1. **Problem statement:** Given that $\sin \alpha \neq \frac{1}{2}$ and $\cos \beta = \frac{1}{2}$, find the value of $\alpha + \beta$. 2. **Recall the values of cosine:** - $\cos 60^\circ = \frac{1}{2}$ - $\cos 300^\circ = \frac{1}{2}$ (or $-60^\circ$ in negative angle measure) Since cosine is positive in the first and fourth quadrants, $\beta$ could be $60^\circ$ or $300^\circ$. 3. **Check the condition on $\sin \alpha$:** - $\sin \alpha \neq \frac{1}{2}$ means $\alpha$ is not $30^\circ$ or $150^\circ$ (since $\sin 30^\circ = \sin 150^\circ = \frac{1}{2}$). 4. **Find $\alpha + \beta$:** - Since $\beta = 60^\circ$ (choosing the principal value), and $\alpha$ is such that $\sin \alpha \neq \frac{1}{2}$, the sum $\alpha + \beta$ can be $90^\circ$ if $\alpha = 30^\circ$ is excluded. - The only option from the choices that fits is $90^\circ$. **Final answer:** $\boxed{90^\circ}$