The real number $\theta$ is an angle measure in degrees that satisfies $0^{\circ}<\theta<360^{\circ}$ and

$$
2022^{2 \sin ^{2} \theta-3 \sin \theta+1}=1
$$

The sum of the possible values of $\theta$ is $k^{\circ}$. What is the value of $k$ ?
If $2022^{x}=1$, then $x=0$. This means we have $2 \sin ^{2} \theta-3 \sin \theta+1=0$.

The expression on the left can be factored to get $(\sin \theta-1)(2 \sin \theta-1)=0$.

Therefore, the given equation is true exactly when $\sin \theta=1$ or $\sin \theta=\frac{1}{2}$.

Since $0^{\circ}<\theta<360^{\circ}$, the only value of $\theta$ with $\sin \theta=1$ is $\theta=90^{\circ}$.

The only values of $\theta$ with $\sin \theta=\frac{1}{2}$ are $\theta=30^{\circ}$ and $\theta=150^{\circ}$.

Therefore, the answer is $90^{\circ}+30^{\circ}+150^{\circ}=270^{\circ}$.

ANSWER: $270^{\circ}$