Use trig identity:
cos (x + y) = cos x.cos y - sin x.sin y (1)
First, find cos y knowing sin y = 0.6
cos^2 y = 1 - sin^2 y = 1 - 0.36 = 0.64cos2y=1−sin2y=1−0.36=0.64
cos y = 0.8cosy=0.8 (because y is in Quadrant 1)
Next, find sin x and cos x, knowing tan x = 2.4.
cos^2 x = 1/(1 + tan^2 x) = 1/(1 + 5.76) = 1/6.76 cos2x=11+tan2x=11+5.76=16.76
cos x = 1/2.6cosx=12.6 (because x is in Q. 1)
sin^2 x = 1 - cos^2 x = 1 - 1/6.76 = 5.76/6.76sin2x=1−cos2x=1−16.76=5.766.76
sin x = 2.4/2.6sinx=2.42.6
Replace all numeric values into equation (1), we get:
cos (x + y) = (1/2.6)(0.8) - (2.4/2.6)(0.6) = 0.8/2.6 - 1.44/2.6cos(x+y)=(12.6)(0.8)−(2.42.6)(0.6)=0.82.6−1.442.6
cos (x + y) = - 0.64/2.6 = - 0.246cos(x+y)=−0.642.6=−0.246
Check by calculator.
sin y = 0.6siny=0.6 --> y = 36^@87y=36∘87
cos x = 1/2.6cosx=12.6 --> x = 67^@38x=67∘38
x + y = 36.87 + 67.38 = 104^@25x+y=36.87+67.38=104∘25 --> cos 104.25 = - 0.246. OK
