use identities to find the exact value. cos 165° a. \frac{ sqrt{6} - sqrt{2} }{4} b. \frac{ sqrt{6} +…

use identities to find the exact value. cos 165° a. \frac{ sqrt{6} - sqrt{2} }{4} b. \frac{ sqrt{6} + sqrt{2} }{4} c. \frac{ - sqrt{6} - sqrt{2} }{4} d. \frac{ sqrt{2} - sqrt{6} }{4}

use identities to find the exact value. cos 165° a. \frac{ sqrt{6} - sqrt{2} }{4} b. \frac{ sqrt{6} + sqrt{2} }{4} c. \frac{ - sqrt{6} - sqrt{2} }{4} d. \frac{ sqrt{2} - sqrt{6} }{4}

Answer

Explanation:

Step1: Express (165^{\circ}) as a sum of known angles

(165^{\circ}=120^{\circ} + 45^{\circ})

Step2: Use the cosine - of - a - sum identity (\cos(A + B)=\cos A\cos B-\sin A\sin B)

Here (A = 120^{\circ}) and (B=45^{\circ}) We know that (\cos120^{\circ}=-\frac{1}{2}), (\sin120^{\circ}=\frac{\sqrt{3}}{2}), (\cos45^{\circ}=\frac{\sqrt{2}}{2}), (\sin45^{\circ}=\frac{\sqrt{2}}{2}) [ \begin{align*} \cos165^{\circ}&=\cos(120^{\circ}+45^{\circ})\ &=\cos120^{\circ}\cos45^{\circ}-\sin120^{\circ}\sin45^{\circ}\ &=\left(-\frac{1}{2}\right)\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} \end{align*} ]

Answer:

C. (\frac{-\sqrt{6}-\sqrt{2}}{4})