# COMPOUND ANGLE FORMULA EXAMPLES

Compound Angle Formula Examples :

Here we are going to see example problems based on the formula for compound angles.

## Formula for sin (x+y) and sin (x-y)

sin (x + y)  =  sin x cos y + cos x sin y

sin (x - y)  =  sin x cos y - cos x sin y

## Formula for cos (x+y) and cos (x-y)

cos (x + y)  =  cos x cos y - sin x sin y

cos (x - y)  =  cos x cos y + sin x sin y

## Formula for tan (x+y) and tan (x-y)

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

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

## Compound Angle Formula Examples

Question 1 :

If sin A = 3/5 and cos B = 9/41 , 0 < A < π/2, 0 < B < π/2,

find the value of (i) sin(A + B) (ii) cos(A − B).

Solution :

(i) sin(A + B)

Formula for sin (A + B) is sin A cos B + cos A sin B.Now we have to find the values of cos A and sin B.

 sin A = 3/5 cos A  =  √(1 - sin2x)    =  √(1 - (3/5)2)   =  √(1 - (9/25))  =  √(25 - 9)/25   =  √(16/25) cos A  =  4/5 cos B = 9/41 sin B  =  √(1 - cos2B)  =  √(1 - (9/41)2)   =  √(1 - (81/1681)  =  √(1681 - 81)/1681   =  √(1600/1681) sin B  =  40/41

sin(A + B)  =  sin A cos B + cos A sin B

=  (3/5)  (9/41) + (4/15) ⋅ (40/41)

=  (27/205) + (160/205)

=  (27 + 160)/205

=  187/205

Hence the value of sin (A + B) is 187/205.

(ii) cos(A − B)

Formula for cos (A - B) is cos A cos B + sin A sin B.

cos (A - B) = cos A cos B + sin A sin B.

=  (4/5)  (9/41) + (3/5) ⋅ (40/41)

=  (36/205) + (120/205)

=  (36 + 120)/205

=  156/205

Hence the value of cos (A - B) is 156/205.

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