Download CHAPTER 20 Trigonometric Functions 185

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
,