669 CHAPTER 6 Test Prep Concepts Examples 6.2 The Unit Circle and Circular Functions Circular Functions Start at the point 11, 02 on the unit circle x2 + y2 = 1 and measure off an arc of length s along the circle, moving counterclockwise if s is positive and clockwise if s is negative. Let the endpoint of the arc be at the point 1x, y2. The six circular functions of s are defined as follows. (Assume no denominators are 0.) sins =y cos s =x tans = y x csc s = 1 y sec s = 1 x cot s = x y The Unit Circle 0 0 8 1808 608 908 150° 2108 3008 3608 3158 3308 2P P 1358 1208 2258 2408 2708 458 308 3P 2 5P 3 7P 4 4P 3 5P 4 7P 6 5P 6 3P 4 2P 3 11P 6 P 3 P 2 P 4 P 6 (0, 1) (1, 0) (0, –1) (–1, 0) x y 1 2 1 2 ( , ) √3 2 1 2 ( , – ) √3 2 1 2 (– , – ) √3 2 ( , ) √2 2 √2 2 1 2 ( ,– ) √3 2 ( , ) √3 2 1 2 1 2 (– , ) √3 2 (– , ) √2 2 √2 2 ( , – ) √2 2 √2 2 1 2 (– ,– ) √3 2 (– , – ) √2 2 √2 2 (– , ) √3 2 0 The unit circle x2 + y2 = 1 Use the unit circle to find each value. sin 5p 6 = 1 2 cos 3p 2 = 0 tan p 4 = 22 222 2 = 1 csc 7p 4 = 1 - 22 2 = -22 sec 7p 6 = 1 - 23 2 = - 223 3 cot p 3 = 1 22 3 2 = 23 3 sin0 = 0 cos p 2 = 0 Find the exact value of s in C 0, p 2 D if cos s = 23 2 . In C 0, p 2 D , the arc length s = p 6 is associated with the point Q 23 2 , 1 2R. The first coordinate is cos s = cos p 6 = 23 2 . Thus we have s = p 6. A belt runs a machine pulley of radius 8 in. at 60 revolutions per min. (a) Find the angular speed v in radians per minute. v = 60 revolutions 1min # 2pradians 1 revolution v = 120pradians per min (b) Find the linear speed v in inches per minute. v = rv v = 81120p2 v = 960pin. permin 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
RkJQdWJsaXNoZXIy NjM5ODQ=