Evaluate \(\cos 75°\), leaving the answer in surd form.
Question 1 Report
Evaluate \(\cos 75°\), leaving the answer in surd form.
Answer Details
We can use the trigonometric identity \(\cos 75^\circ = \cos(45^\circ+30^\circ) = \cos 45^\circ \cos 30^\circ - \sin 45^\circ \sin 30^\circ\). We know that \(\cos 45^\circ = \frac{\sqrt{2}}{2}\) and \(\sin 45^\circ = \frac{\sqrt{2}}{2}\), and we can find \(\cos 30^\circ\) and \(\sin 30^\circ\) using a 30-60-90 triangle or the unit circle. In either case, we get \(\cos 30^\circ = \frac{\sqrt{3}}{2}\) and \(\sin 30^\circ = \frac{1}{2}\). Substituting these values, we have:
\begin{align*}
\cos 75^\circ &= \cos(45^\circ+30^\circ) \\
&= \cos 45^\circ \cos 30^\circ - \sin 45^\circ \sin 30^\circ \\
&= \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} \\
&= \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} \\
&= \frac{\sqrt{6} - \sqrt{2}}{4}
\end{align*}
Therefore, the answer is \(\frac{\sqrt{2}}{4}(\sqrt{3} - 1)\).
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