# How do you use the angle sum or difference identity to find the exact value of cos((11pi)/12)#?

Nov 25, 2016

$\cos \left(\frac{11 \pi}{12}\right) = - \cos \left(\frac{\pi}{12}\right)$

#### Explanation:

$\cos \left(\frac{11 \pi}{12}\right)$

= $\cos \left(\pi - \frac{\pi}{12}\right)$

Now using identity $\cos \left(A - B\right) = \cos A \cos B + \sin A \sin B$, this becomes

$\cos \pi \cos \left(\frac{\pi}{12}\right) + \sin \pi \sin \left(\frac{\pi}{12}\right)$

= $- 1 \times \cos \left(\frac{\pi}{12}\right) + 0 \times \sin \left(\frac{\pi}{12}\right)$

= $- \cos \left(\frac{\pi}{12}\right)$

Nov 25, 2016

$- \frac{\sqrt{2 + \sqrt{3}}}{2}$

#### Explanation:

Trig unit circle -->
$\cos \left(\frac{11 \pi}{12}\right) = \cos \left(- \frac{\pi}{12} + \pi\right) = - \cos \left(\frac{\pi}{12}\right)$ (1)
Evaluate $\cos \left(\frac{\pi}{12}\right)$ by the trig identity:
$2 {\cos}^{2} a = 1 + \cos 2 a$
$2 {\cos}^{2} \left(\frac{\pi}{12}\right) = 1 + \cos \left(\frac{\pi}{6}\right) = 1 + \frac{\sqrt{3}}{2} = \frac{2 + \sqrt{3}}{2}$
${\cos}^{2} \left(\frac{\pi}{12}\right) = \frac{2 + \sqrt{3}}{4}$
$\cos \left(\frac{\pi}{12}\right) = \pm \frac{\sqrt{2 + \sqrt{3}}}{2}$
Since $\cos \left(\frac{\pi}{12}\right)$ is positive, then only the positive value is accepted.
Finally, from (1):
$\cos \left(\frac{11 \pi}{12}\right) = - \cos \left(\frac{\pi}{12}\right) = - \frac{\sqrt{2 + \sqrt{3}}}{2}$

Check by calculator.
$\cos \left(\frac{11 \pi}{12}\right) = \cos 165 = - 0.97$
$- \frac{\sqrt{2 + \sqrt{3}}}{2} = \frac{\sqrt{3.73}}{2} = \frac{1.93}{2} = 0.97 .$ OK