Example 11.4.2.
Find \(\cos(75^\circ)\) and \(\sin(75^\circ)\) using the addition formulas and familiar trigonometric function values from Section 9.3.
Solution.
Note that \(75^\circ = 45^\circ + 30^\circ\) and both of these are familiar angles whose values we know exactly from special triangles.
\begin{align*}
\cos(75^\circ)
\amp= \cos 45^\circ \cos 30^\circ - \sin 45^\circ \sin 30^\circ\\
\amp= \frac{\sqrt{2}}{2} \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \frac{1}{2}\\
\amp= \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}\\
\amp= \frac{\sqrt{6}-\sqrt{2}}{4}
\end{align*}
Similarly,
\begin{align*}
\sin(75^\circ) \amp=
\sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ\\
\amp= \frac{\sqrt{2}}{2} \frac{1}{2} + \frac{\sqrt{2}}{2} \frac{\sqrt{3}}{2}\\
\amp= \frac{\sqrt{2}+\sqrt{6}}{4}.
\end{align*}
