Question

If cos X COS + sin xsin -- then x can equal: Check all that apply. 7 2 A. 7T T +2977 7 B. 3+5+2017 + 2T 7 D C. C++2016 +1 2nT 57 OD + + Zot 7

Answer 1

SOLUTION

Given the question in the question tab, the following are the solution steps to solve the problem

Step 1: Write out the given equation

$\cos x\cos (\frac{\pi}{7})+\sin x\sin (\frac{\pi}{7})=-\frac{\sqrt[]{2}}{2}$

Step 2: We use the formula for sum angles for cosine which states that:

$\begin{gathered} CosAcosB+\sin A\sin B=cos(A-B) \\ \text{Letting }A=x\text{ and }B=\frac{\pi}{7} \\ \cos x\cos (\frac{\pi}{7})+\sin x\sin (\frac{\pi}{7})=\cos (x-\frac{\pi}{7})=-\frac{\sqrt[]{2}}{2} \\ (x-\frac{\pi}{7})=\cos ^{-1}(-\frac{\sqrt[]{2}}{2}) \\ x-\frac{\pi}{7}=\frac{3\pi}{4}+2n\pi,\frac{5\pi}{4}+2n\pi \\ x-\frac{\pi}{7}=\frac{3\pi}{4}+2n\pi,x-\frac{\pi}{7}=\frac{5\pi}{4}+2n\pi, \\ x=\frac{3\pi}{4}+\frac{\pi}{7}+2n\pi,x=\frac{5\pi}{4}+\frac{\pi}{7}+2n\pi \end{gathered}$

Hence, the solutions that apply to the given question are Option B and Option D

$x=\frac{3\pi}{4}+\frac{\pi}{7}+2n\pi,x=\frac{5\pi}{4}+\frac{\pi}{7}+2n\pi$