Solve 2 sin(x-2) = 3 cos x

Question

2 sin(x-2) = 3 cos x. Thanks in advance!!

Answer 1

The trigonometric equation is 2 sin(x - 2) = 3cos x.

Composite formula : sin(A - B) = sin Acos B - cos Asin B.

2[sin x cos 2 - cos x sin 2] = 3 cos x

[sin x cos 2 - cos x sin 2] / cos x = 3/2

tan x * cos 2 - sin 2 = 1.5

tan x * cos 2 = 1.5 + sin 2

tan x * cos 2 = 1.5 + 0.0345 = 1.5348

tan x = 1.5348/cos 2 = 1.5348/0.9993

tan x = 1.5358

tan x = tan(tan^{- 1} (1.5358))

tan x = tan(56.93⁰) = tan(0.316π)

The genaral solution of tan(θ) = tan(α) is θ = nπ + α, where n is an integer.

⇒ x = nπ + 0.316π .

The solution is x = nπ + 0.316π