Question

Suppose sin x = 5/7 
cos x > 0, sin y = − 1/5 
and cos y < 0. Then 

cos x = 
cos y = 

Find each of the following quantities: 

sin(x + y) = 

cos(x + y) = 

tan(x + y) = 

Answer 1

The values of sin x = 5 / 7 and sin y = -1 /  5.

Sin x = 5/7 =opposite /hypotenuse

hypotenuse² = opposite side²  + adjacent side ²

49 = 25 + adjacent side²= > adjacent side ² = 49 - 25 = 16

adjacent side = 4.

cos x = adjacent side /hypotenuse = 4 / 7.

Tan x = sinx /cos x= (5 / 7) / (4/7)= 5 / 4.

sin y = -1 / 5 = opposite side /hypotenuse

5 ² = (-1)² + adjacent side²

adjacent side ² = 25-1 = 24 =>adjacent side =  = >cos y =  / 5.

tan y = sin y/ cos y = (-1/ 5)/( /5) = -1 / .

sin(x +y) = sin x cos y +cosx sin y= (5 / 7) ( /5 )+  (4 / 7)( -1 / 5) = / 7 - 4 / 5=

cos (x + y)= cos x cos y -sin x siny = (4 /7)( / 5)  -  (5 / 7) (- 1 /  5) =  

tan (x +y ) = tan x +tan y/(1 -tan x tan y)

 = (5 /4) - 1 /  / (1 - (5 /4)(1 / )

tan (x +y)= (5 - 4) /  (4 + 5) .