Question

The number of Solutions in `[0,2pi]` such that `(Sin (2x))^4 = 1/8` is?

Answer 1

There are 4 solutions.

Explanation:

Right now, we have: `sin^4(2x)=1/8`

First, we must remove the `""^4` by taking the fourth root of each side:
`root(4)(sin^4(2x))=sin(2x)`

`sin(2x)=root(4)(1/8)=(root(4)(1))/(root(4)(8))=+-1/1.682=+-0.594530321`

Now, we must take the arcsin of each side:
`arcsin(sin(2x))=arcsin(+-0.594530321)-=2x=sin^(-1)(+-0.594530321)`

Since our coefficient of `x` is 2, our range must be doubled to `[0,4\pi]`

`sin^(-1)(0.594530321)=2x=0.20266198\pi and 2.20266198\pi`
`sin^(-1)(-0.594530321)=2x=1.79733802pi and 3.79733802pi\pi`

Therefore, `x=0.10133099\pi, 0.89866901\pi, 1.89866901\pi and 1.10133099\pi`, so there are 4 solutions.