Answer
The solution set is $$\{180^\circ+720^\circ n,n\in Z\}$$
Work Step by Step
$$\sin\frac{\theta}{2}=1$$
1) First, we solve the equation over the interval $[0^\circ,360^\circ)$
- For $\sin\frac{\theta}{2}=1$, over the interval $[0^\circ, 360^\circ)$, there is one value of $\theta$ where $\sin\frac{\theta}{2}=1$, which is $90^\circ$.
Therefore, $$\frac{\theta}{2}=\{90^\circ\}$$
(Be careful that the angle we are solving the equation for is $\frac{\theta}{2}$, not $\theta$)
2) Solve the equation for all solutions
Sine function has period $360^\circ$, so we would add $360^\circ$ to all solutions found in part 1) for $\frac{\theta}{2}$.
$$\frac{\theta}{2}=\{90^\circ+360^\circ n,n\in Z\}$$
Finally, we find the solutions for $\theta$, which is also the solution set:
$$\theta=\{180^\circ+720^\circ n,n\in Z\}$$
