Solve for x in sin 3x + cos 2x = 0

Sine and cosine are two basic trigonometric functions and are related to each other. We will use trigonometric function identities to prove that sin 3x + cos 2x = 0.

Answer: Solving for x in sin 3x + cos 2x = 0, we get x = 2nπ - π/2, (2n + 1) π/5 + π/10

Explanation:

Given, sin 3x + cos 2x = 0

Using formula cos θ = sin ( π/2 - θ), we can rewrite the given equation as

 sin 3x + sin ( π/2 - 2x) = 0 --- (1)

applying  sum and product formula of trigonometry (sin A + sin B = 2 sin(A + B) / 2 cos (A - B) / 2)  in  (1)

we get,

2 [sin (3x + π/2 - 2x) /2 cos (3x -  π/2 + 2x) / 2] = 0

solving it,

⇒ 2 sin ( x/2 + π/4 ) cos ( 5x/2 - π/4 ) = 0

From the above equation, we can say that,

 sin ( x/2 + π/4 ) = 0  and cos ( 5x/2 - π/4 ) = 0

The value of sin ( x/2 + π/4 ) = 0 at nπ and the value of cos ( 5x/2 - π/4 ) = 0 at (2n + 1) π/2, where n is any integer.

⇒ x/2 + π/4  = nπ  and  5x/2 - π/4 = (2n + 1) π/2

⇒ x/2 = nπ - π/4  and  5x/2 = (2n + 1) π/2 + π/4 

⇒ x = 2nπ - π/2  and  x = (2n + 1) π/5 + π/10 

Thus the value of x is 2nπ - π/2  and (2n + 1) π/5 + π/10