Find exact solution of 16 sin^-1 (x) = 4pi

Question

I have an answer of {sqrt 2/2, sqrt2/2} but I am not sure if I did the work correct?

 

Answer 1

The trigonometric equation is 16sin^{- 1} [x] = 4π.

Divide each side by 16.

sin^{- 1} [x] = π/4

sin^{- 1} [x] = sin^{- 1} (sin π/4)

sin(sin^-1(x)) = x     for every x in the interval [-1, 1]

sin^-1(sin(x)) = x     for every x in the interval [0, π].

x = sin π/4

⇒ x = 1/√2 = √2/(√2 * √2) = √2 / 2.

The exact solution is  x = √2/2.

You are correct.