How do you solve sin(3x) + sin^2 (x)= 2?

1 Answer
Nghi N.
Oct 17, 2015

Solve sin 3x + sin^2 x = 2

Ans: x = (3pi)/2

Explanation:

sin 3x + sin^2 x = 2
Trig identity --> sin (3x) = 3sin x - 4sin^3 x. Call sin x = t, we get
3t - 4t^3 + t^2 = 2
f(t) = 4t^3 - t^2 - 3t + 2 = 0
Since (a - b + c - d = 0), one factor is (t + 1). By division, we get:
f(t) = (t + 1)(4t^2 - 5t + 2) = 0. Solve this product.

a. t = sin x = - 1 --> x = (3pi)/2
b. (4t^2 - 5t + 2) = 0
D = b^2 - 4ac = 25 - 32 < 0. There are no real roots.
Therefor, there is unique answer: x = (3pi)/2
Check.
x = (3pi)/2 --> sin 3x = sin ((9pi)/2) = sin (pi/2) = 1 --> sin^2 x = 1.
sin 3x + sin^2 x = 1 + 1 = 2. OK

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