Question

Calculate \(\sin 75^{\circ}\) without the use of a calculator.

Answer 1

\(\sin 75^{\circ}=\sin \left(45^{\circ}+30^{\circ}\right) \quad\) (write \(75^{\circ}\) as the sum of two special angles)

\(\therefore \sin 75^{\circ}=\sin 45^{\circ} \cdot \cos 30^{\circ}+\cos 45^{\circ} \cdot \sin 30^{\circ}\)

\(\therefore \sin 75^{\circ}=\left(\frac{\sqrt{2}}{2}\right) \cdot\left(\frac{\sqrt{3}}{2}\right)+\left(\frac{\sqrt{2}}{2}\right) \cdot\left(\frac{1}{2}\right)=\frac{\sqrt{6}}{4}+\frac{\sqrt{2}}{4}=\frac{\sqrt{6}+\sqrt{2}}{4}\)