Question

Use an Addition or Subtraction Formula to find the exact value of the expression, as Demonstrated.

$\cos \frac{11\pi }{12}$

Short answer

Therefore, the exact value of the expression is$\mathbf{-}\frac{\sqrt{\mathbf{2}}\mathbf{+}\sqrt{\mathbf{6}}}{\mathbf{4}}$.

Step-by-step solution

Step 1. Given information.

The given function is, $\cos \frac{11\pi }{12}$.

Step 2. Determining the value.

Note that $\frac{11\pi }{12}=\frac{8\pi }{12}+\frac{3\pi }{12}=\frac{2\pi }{3}+\frac{\pi }{4}$

Therefore,

$\cos \frac{11\pi }{12}=\cos \left(\frac{2\pi }{3}+\frac{\pi }{4}\right)$

$$\begin{gathered} =\cos \frac{2\pi }{3}\cos \frac{\pi }{4}-\sin \frac{2\pi }{3}\sin \frac{\pi }{4} \\ =-\frac{1}{2}\times \frac{\sqrt{2}}{2}-\frac{\sqrt{3}}{2}\times \frac{\sqrt{2}}{2} \\ =-\frac{\sqrt{2}}{4}-\frac{\sqrt{6}}{4} \\ =\frac{-\sqrt{2}-\sqrt{6}}{4} \\ =-\frac{\sqrt{2}+\sqrt{6}}{4} \end{gathered}$$