Download adj ~5

192
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
Solutions to Odd.Numbered Exercises
1.
adj = ~/5z-3z= ~=4
3.
hyp 5
opp 3
sin =0~opp
---- 3
hyp 5
CSC 0-- ~---
cos =0~ adj
=-- 4
hyp 5
sec 0 = ~ -’-
tan 0 = op___p_p = 3
adj 4
cot 0 = adj = _4
opp 3
hyp 5
adj 4
hyp = ~/82 + 15z = 17
sin 0 = op__p_p _ 8
hyp 17
csc 0 = hy___p_p = 17
opp 8
adj ~5
hyp_
17
sec 0 = adj
- 1-~
cos 0 -
hyp 17
tan 0 = op__p_p _ 8
adj 15
adj 15
cot 0 = -opp 8
hyp = ~/i8~ + 12:~ = ~ = 6./~
sin 0 = op___p_p 18
3 3~
hyp = ~~ = "~= 13
cos 0 = adj 12
2
tiyp = ~~ =-~= 13
2.fi-~
tan 0 = op___p.p18
__ 3
adj 12 2
sin 0 = op___p_p
-,.. 1
hyp 3
cosO= adj _2
hyp 3
tanO= opp_ 1
adj 2,~- 4
cot0= adj =2
opp 3
csc 0 = hy.___p_p = 3
opp
sec 0 = hy___p_p 3
adj - 2.,/~ cot 0 - adj _ 2
opp
adj = ,/6z -22 = ,f~ = 4~/~
sin 0
opp
=~- 21
hyp 6 3
¯
cos 0
=~-adj 4,/~ 2~/~
6
3
hyp
2
1 ,/~
tan 0 = op__p_p
adj - 4,f~ - 2~,/~ - 4
hyp 6
csc0-
- -3
opp 2
6
3
sec 0 - hyp
adj - 4,~/~ - 2~/~ ~
cot 0 adj_ 4~/~
opp 2
- 2,/-~
The function values are the same since the triangles are similar and the corresponding sides are proportional.
193
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
opp = ./102 - 82 = 6
sin 0 - opp
6- 3
hyp 10 5
cos 0 - adj 8 -4
hyp 10 5
tan 0
6 3= --opp
adj 8 4
0
8
10 5
csc 0 - hyp _ _
opp 6 3
10 5
sec 0 - hyp _ _
adj 8 4
iadj 8cot
4 0-
- opp 6 3
opp = ~/2.52 -22 = 1.5
sin 0 =-opp----’=-1.5 3
hyp 2.5 5
hyp 2.5 5
-opp 1.5 3
CSC 0 :----
2.5 5
-2 4
sec 0 - hyp _ _
hyp 2.5 5
adj 2 4
tan 0 = opp 1.5 3
adj 2 4
cot 0 - - adj
4
opp 1.5 3
2
The function values are the same since the triangles are similar and the corresponding sides are proportional.
cos 0 - adj
-
11. Given: sin 0 = 5 = op_._p_p
6 hyp
2
2
5 + (adj) = 62
,cos 0 = adj = .!if
tan 0 = op___p_p =
adj ./~1 11
cos0= adj=-1
hyp 4
tan 0 = op___p_p = ~
cot 0 -" adj __ -,/i-i
opp 5
adj
1
adj
6./]-]"
,~ = 11
~
15
cot0= adj _ 1
opp- ~
csc 0 = hy__p_p 4
opp = "~ = 15
csc 0 = hy___p_p 6
opp 5
3 opp
15. Given: tan 0 = 3- -
1 adj
17. Given: cot 0 = _9 = adj
4 opp
32 + 12
hyp
sin 0 =
sin 0 = op__p_p _
hyp ~ 97
cos0= adj _ 1
hyp- ~cot0= adj_ 1
opp 3
cos0= adj_ 9
hyp ~
sec0=hyp- ~
sec 0 = hy_____p_p_ ~
adj 9
adj
csc 0 = hy___p_p = ,/~
opp 3
/
/
sin 0 = op___p_p _ ~
hyp 4
0~
hyp 6
see 0 = --~ = 6
13. Given: sec 0 = 4 = 4 _ hyp
1 adj
(opp)2 + 12 = 42
tan 0 = opp = 4
adj 9
csc 0 = hy___p_p _
opp
4
9.,/~
97
194
19. sin 60° -
45
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
cos 60°=2
2’
(a) tan 60°- sin 60°° _ v/~
cos 60
1
(b) sin 30° = cos 60°
2
(c) cos 30° = sin 60° -
2
°
cos 60
1 ,~
(d) cot 60° sin 60° ,~ - 3
21. cscO= 3, secO-
4
(a) sin 0 = 1_ 1
csc 0 3
’1
2.,/~
(b)
~ cos 0 - sec 0 - 3
sin 0
1/3
(c) tan 0 -
oos 0 -
(d) sec(90° - O) = csc 0 = 3
23. cos a = 4
(a) seca-
1
COS O!
