Question

What's 2sin(2theta)=1? Need it for webassign.

Answer 1

Solve: $2sin(2x) = 1$

Divide both sides by 2.
$sin(2x) = \frac{1}{2}$

Take the sine inverse of both sides:
$2x = sin^{-1}(\frac{1}{2})$
$2x = \frac{\pi}{6}, \frac{5\pi}{6}, 0 \leq x \leq 2\pi$
$x = \frac{\pi}{12}, \frac{5\pi}{12}$

But we know there are more solutions if we extend the domain. In fact, there are infinitely more solutions since the domain of sine is all real x values.
Thus, we can develop a general solution:

For every $\pi$ units, there is another solution for both $\frac{\pi}{12}$ and $\frac{5\pi}{12}$

General solutions:
$x = \pi n + \frac{\pi}{12}, n \in \mathbb{Z}$
$x = \pi n + \frac{5\pi}{12}, n \in \mathbb{Z}$