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5
Analytic Trigonometry
Using Fundamental Identities
Section 5.1
[] You should know the fundamental trigonometric identities.
(a) Reciprocal Identities
1
1
sin u csc u csc u
sin u
1
1
COS /,/ --
seg/~ = ~
sec/~
cos/A
1
sin u
tanu = cot u cos u
1
cos u
cotu = ~ tan u sin u
(b) Pythagorean Identities
Sin2u + cos2u = 1
1 + tan2 u = sec2 u
1 +cot2u=csczu
(c) Cofunction Identities
sin
-- /,
= cos/~
cos
- u = sin u
tan
- u = cot u
cot
- u = tan u
sec
-- /,~
"-- CSC /g
CSC
-- R
= sec /A
(d) Negative Angle Identifies
csc(-x) = -csc x
sin(-x) = -sin x
sec(-x) = sec x
cos(-x) = cos x
cot(-x) = -cot x
tan(-x) = -tan x
[] You should be able to use these fundamental identities to find function values.
[] You should be able to convert trigonometric expressions to equivalent forms by using the fundamental
identities.
You should be able to check your answers with a graphing utility.
244
245
PART 1: Solutions to Odd-Numbered Exercises and Practice Tests
Solutions to Odd-Numbered Exercises
3. secO=~, sinO=-
==> Oisin
2
Quadrant IV.
tan x -
sinx =m2 = ~/~
1
2
cos 0 tan 0 cot 0 -
1
1
1 42
see 0 - 42 - 2
sin0
-,/~/2 _ _ 1
cos 0 - 42/2/2
1
tan 0
COS X
7
-25
5. tan x = ~-~, sec x - 24 ~ x is in Quadrant III
24
7
cot X = --
24
25
COS X ~
1
CSC X "" -
7. sec 4) = -!, sin 4)
cos 4) = - 1
tan ~b = 0
cot 4) is undefined.
csc 4) is undefined.
25
--
7
2
2
9. sin(-x)=-sinx= 3 ==> sinx=-~
2
sinx=~, tanx=
11. tan 0 = 4, sin 0 < 0 =, 0 is in Quadrant III
secO-- -~/tan20+ 1 = -~
1 ./Tq
5 ==> x is in
Quadrant II.
cos 0 = --~ = 17
1
cot 0 = 4
sin 0 = -~/i - cos2 0 = -
CSC x --
13
-sin x 2
13. sin0=-l, cot0=0 =:> 0=3¢r
2
zO=O
cosO= ~/1-sin
sec 0 is undefined.
tan 0 is undefined.
csc 0 = -1
CSC 0 =
1
1
17
4
18. By looking at the basic graphs of sin x and
"/r-
csc x, we see that as x --> ~ , sin x --> 1 and
csc x --> 1.
4
246
PART l: Solutions to Odd-Numbered Exercises and Practice Tests
17. By looking at the basic ~aphs of tan x and cot x, we see that as x ----> ~- , tan x ----> ~ and cot x ---> 0.
1
19. csc x sin x sin
= ~x sin x = 1. Matches (d)
21. tan2 x - sec2 x -- tan2 x - (tan2 x + 1) = - 1
The expression is matched with (a).
23. sin(-x)- --sin x
tan x
cos(-x) cos x
The expression is matched with (e).
25. cos x csc x -
cos x
. = cot x. Matches (b)
sin x
27. sec4 x - tan4 x = (sec2x + tan2 x)(sec2 x - tan2 x)
= (sec2x + tanEx)(1) = sec2x + tan2x
The expression is matched with (f).
29.
sec2 x -- 1 tan2 x sin2 x
-sin2 x
sin2 x cos2 x
1
-- sec2 x
sin2 x
cos x .
31. cot x sin x = ~ sin x = cos x
sin x
The expression is matched with (e).
33. sin ¢(csc ¢ - sin ¢) = sin csc - sin2
1
= sin 4’ ° ~- sin2
sin ¢
= 1 - sin2
35.
