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