We have seen that angles can be measured in degrees or in degrees, minutes, and seconds. A...

We have seen that angles can be measured in degrees or in degrees, minutes, and seconds. Another unit of angle measure that is especially useful in calculus is the radian. To define one radian, consider an angle situated so that its vertex is at the center of a circle. Such an angle is called a central angle. One radian is the measure of the central angle, θ, that cuts off a portion of the distance around the circle that is the same length as the radius of the circle. The figure shows a central angle θ with a measure of 1 radian.

The radian measure of one full revolution is 2π, so 2π = 360°. It follows that an angle with a measure of π radians corresponds to 180°, one-half of a revolution.

a. Use a compass to draw a circle such as the one shown centered on a set of axes. Use a protractor to measure and mark the following angles: 30°, 45°, 60°, 90°, 135°, 180°, 225°, 270°, 315°, and 360°. For each angle marked in degrees, identify the radian measure and express it as a fraction in lowest terms.

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b. Two angles are complementary. One of them has a measure of  radians. Find the measure of the other angle in radians.

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c. One angle measure is twice another. The angles are supplementary. Find the measure of the two angles expressed in radians.