Fig. 10-1
10.2
Give the equations relating degree measure and radian measure of angles.
I 2-rr radians is the same as 360 degrees. Hence, 1 radian = 180/Tr degrees, and 1 degree = 77/180
radians. So, if an angle has a measure of D degrees and/? radians, then D = (180/7r).R and R = (77/180)D.
10.3
Give the radian measure of angles of 30°, 45°, 60°, 90°, 120°, 135°, 180°, 270°, and 360°.
I We use the formula R = (?r/180)D. Hence 30° = 77/6 radians, 45° = 77/4 radians, 60° = 77/3 radians,
90° = 77/2 radians, 120° = 27T/3 radians, 135° = 377/4 radians, 180° = 77 radians, 270° = 377/2 radians,
360° = 277 radians.
10.4
Give the degree measure of angles of 377/5 radians and 577/6 radians.
I We use the formula D = (180 ITT)R. Thus, 377/5 radians = 108° and 577/6 radians = 150°.
10.5
In a circle of radius 10 inches, what arc length along the circumference is intercepted by a central angle of 77/5
radians?
I The arc length s, the radius r, and the central angle 6 (measured in radians) are related by the equation
s = r6. In this case, r = 10 inches and 0 = 77/5. Hence, 5 = 277 inches.
10.6
If a bug moves a distance of 377 centimeters along a circular arc and if this arc subtends a central angle of 45°, what
is the radius of the circle?
I s = rO. In this case, s = 3ir centimeters and 0 = 77/4 (the radian measure equivalent of 45°). Thus,
377 = r • 77/4. Hence, r = 12 centimeters.
10.7
Draw a picture of the rotation determining an angle of -77/3 radians.
I See Fig. 10-2. 77/3 radians = 60°, and the minus sign indicates that a 60° rotation is to be taken in the
clockwise direction. (Positive angles correspond to counterclockwise rotations.)
62
CHAPTER 10
Trigonometric Functions and
Their Derivatives
10.1
Define radian measure, that is, describe an angle of 1 radian.
Consider a circle with a radius of one unit (Fig. 10-1). Let the center be C, and let CA and CB be two radii
for which the intercepted arc AB of the circle has length 1. Then the central angle /LACE has a measure of one
radian.
10.2
Give the equations relating degree measure and radian measure of angles.
I 2-rr radians is the same as 360 degrees. Hence, 1 radian = 180/Tr degrees, and 1 degree = 77/180
radians. So, if an angle has a measure of D degrees and/? radians, then D = (180/7r).R and R = (77/180)D.
10.3
Give the radian measure of angles of 30°, 45°, 60°, 90°, 120°, 135°, 180°, 270°, and 360°.
I We use the formula R = (?r/180)D. Hence 30° = 77/6 radians, 45° = 77/4 radians, 60° = 77/3 radians,
90° = 77/2 radians, 120° = 27T/3 radians, 135° = 377/4 radians, 180° = 77 radians, 270° = 377/2 radians,
360° = 277 radians.
10.4
Give the degree measure of angles of 377/5 radians and 577/6 radians.
I We use the formula D = (180 ITT)R. Thus, 377/5 radians = 108° and 577/6 radians = 150°.
10.5
In a circle of radius 10 inches, what arc length along the circumference is intercepted by a central angle of 77/5
radians?
I The arc length s, the radius r, and the central angle 6 (measured in radians) are related by the equation
s = r6. In this case, r = 10 inches and 0 = 77/5. Hence, 5 = 277 inches.
10.6
If a bug moves a distance of 377 centimeters along a circular arc and if this arc subtends a central angle of 45°, what
is the radius of the circle?
I s = rO. In this case, s = 3ir centimeters and 0 = 77/4 (the radian measure equivalent of 45°). Thus,
377 = r • 77/4. Hence, r = 12 centimeters.
10.7
Draw a picture of the rotation determining an angle of -77/3 radians.
I See Fig. 10-2. 77/3 radians = 60°, and the minus sign indicates that a 60° rotation is to be taken in the
clockwise direction. (Positive angles correspond to counterclockwise rotations.)
62
CHAPTER 10
Trigonometric Functions and
Their Derivatives
10.1
Define radian measure, that is, describe an angle of 1 radian.
Consider a circle with a radius of one unit (Fig. 10-1). Let the center be C, and let CA and CB be two radii
for which the intercepted arc AB of the circle has length 1. Then the central angle /LACE has a measure of one
radian.