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185 CHAPTER 20 Trigonometric Functions 2. Cosine. From the reciprocal identity, 1 1 cos t ! sec t ! 25 sin t 3. Sine. From the quotient identity, tan t ! cos t ; hence, 1 2 !" sin t ! tan t cos t ! ("2) 25 25 4. Cotangent. From the reciprocal identity, cot t ! 1 1 1 ! !" tan t "2 2 5. Cosecant. From the reciprocal identity, csc t ! 25 1 1 !" ! sin t 2 "2 / 25 20.24. For an arbitrary value of t express the other trigonometric functions in terms of sint. 1. Cosine. From the Pythagorean identity, cos2 t ! 1 " sin2 t. Hence cos t ! # 21 " sin2 t. sin t 2. Tangent. From the quotient identity, tan t ! cos t . Using the previous result, tan t ! # sin t 21 " sin2 t . 21 " sin2 t cos t . Hence cot t ! # . sin t sin t 3. Cotangent. From the quotient identity, cot t ! 1 4. Secant. From the reciprocal identity, sec t ! cos t . Hence sec t ! 5. Cosecant. From the reciprocal identity, csc t ! 1 # 21 " sin2 t !# 1 21 " sin2 t . 1 . sin t SUPPLEMENTARY PROBLEMS 5 12 20.25. If t is a point on the unit circle with coordinates A "13, "13 B , find the six trigonometric functions of t. Ans. sin t ! "12/13, cos t ! "5/13, tan t ! 12/5, cot t ! 5/12, sect ! "13/5, csct ! "13/12 20.26. If t is a point on the unit circle with coordinates a Ans. 2 25 ," 1 25 b, find the six trigonometric functions of t. sin t ! "1/ 25, cos t ! 2 / 25, tan t ! "1 / 2, cot t ! "2, sec t ! 25 / 2, csc t ! "25 20.27. Find the six trigonometric functions of p. Ans. sin p ! 0, cos p ! "1, tan p ! 0, cot p is undefined, sec p ! "1, csc p is undefined. 20.28. Find the six trigonometric functions of "p /2. Ans. sin("p /2) ! "1, cos("p /2) ! 0, tan("p /2) is undefined, cot("p /2) ! 0, sec("p /2) is undefined, csc("p /2) ! "1 20.29. Find the six trigonometric functions of 7p /4. Ans. sin(7p /4) ! "1/ 22, cos(7p/4) ! 1/ 22, tan(7 p /4) ! "1, cot(7p/4) ! "1, sec(7p /4) ! 22, csc(7p /4) ! " 22 186 CHAPTER 20 Trigonometric Functions 20.30. Prove that for all t, sin(t ! 2pn) " sin t for any integer value of n. 20.31. Prove the Pythagorean identity cot2 t ! 1 " csc2 t. 20.32. Given cos t " 2/5 and t in quadrant I, find the other five trigonometric functions of t. Ans. sin t " 221 / 5, tan t " 221 / 2, cot t " 2 / 221, sec t " 5 / 2, csc t " 5 / 221 20.33. Given tant " #2/3 and t in quadrant IV, find the other five trigonometric functions of t. Ans. sin t " #2 / 213, cos t " 3 / 213, cot t " #3 / 2, sec t " 213 / 3, csc t " # 213 / 2 20.34. Given cott " 25 and t in quadrant III, find the other five trigonometric functions of t. Ans. sin t " #1 / 26, cos t " # 25 / 26, tan t " 1 / 25, sec t " # 26 / 25, csc t " # 26 20.35. Given sec t " #135 and t in quadrant II, find the other five trigonometric functions of t. Ans. sin t " 5 13 12 5 12 , cos t " # , tan t " # , cot t " # , csc t " 13 13 5 12 12 20.36. Given sin t " a and t in quadrant II, find the other five trigonometric functions of t. Ans. cos t " # 21 # a2, tan t " # a 21 # a2 , cot t " # 21 # a2 , sec t " # a 1 21 # a2 1 , csc t " a 20.37. Given cos t " a and t in quadrant IV, find the other five trigonometric functions of t. Ans. sin t " # 21 # a2, tan t " # 21 # a2 , cot t " # a a 21 # a2 1 , sec t " a , csc t " # 1 21 # a2 20.38. Given tan t " a and t in quadrant II, find the other five trigonometric functions of t. Ans. sin t " # a 2a2 ! 1 , cos t " # 1 2a2 ! 1 # 2a2 ! 1 1 , cot t " a , sec t " # 2a2 ! 1, csc t " a 20.39. For an arbitrary value of t, express the other trigonometric functions in terms of tan t. Ans. sin t " $ tan t 21 ! tan2 t , cos t " $ 1 21 ! tan2 t , cot t " 1 , tan t 21 ! tan2 t sec t " $ 21 ! tan2 t, csc t " $ tan t 20.40. For an arbitrary value of t, express the other trigonometric functions in terms of cos t. Ans. sin t " $ 21 # cos2 t, tan t " $ 1 sec t " cos t , csc t " $ 21 # cos2 t , cot t " $ cos t 1 21 # cos2 t 20.41. Show that cosine and secant are even functions. 20.42. Show that sine, tangent, cotangent, and cosecant are odd functions. cos t 21 # cos2 t ,