594 CHAPTER 6 The Circular Functions and Their Graphs 6.1 Radian Measure ■ Radian Measure ■ Conversions between Degrees and Radians ■ Arc Length on a Circle ■ Area of a Sector of a Circle x y r r 0 u = 1 radian U Figure 1 Radian Measure We have seen that angles can be measured in degrees. In more theoretical work in mathematics, radian measure of angles is preferred. Radian measure enables us to treat the trigonometric functions as functions with domains of real numbers, rather than angles. Figure 1 shows an angle u in standard position, along with a circle of radius r. The vertex of u is at the center of the circle. Because angle u intercepts an arc on the circle equal in length to the radius of the circle, we say that angle u has a measure of 1 radian. Radian An angle with its vertex at the center of a circle that intercepts an arc on the circle equal in length to the radius of the circle has a measure of 1 radian. It follows that an angle of measure 2 radians intercepts an arc equal in length to twice the radius of the circle, an angle of measure 1 2 radian intercepts an arc equal in length to half the radius of the circle, and so on. In general, if U is a central angle of a circle of radius r, and U intercepts an arc of length s, then the radian measure of U is s r . See Figure 2. x y 2r r 0 u = 2 radians U Figure 2 x y r r 0 u = radian U 1 2 1 2 x y C = 2Pr r 0 u = 2p radians U Conversions between Degrees and Radians The circumference of a circle — the distance around the circle — is given by C = 2pr, where r is the radius of the circle. The formula C = 2pr shows that the radius can be measured off 2p times around a circle. Therefore, an angle of 360°, which corresponds to a complete circle, intercepts an arc equal in length to 2p times the radius of the circle. Thus, an angle of 360° has a measure of 2p radians. 360° =2P radians An angle of 180° is half the size of an angle of 360°, so an angle of 180° has half the radian measure of an angle of 360°. 180° = 1 2 1 2P2 radians =P radians Degree / radian relationship The ratio s r is a pure number, where s and r are expressed in the same units. Thus, “radians” is not a unit of measure like feet or centimeters.
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