Trigonometria - exercicios

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1) Determine os valores das funções trigonométricas de um arco x quando:

sen 2  cos 2   1 tg 2  1  sec 2  tg 
sec  

1 cos  sen cot g   tg sen cos 

1 1 cot g 2  1  cos sec 2  cos sec   sen cos 

a) senx  

1 3 e  x  2 . 2 2
2

1  1 sen 2 x  cos 2 x  1      cos 2 x  1  cos 2 x  1   cos x  4  2
1 senx 1 2 1 1 3 3 tgx   2  .  .  cos x 2 3 3 3 3 3 3 2 

3 3  4 2

sec x 

1 1 2 2 3 2 3     . cos x 3 3 3 3 3 2 1  tgx 1  3 3  3 3  3 3 . 3 3  3 3  3 3

cot gx 

cos sec x 

1 1   2 1 senx  2

b) cos x 

 1 e 0 x . 3 2
2 2 2

1 8 2 2 1  sen x  cos x  1  sen x     1  sen 2 x  1   senx  9 9 3  3
2

2 2 senx 2 2 3 tgx   3  . 2 2 1 cos x 3 1 3
sec x  1 1 3 cos x 1 3

cot gx 

1 1 1 2 2   .  tgx 2 2 2 2 2 4
1 1 3 3 2 3 2     . senx 2 2 2 2 2 2 2 4 3

cos sec x 

c) cos sec x   2 e   x 

3 . 2

senx 

1 1 1 2 2   .  cos sec x  2  2 2 2
2 2

 2 2 2 2  sen x  cos x  1     2   cos x  1  cos x  1  4  cos x   
2

2 2  4 2

senx tgx   cos x

2 2 1 2  2 

sec x 

1  cos x1  2 2



2 2



2 2

.

2 2



2 2  2 2

cot gx 

1 1  1 tgx 1

d) tgx  3 e 0  x 


2

.

sec 2 x  1  tg 2 x  sec 2 x  1  cos x  1 1  sec x 2

 3

2

 sec 2 x  1  3  sec x  4  2

1 1 sen x  cos x  1  sen x     1  sen 2 x  1   senx  4 2
2 2 2

2

3 3  4 2

cos sec x 

1 1 3 2 3 2 2    .  senx 3 3 3 33 2

cot gx 

1 1 1 3 3   .  tgx 3 3 3 3

2) Sendo cos x 

 4 e 0  x  , calcule sen 2 x  3senx . 5 2
2

16 4 sen 2 x  cos 2 x  1  sen 2 x     1  sen 2 x  1   senx  25 5 9 36  3  9 9 9  45 Logo : sen 2 x  3senx   3      25  5  25 5 25 25

9 3  25 5

3) Sabe-se que cos a  

5  e  a   , calcule o valor de 1  sena 1  sena  . . 2 52 2 2

  1  sena 1  sena   1  sen a  cos a    5   5  1 . .  5  25 5  
4) Dado cos x 

2  , com 0  x  , determine sec x  cos sec x . 2 2
2 2

 2 2 2  sen x  cos x  1  sen x    2   1  sen x  1  4  senx   
2 2

2 2  4 2

sec x  cos sec x 

1 1 2 2 4 4 2 4 2      .  2 2 cos x senx 2 2 2 2 2 2

5) Se cos a 

 1 cos sec a sena e 0  a  , qual é o valor da expressão y  ? 2 2 sec a  cos a
2

1 1 sen 2 a  cos 2 a  1  sen 2 a     1  sen 2 a  1   sena  4 2

3 3  4 2

2 3 43 1 1   sena cos sec a  sena sena 1 2 1 1 3 3 3 2 2 3 2 3 .  y        . 1 1 4 1 3 sec a  cos a 9 2 3 3 3 3 3 3 3 2  cos a cos a 2 2 2
6) Simplifique: a) y 

sec x  cos sec x 1  cot gx

. . b) y  sec x cos x cos sec x  senx tgx  cot gx 

senx  cos x 1 1  senx senx 1 cos x senx  senx. cos x  senx  cos x .    sec x a) y  senx  cos x cos x senx. cos x senx  cos x senx. cos x cos x 1 senx senx

1 senx cos x  1  1  senx cos x   senx cos x    senx   cos x  y     senx cos x      senx  cos x  cos x senx   cos x senx   senx cos x cos x senx 1  sen 2 x  cos 2 x  sen 2 x cos 2 x   senx cos x   cos 2 x  cos 2 x  sen 2 x cos 2 x   senx cos x  . . y   cos x  senx    cos x  senx    b) senx cos x senx cos x      
2 2  sen 2 x cos 2 x   senx cos x   senx cos x   sen x cos x senx cos .   y     senx cos x .    senx  cos x senx   cos x  senx cos x   cos x senx  2 2 y  sen x cos x  1





x   

7) Determine o valor de A 

1 cot gx  1 , dado cos x  . 2 cos sec x  sec x

cot gx  1  A cos sec x  sec x

cos x cos x  senx 1 senx. cos x cos x  senx senx. cos x 1 senx senx   .   cos x  1 1 cos x  senx senx senx cos x  senx 2  senx cos x senx. cos x
2 2

1 sen x  cos x  1  sen x     1  senx  2
2 2

3 3  4 2...
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