Answer
$41.8^\circ$ and $138.2^\circ$
Work Step by Step
Given equation is-
$\sin\theta$ = $\frac{2}{3}$
To find one solution within domain, taking $\sin^{-1}$ of each side-
$\theta$ = $\sin^{-1}\frac{2}{3}$
Using calculator in degree mode-
$\sin^{-1}$ → $2\div3$ = 41.8103148958
i.e. $\theta$ = $\sin^{-1}(\frac{2}{3}) \approx 41.8^\circ$
(Rounded to one decimal place)
Another solution with $\theta$ between $0^\circ$ and $180^\circ$ is obtained by taking the supplement of the angle as this will be the reference angle of desired angle. Therefore-
Second solution with $\theta$ between $0^\circ$ and $180^\circ$
= $180^\circ - 41.8^\circ$ = $138.2^\circ$
So desired angles between $0^\circ$ and $180^\circ$ satisfying the given equation are $41.8^\circ$ and $138.2^\circ$
