Exercicios de concurso publico (1654)

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225
Resolução dos exercícios
b) Mostrar que: tg 60© 2 tg 45© i tg 15©
60 45© © 152 2tg tg 3
(
45
45
15 60 )
1 60
60
© © ©
© ©
© ©
8
2
25 5
1
5tg tg 45
tg
tg
tg
tg
2 35 2
, ã :n3 1 2 32 i 2Como e t o
60 45 15© © ©2 itg tg tg
62. a) cos 105© 5 cos (60© 1 45©) V
V cos 105© 5 cos 60© 8 cos 45© 2 sen 60© 8 sen 45© V
105© 5 8 2 8 VcosV
2
1
2
2
2
3
2
2
105© 5cos
6
V
2
4
2
b) 
( )
© 5
©
5
© 2 ©
Vcossec15
15 60sen
1
sen 45
1
© 5
© 8 © 2 © 8 ©
V
cossec15
60 45 45 60
V
sen cos sen cos
1=
15© 5
8 2 8
VcossecV
2
3
2
2
2
2
2
1
1
© 5cossec15
4
6
4
2 2
6 2
V
2
2 1
1
2
1
6 6
4
=
5 V
_ _
_
i i
i
15©cossec 25V 16
c)  75
( )
© 5
©
5
© 1 ©
V
75 45
cotg
tg
1
tg 30
1
© 5
© ©
© 1 ©
© 1 ©
© 8 ©
75
1 45 30
45 30
45
45 30
V
2 8
2
cotg
tg tg
tg tg
1
tg tg 30
1 tg tg
=
5 V
75© 5 5
1
2
V
3
3
3
3
V
1
2
cotg
1
1
3 3
3 3
75© 5 3V 2cotg 2
d)  105
(60 )
©
© © ©
sec V5 5
1cos105
1
cos 45
1
60
© 5
© 8 © 2 © 8 ©
V
sec105
45 60 45
V
cos cos sen sen
1=
© 5
8 2 8
Vsec105
2
V
2
1
2
2 3
2
2
1
105©sec
6
65 5 2V
2
2
2
1 2
63. cos 165© 5 cos (135© 1 30©) 5
5 cos 135© 8 cos 30© 2 sen 135© 8 sen 30© 5
2
2
2
3
2
2
2
1
4
6 2
5 2 8 2 8 5
2 2
165 135 )
1 135
135 30
© © ©
© ©
© ©
5 1
8
1
5
2
5tg tg ( 30
tg tg 30
tg tg
1 1
1
3
3
2
3
3
3
3
3
3
35
2 2 8
2 1
5
1
2 1
5 2 1
f p
64.  1 1
1
cos cos
cos cos
y y y
y y
5
4
25
16
25
9
1 5 V 1 5 V
V 1 5 V 5
sen2 2
2
2
2 2
e o
: cos y 5 2,
π
πy
2 5
3, ,Como temos
Logo:
2
cosy y
y
5
3
5
4
3
45 5 5 2tg
sen
a)  ( )x y
x y
x y
1
1
3
2
3
4
3
2
3
4
1
9
8
3
2
9
17
3
2
17
6
1 5
2 8
1
5
2 8 2
1 2
5
5
1
2
5
2
5 2
tg
tg tg
tg tg
e
e
o
o
b)  ( )
2
2x y
x y
x y
1
1
3
2
3
4
3
2
3
4
1
9
8
2
9
1
2 18
5
1 8
5
1 8 2
2 2
5
5
2
5 5
tg
tg tg
tg tg
e
e
o
o
65. tg(x 1 y) 5 30 e tg x 5 4
Então:
( )x y
x y
x y
y
y
y y
30
1
30
1 4
4
30 4 30 120
1 5 V
2 8
1
5 V
V
2
1
5 V 1 5 2 V
tg
tg tg
tg tg
tg
tg
tg tg
121 26y y
121
26V 5 V 5tg tg
Portanto:  y
121
265tg