386 CHAPTER 6 Trigonometric Functions Figure 12 Terminal side Initial side (a) r r r 1 radian Terminal side Initial side 1 3 (b) 1 3 1 radian 3 Find the Length of an Arc of a Circle Now consider a circle of radius r and two central angles, θ and ,1θ measured in radians. Suppose that these central angles subtend arcs of lengths s and s ,1 respectively, as shown in Figure 13. From geometry, the ratio of the measures of the angles equals the ratio of the corresponding lengths of the arcs subtended by these angles; that is, s s 1 1 θ θ = (2) Suppose that 1 1θ = radian. Refer again to Figure 12(a). The length s1 of the arc subtended by the central angle 1 1θ = radian equals the radius r of the circle. Then s r, 1 = so equation (2) reduces to s r s r 1 or θ θ = = (3) Figure 13 θ θ = s s 1 1 s u u1 r s1 THEOREM Arc Length For a circle of radius r, a central angle of θ radians subtends an arc whose length s is s rθ = (4) In many applications, such as describing the exact location of a star or the precise position of a ship at sea, angles measured in degrees, minutes, and even seconds are used. For calculation purposes, these are transformed to decimal form. In other applications, especially those in calculus, angles are measured using radians . Radian Measure A central angle is a positive angle whose vertex is at the center of a circle. The rays of a central angle subtend (intersect) an arc on the circle. If the radius of the circle is r and the length of the arc subtended by the central angle is also r, then the measure of the angle is 1 radian . See Figure 12(a). For a circle of radius 1, the rays of a central angle with measure 1 radian subtend an arc of length 1. See Figure 12(b). For a circle of radius 3, the rays of a central angle with measure 1 radian subtend an arc of length 3.
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