Find the exact value of sin 105°. Then give it to three decimal places.

Step 1. Split the angle. $\sin 105^\circ = \sin(60^\circ + 45^\circ)$.

Step 2. The formula. $\sin 60^\circ \cos 45^\circ + \cos 60^\circ \sin 45^\circ = \dfrac{\sqrt{3}}{2} \cdot \dfrac{1}{\sqrt{2}} + \dfrac{1}{2} \cdot \dfrac{1}{\sqrt{2}} = \dfrac{\sqrt{3} + 1}{2\sqrt{2}}$.

Step 3. Tidy and check. Multiply the top and the bottom by $\sqrt{2}$: $\dfrac{\sqrt{6} + \sqrt{2}}{4} \approx 0.966$.