Question

Find the exact value of the expression:

$\cos \left(\frac{7\mathrm{\pi }}{12}\right)$

Short answer

$\frac{1}{4}(\sqrt{2}-\sqrt{6})$

Step-by-step solution

Step 1. Given information 

We have been given an expression $\cos \left(\frac{7\mathrm{\pi }}{12}\right)$.

We have to find its exact value.

Step 2. Simplify the expression

We can also write the expression as:

$$\begin{gathered} \cos \left(\frac{7\mathrm{\pi }}{12}\right)=\cos \left(\left(\frac{3\mathrm{\pi }}{12}\right)+\left(\frac{4\mathrm{\pi }}{12}\right)\right) \\ =\cos \left(\left(\frac{\mathrm{\pi }}{4}\right)+\left(\frac{\mathrm{\pi }}{3}\right)\right) \end{gathered}$$

Step 3.  Apply the compound angle formula

$\cos \left(\left(\frac{\mathrm{\pi }}{4}\right)+\left(\frac{\mathrm{\pi }}{3}\right)\right)$has the form role="math" localid="1646467605260" $\cos \left(\alpha +\beta \right)$

Therefore, apply

$\cos \left(\alpha +\beta \right)=\cos \alpha \cos \beta -\sin \alpha \sin \beta$

role="math" localid="1646467708680" $\begin{array}{r}\cos \left(\frac{\pi }{4}\right)\cos \left(\frac{\pi }{3}\right)-\sin \left(\frac{\pi }{4}\right)\sin \left(\frac{\pi }{3}\right)=\left(\left(\frac{\sqrt{2}}{2}\right)\cdot \left(\frac{1}{2}\right)\right)-\left(\left(\frac{\sqrt{2}}{2}\right)\cdot \left(\frac{\sqrt{3}}{2}\right)\right)\\ =\left(\frac{\sqrt{2}}{4}\right)-\left(\frac{\sqrt{6}}{4}\right)\\ =\left(\frac{\sqrt{2}-\sqrt{6}}{4}\right)\\ =\frac{1}{4}(\sqrt{2}-\sqrt{6})\end{array}$