1. **Problem:** Find $\sin 30^\circ + \cos 45^\circ$ without using a calculator. 2. **Recall the exact values:** - $\sin 30^\circ = \frac{1}{2}$ - $\cos 45^\circ = \frac{\sqrt{2}}{2}$ 3. **Add the values:** $$\sin 30^\circ + \cos 45^\circ = \frac{1}{2} + \frac{\sqrt{2}}{2} = \frac{1 + \sqrt{2}}{2}$$ 4. **Compare with options:** - a) $1 + \frac{\sqrt{3}}{2}$ - b) $1 + \frac{1}{2}$ - c) $\frac{\sqrt{3} - \sqrt{2}}{2}$ - d) $\frac{\sqrt{3}}{2} + \frac{1}{2}$ Our answer $\frac{1 + \sqrt{2}}{2}$ matches none exactly, but closest is none. So the correct simplified form is $\frac{1 + \sqrt{2}}{2}$. **Final answer:** $\boxed{\frac{1 + \sqrt{2}}{2}}$