cotx
csc x
cosx/sinx
l/sin x
cos x
sin x
sin x
= COS X
1
= COS2 ¢
37. sec a
1
sin a
(sin a) cot a
tan a cos a
1 (sin
cos a
\ si-~ /
sin(-x)
39. -- COS X
sin x
COS X
= -tan x
=1
~ = COt X
-x cscx=cosx °sinx
sin ¢ 1
1
45. tan csc .... sec
cos¢ sine cos
43,
cosz y
1 - sin y
1 - sinz y
1 - sin y
(1 + sin y)(1 - sin y)
1 - sin y
= 1 +siny
cos~0
cos00sin
cos00sin
+
47.csc
~ +0sin
0 = cot 0 + cot 0
= 2 cot 0
49. 1
sin2 0
1 -- cos 0 -- sin2 0 cos2 0 -- cos 0
= 1 - cos 0
1 - cos 0
1 - cos 0
cos 0(cos 0 - 1) = --COS 0
1 - cos 0
247
cot(-O)
csc 0
PART l: Solutions to Odd-Numbered Exercises and Practice Tests
cos(-O) sin 0 - cos_~_O sin 0 = -cos 0
~sin 0
sin(-O)
53. sin O + cos 0 cot 0 = sin 0 + cos 0 c.os 0
sin 0
cos 0
cos 0 1 + sin 0
55. 1-sinO= 1-sinO" l+sinO
sin2 0 + cos2 0
sin 0
cos 0(1 + sin O)
1 - sin~ 0
1
sin 0
cos 0(1 + sin O)
cos~ 0
= csc 0
1 + sin 0
cos 0
= sec 0 + tan 0
sin 0 cos 0
57. c-~cO + ~sec 0 = sin2 0 + COS2 0 -" 1
1 + 2cos 0 + cos20 + sin2 0
sin 0(1 + cos O)
1 + cos 0 sin 0
+
sin 0
1 + cos 0
2 + 2 cos 0
sin 0(1 + cos O)
_._
-
2(1 + cos O)
sin 0(1 ÷ cos O)
2
- 2 csc 0
sin 0
61. lnlcsc O[ = lnlsi~O[ = In [sin 0[-1 = -ln[sin O[
63. COt2 x -- cot~ x cos2 x = cot: x(1 - cos2 x) = c°s~---~x2sin2 x = COS2 x
sin x
65; sin2 x sec2 x - sin2 x sin2 x(sec2 x - 1)
= sin2 x tan2 x
67. tanax + 2 tan2x + 1 = (tan~x + 1)~
= (sec x)
= see4 x
69. sin’~ x - cos4 x = (sin2 x + cos2 x)(sin2 x - cos2 x)71. (sin x + cos x)2 = sin2x + 2 sin x cos x + cos2x
= (sin2 x + cos: x) + 2 sin x cos x
= (1)(sin~ x - cos~ x)
= l+2sinxcosx
= sin2 x - cos~ x
73. (sec x + 1)(sec x - 1) = sec2 x - 1 = tan~ x
248
75.
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
1
1
1-cosx+ l+cosx
+
77.
+
1 + cos x 1 - cos x (1 + cos x)(1 - cos x)
2
cosx
l+sinx
1 + sin x
cos x
1 -- COS2 X
2
sin2x
= 2 cscz x
cos2 x + (1 + sin x)2
cos x(1 + sin x)
2 + 2 sin x
cos x(1 + sin x)
2(1 + sinx)
cos x(1 + sin x)
2
COS X
= 2 sec x
79.
sin2y_ 1-cos2y
1 - cos y 1 - cos y
81.
3
o secx+tanx
sec x - tan x sec x + tan x
(1 + cos y)(1 - cos y)
1 - cos y
_._
= 1 + cos y
x
0.2
0.4
¸0.6
0.8
1.0
1.2
1.4
Yl
0.1987
0.3894
0.5646
0.7174
0.8415
0.9320
0.9854
Y2
0.1987
0.3894
0.5646
0.7174
0.8415
0.9320
0.9854
1.0
1.2
1.4
1,0
Conjecture: Yl = Y2
cos x
1 + sin x
85. Yx- 1 -sinx’ Y2- COS X
Yl
,.,
0.2
0.4
0.6
1.2230
1.5085
1.8958
2.4650
3.4082
, ,,
5.3319
11.6814
1.2230
1.5085
1.8958
2.4650
3.4082
5.3319
11.6814
12,0
Conjecture: Ya = Y2
o
o
3(sec x + tan x)
sec2 x - tan2 x
3(sec x + tan x)
1
= 3(sec x + tan x)
83° Yl = COS --X , Y2 "- sJnx
~1.5
__
249
8’7, Yl "-
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
COS x cot X + sin x = cSC x
-6.28
!
i
COS x
89. y~ = sec x 1 + sin x tan x
6.28
91. ~/25 - x2 = ~/25 - (5 sin 0)2, x = 5 sin 0
= ~/25 - 25 sin2 0
= ~/25(1 - sin2 O)
= -,/25 cos2 0
= 5 cos 0
93. v/~-9= ~/(3secO)2-9, x=3secO
= ~/9 sec2 0 - 9
= ~/9 (sec2 0 - 1)
= ~/9 tan2 0
= 3 tan 0
95. ~/x2 + 25 = ~/(5 tan 0)2 + 25, x = 5 tan 0
= ~/25 tan2 0 + 25
= ~/25(tan2 0 + 1)
= ~/25 sec2 0
= 5 sec 0
97. sin 0 = ~/1 - cos2 0
2
Let Yl = sin x and Y2 = ~/1 - cos2x, 0 --< X < 2¢r.
