612 CHAPTER 6 The Circular Functions and Their Graphs x y (1, 0) P = 3p 2 p 5P 4 P 4 P 4 0 ( , ) √2 2 √2 2 (– , – ) √2 2 √2 2 Figure 17 (b) Recall that tan p 4 = 1, and in quadrant III tan s is positive. tanap + p 4b = tan 5p 4 = 1 Thus, s = 5p 4 . See Figure 17. S Now Try Exercises 65 and 73. CAUTION Remember that a calculator must be in radian mode when used to find a circular function value of a real number. Formulas for Angular and Linear Speed Angular Speed V Linear Speed v V= U t (v in radians per unit time t, u in radians) v = s t v = rU t v =rV This screen supports the result in Example 4(b) with calculator approximations. Linear and Angular Speed There are situations when we need to know how fast a point on a circular disk is moving or how fast the central angle of such a disk is changing. Some examples occur with machinery involving gears or pulleys or the speed of a car around a curved portion of highway. Suppose that point P moves at a constant speed along a circle of radius r and center O. See Figure 18. The measure of how fast the position of P is changing is the linear speed. If v represents linear speed, then speed = distance time , or v = s t , where s is the length of the arc traced by point P at time t. (This formula is just a restatement of r = d t with s as distance, v as rate (speed), and t as time.) Refer to Figure 18. As point P in the figure moves along the circle, ray OP rotates around the origin. Because ray OP is the terminal side of angle POB, the measure of the angle changes as P moves along the circle. The measure of how fast angle POB is changing is its angular speed. Angular speed, symbolized v, is given as V= U t , where Uis in radians. Here u is the measure of angle POB at time t. As with earlier formulas in this chapter, Umust be measured in radians, withVexpressed in radians per unit of time. The length s of the arc intercepted on a circle of radius r by a central angle of measure u radians is s = ru. Using this formula, we can write the formula for linear speed, v = s t , in several useful forms. v = s t Formula for linear speed v = ru t s = ru v = r # u t ab c = a # b c v = rv v = u t x y s r u O B P P moves at a constant speed along the circle. Figure 18
RkJQdWJsaXNoZXIy NjM5ODQ=