-4
(b) sin2a + cosZa = 1
sin2 a +
sin2 ot = -16
sin a = +~
4
cos a
1
1/4
~
(c) cot a- sin a
±
.,/~/4- +---~’~/1~
15
1
(d) sin(90° a) =cosa=4
25. tan Ocot 0 = tan
O(ta+O)
=1
27. tan a cos a =
sin a] cos a = sin a
cos a/
29. (1 + cos 0)(1 - cos O) = 1 - cos2 0
= (sinzO+cos20)-cos20
= Sin2 0
sinO cosO sin:~O+coszO
31. ~ + cos 0 sin 0
sin 0 cos 0
1
sin 0 cos 0
1 1
sin 0 cos 0
= csc 0 sec 0
1
33. (a) cos 60°= 2
195
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
35. (a) cot~-= cgt 45°= 1
(b) cos 45°= ~- 2
39. (a) sin 25° ~ 0.4226
(b) cos 65° = 0.4226
Note: sin 25° = cos(90° - 25°) = cos 65°
37. (a) cos ~ = cos 30° = -~1
(b) sec60°=~=2 °
cos 60
1
41. (a) sec 42° 12’ = sec 42.2° - cos 42.2° -~ 1.3499
1
(b) csc 4807’ sin(48 + ~)o 1.3432
43. Make sure that your calculator is in radian mode.
1
"rr
(a) cot 16 - tan(,rr/16) ~ 5.0273
(b) tan ~ = 0.1989
45, Make sure that your calculator is in radian mode.
1
(a) csc 1 sin 1 1.1884
1
(b) tan- ~ 0.5463
2
47.
49. (a) secO=2 =* 0=60°- 3
(a) sin0=2 ~ 0 30° 6
(b) cscO=2 ==, O= 30° 6
51. (a) csc0- 3 ~ 0=60°- 3
(b) sin0 2 ~ 0 45° 4
(b) cot0= 1 ~ 0=45° 4
53. (a) sin 0 = 0.8191
(b) cos 0 = 0.0175
55. (a) tan 0 = 1.1920 :=. 0 = 50° ~ 0.873 radian
(b) tan 0 = 0.4663 ~ 0 ~ 25° ~ 0.436 radian
57. tan 30° -
y
105
x
59. cot 60°=38
x
y = 105 ¯ tan 30° = 105--~-
38
0 = 55° = 0.960 radian
0 ~ 89° = 1.553 radians
196
PART I: Solutions to Odd-Numbered Exercises and Practice-Tests
61. sin 50° = y
15
y- 15 . sin 50°~-, 11.4907
6
h
(b) tan 0 = 2- and tan 0 ~
135
63. (a)
135 ¯ 6
(c) --- h = 270 feet
3
~ 132-----~
3
Not drawn to scale
65.
tan 0 = op___p_p
adj
67. (a) ~3½
w
tan 58° = ~
100
(b) sin 0 - opp
hyp
w = 100 tan 58° ~ 160.0 feet
sin 0 =
lO/3- 1
20 6
1
(c) sin 0 = : ~ 0 ~-- 9.59°
(d) o ~ 9.59°
69.
71. tan 3° =xM
15
x = 15 tan 3°
d=5+2x
= 5 + 2(15 tan 3°)
~ 6.57 centimeters
opp
tan 0 _- --=.
adj
h
tan 80° = -75
h = 75 tan 80° -~ 425.3 meters
h
cos 0 = adj
hyp
75
d
COS 80° = --
1
d = 75 ~ ~ °431.9 meters
cos 80
73. x ~ 2.588, y ~ 9.659
y
sin 0 = y--- ~ 0.97
10
lO
csc o = -- = 1.o4
y
cos 0 = -- ~ 0.26
10
x
10
sec 0 =-- = 3.86
tan 0 = y ~ 3.73
cot 0 = x
- ~0 0.27
Y
x
x
(x, y)
197
PART I: Solutions to Odd-Numbered Exercises and Practice Tests
10° - csc2 10° = -1
True, because 1 + cotz 0 = csc~ 0
cot20 = csc2 0- 1
77. cot2
75. sec 30° = csc 60°
True, because sec(90° - 0) = csc 0.
cot2 0 -- CSC~ 0 = --1.
1
20°
0.9397
40°
0.7660
60°
0.5000
80°
0.1736
sin(90°- O) 1
0.9397
0.7660
0.5000
0.1736.
79.I
o
cos 0
It seems that cos 0 = sin(90° - 0) for all 0.
0 and 90° - 0 are called complementary angles.
83. -3x+8y= 16
Intercepts: (0, 2), (-~q, 0)
81. y= -x-9
Intercepts: (0,-9), (-9, O)
Y
y
3,
¯
-2,
-3-4-
87. - 310° 30’ lies in quadrant I
85. 146° lies in quadrant II
Section 4.4 Trigonometric Functions of Any Angle
Know the Definitions of Trigonometric Functions of Any Angle~
If 0 is in standard position, (x, y) a point on the terminal side and r = ~/x2 + y2 4= O, then:
sinO=-Y
r
x
cos 0 = r
cscO= r-, y:i:0
y
r
sec 0 = -, x g: 0
x
cotO=-,y4:oX
tano=-Y, x4:0
x
y
[] You should know the signs of the trigonometric functions in each quadrant.
’n"
3,tr
[] You should know the trigonometric function values of the quadrant angles O, ~-, ,rr, and --~-.
[] You should be able to find reference angles.
[] You should be able to evaluate trigonometric functions of any angle. (Use reference angles.)
[] You should know that the period of sine and cosine is 2,tr.
You should know which trigonometric functions are odd and even.
Even: cos x and sec x
Odd: sin x, tan x, cot x, csc x