 180 180
Trigonometric Functions of Angles
We can think of the input of any trigonometric function as either a directed distance along the unit circle or as an angle, because an angle in standard position will determine a point on the unit circle.
EXAMPLE 19.4 Suppose angle # is in standard position and the point %, ! lies on the terminal side of # but does not lie on the unit circle. Determine the sine and cosine of #.
v (a, b) (a, b
1 u
Figure 19.24
SOLUTION Compute %2 !2 to determine the distance from point %, ! to the origin. LetÕs denote this distance by . The simplest approach is to just scale to a unit circle. If the point %, !
lies on the terminal side of the angle, so does , , and the latter point lies on the unit circle as well.
19.4 Angles and Arc Lengths 623
Therefore,
cos # % and sin # ! , where %2 !2.
Tapping Circle Symmetry for Trigonometric Information
Supposeweknowthat #0.8,0.6,where# [0,2].Thennotonlydoweknow the sine and cosine of #, but we know the trigonometric functions of any angle coterminal with#,thatis,# 2foranyinteger.Weactuallyknowsubstantiallymorethanthis.The cosine and sine of # are both positive, so # must be an acute angle. Given the coordinates of point , we also know the coordinates of points &, , and ’ (one in each of the remaining three quadrants), as shown in the Figure 19.25.
v v
(—.8, .6) Q
P () = (.8, .6)
u
Q ( — ) P
u
(—.8, —.6) R S (.8, —.6) R (+ ) S (2— )
Figure 19.25
Not only do we know the trigonometric functions of # and any angle coterminal to #, but we also know the trigonometric functions of
#
#, and any angle coterminal to any of these.11
2 #,
Use Figure 19.25 to verify the following.
sin # 0.6
sin #0.6 sin #0.6
sin2 #0.6
cos # 0.8
cos #0.8 cos #0.8
cos2 #0.8
EXERCISE 19.13 Usingonlythesymmetryoftheunitcircleandtheinformationcos10.54andsin10.84, approximate all solutions to the following equations.
(a) cos 0.54 (b) cos 0.54
11 Actually we know more than this. By interchanging the and coordinates, we can obtain the trigonometric functions of 2 #, #, 3 #, and 3 #.
624 CHAPTER 19 Trigonometry: Introducing Periodic Functions
(c) sin 0.84 (d) sin 0.84
v
1 P(1)
1 u
Figure 19.26
Answers
(a) 1 2, where is any integer or
2 12, where is any integer.
(b) 12, where is any integer or 12, where is any integer.
(c) 1 2, where is any integer or
12, where is any integer.
(d) 2 12, where is any integer or 12, where is any integer.
P R O B L E M S F O R S E C T I O N 1 9 . 4
1. (a) Convert the following to radians.
(i) 60 (ii) 30 (iii) 45 (iv) 120
(b) Convert 2 radians to degrees.
2. Convert these angles to radian measure.
(a) 60
(d) 40
(b) 45 (c) 270
(e) 120
3. Convert these angles given in radians to degrees.
(a) 3 (b) 3
(d) 3 (e) 5
(c) 5
(f) 3.2 (g) 4
4. A second hand of a clock is 6 inches long.
(a) How far does the pointer of the second hand travel in 20 seconds?
(b) How far does the pointer of the second hand travel when the second hand travels through an angle of 70?
19.4 Angles and Arc Lengths 625
(c) In one hour the minute hand of the clock moves through an angle of 2 radians. In this amount of time, through what angle does the second hand travel? The hour hand? Give your answers in radians.
5. A bicycle gear with radius 4 inches is rotating with a frequency of 50 revolutions per minute. In 2 minutes what distance has been covered by a point on the corresponding chain?
6. A nautical mile is the distance along the surface of the earth subtended by an angle with vertex at the center of the earth and measuring 60 .
(a) The radius of the earth is about 3960 miles. Use this to approximate a nautical mile. Give your answer in feet. (One mile is 5280 feet.)
(b) The Random House Dictionary deÞnes a nautical mile to be 6076 feet. Use this to get a more accurate estimate for the radius of the earth than that given in part (a).
7. The earth travels around the sun in an elliptical orbit, but the ellipse is very close to circular. The earthÕs distance from the sun varies between 147 million kilometers at perihelion (when the earth is closest to the sun) and 153 million kilometers at aphelion (when the earth is farthest from the sun).
Use the following simplifying assumptions to give a rough estimate of how far the earth travels along its orbit each day.
Simplifying assumptions:
Model the earthÕs path around the sun as a circle with radius 150 million km. Assume that the earth completes a trip around the circle every 365 days.
8. A bicycle wheel is 26 inches in diameter. When the brakes are applied the bike wheel makes 2.2 revolutions before coming to a halt. How far has the bike traveled? (Assume the bike does not skid.)
9. A system of gears is set up as drawn.
Q Chain
R
5" 2"
Consider the height of the point &on the large gear. The height is measured as vertical position (in inches) with respect to the line through the center of the gears and is given as a function of time by
5 cos,
where is measured in seconds.
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