Yl = Y2 for 0 < x < ~r, so we have
sin 0 = ~/1 - cosz 0 for 0 < 0 _< ~.
0 ~ 6,28
-2
99. sec 0 = ~/1 + tan20
Let Yl -
1
COS X
4
and Y2 = ~/1 + tan2x, 0 < x < 2~r.
,n" 3"rr
yl = yz for 0 < x < ~ and ~ < x < 2,n’, so we have
"rr 3-a"
sin 0 = ~/1 + tan2 0 for 0 < 0 < ~ and 2 < 0 < 2,rr.
- lnlcot
.101. lnlcos 01 - lnlsin 01 = ~
In isin
Oi - 0I
103. ln(1 + sinx) -lnlsecxl =
+
lnlI1’ sinai = lnlcosx + cosx. sinxI
105. (a) CSC2-132° -- cot2132° ~ 1.8107 - 0.8107 = 1
2,rr
(b) csc2 ~
- cot ~2 2~r
~-- 1.6360 - 0.6360 = 1
6.28
25O
PART 1: Solutions to Odd-Numbered Exercises and Practice Tests
1 cOS x
109. csc x cot x - cos x ..... cos x
sin x sin x
= sin 0 ........
107. cos -
= cos x(csc2 x - 1)
(a) 0=80°
cos(90* - 80°) = sin 80°
0.9848 = 0.9848
= COSX" cot2x
0~) 0 = 0.8
= sin 0.8
0.7174 = 0.7174
11
111. False. 5 cos 0 5 sec 0
113. False. sin 0 csc ~b :/: 1 unless
115. cos 0 sin 0 = + ~/1 - cos20
sin 0
~/1 - cos20
tan0=+
cos 0
cos 0
117. 0 = 341°
1
sinO
1
sec 0 - ’
cos 0
CSC
, 0’= 360° - 341° = 19°
1
-~/1-cos:~
cos 0
1
cot 0 = ~ = +
tan0 -.,/I-cos20
The sign + or - depends on the choice of 0.
35"rr.
ll’n"
121. 0 = ~ is coterminal with ~
119. 0 = -212° is coterminal with 148°
0’= 180° - 148° = 32°
ll~r ~r
O’ = 2"n-- ~
66
y
Period: --=2
~r
2
3
125. f(x)= ~cos(x - ,r) + 3
y
123. f(x) = - 2 tan -~I
l
I
"\’
"\t
"\
~\
’ \-r ,\
I~
I~
I,I,-~T I,
l
I
|
" ’k
y
,,~
Amplitude: ~
I~ I
1,
I
I
I
2~
251
PART 1: Solutions to Odd-Numbered Exercises and Practice Tests
a
127. sin A = - ==~ a = c - sin A = 20 sin 28° ~ 9.39
129. a = ~/c2 - b2 = x/1-2.542 - 6.22 ~ 10.90
c
b 6.2
sin B .... ==~ B = 29.63°
c 12.54
A = 90° - 29.63° = 60.37°
B = 90° -A° = 62°
b
cosA=- ==~ b=c.cosA~- 17.66
c
Section 5.2
Verifying Trigonometric Identities
[] You should know the difference between an expression, a conditional equation, and an identity.
[] You should be able to solve trigonometric identities, using the following techniques.
(a) Work with one side at a time. Do not "cross" the equal sign.
(b) Use algebraic techniques such as combining fractions, factoring expressions, rationalizing denominators,
and squaring binomials.
(c) Use the fundamental identities.
(d) Convert all the terms into sines and cosines.
Solutions to Odd-Numbered Exercises
csc2 x 1 sin x
1
cot x sin2 x cos x sin x ¯ cos x
1. sin tcsct = sin t(s-~nt) = 1
= csc x ¯ sec x
7. tan2 0 + 6 =(sec2/9- 1) + 6
= sec2 O + 5
5. cos2/3 - sin2/3 = (1 - sin2/3) - sin2/3
-- 1 - 2 sin2/3
sin x
9. cos x + sin x tan x = cos x + sin x ¯ ~
COS X
._
cos2 x + sin2 x
COS X
1
COS X
11.
x
Yl
¸Y2
0~1.5
0
0.2
4.835
4.835
0.4
0.6
0.8
1.0
1.2
1.4
2.1785 1.2064 0.6767 0.3469 0.1409 0.0293
2.1785 1.2064 O.6767 0.3469
0.1409 0.0293
,-
1
sec x tan x
-- COS X ¯
COS x
sin x
COS2 x
sin x
1 - sin2 x
sin x
1
sin x
sin x
= CSCX-- sinx