Question

If cos(x)cos(π/7)+sin(x)sin(π/7)= - (√2)/2, then x can equal:

(Check all that apply)
A. (π/4)+(π/7)+2nπ

B. (5π/4)+(π/7)+2nπ

C. (7π/4)+(π/7)+2nπ

D. (3π/4)+(π/7)+2nπ

Answer 1

`\bf \textit{Sum and Difference Identities} \\\\ sin(\alpha + \beta)=sin(\alpha)cos(\beta) + cos(\alpha)sin(\beta) \\\\ sin(\alpha - \beta)=sin(\alpha)cos(\beta)- cos(\alpha)sin(\beta) \\\\ cos(\alpha + \beta)= cos(\alpha)cos(\beta)- sin(\alpha)sin(\beta) \\\\ cos(\alpha - \beta)= cos(\alpha)cos(\beta) + sin(\alpha)sin(\beta) \\\\[-0.35em] ~\dotfill\\\\`

`\bf cos(x)cos\left( \cfrac{\pi }{7} \right)+sin(x)sin\left( \cfrac{\pi }{7} \right)=-\cfrac{√(2)}{2}\implies cos\left( x-\cfrac{\pi }{7} \right)=-\cfrac{√(2)}{2} \\\\\\ x-\cfrac{\pi }{7}=cos^(-1)\left( -\cfrac{√(2)}{2} \right)\implies x-\cfrac{\pi }{7}= \begin{cases} (3\pi )/(4)\\\\ (5\pi )/(4) \end{cases} \\\\[-0.35em] ~\dotfill`

`\bf x-\cfrac{\pi }{7}=\cfrac{3\pi }{4}\implies x=\cfrac{3\pi }{4}+\cfrac{\pi }{7}~\hfill \stackrel{n ~\in~ \mathbb{Z}}{x=\cfrac{3\pi }{4}+\cfrac{\pi }{7}~~~~+2\pi n} \\\\[-0.35em] ~\dotfill\\\\ x-\cfrac{\pi }{7}=\cfrac{5\pi }{4}\implies x=\cfrac{5\pi }{4}+\cfrac{\pi }{7}~\hfill \stackrel{n ~\in~ \mathbb{Z}}{x=\cfrac{5\pi }{4}+\cfrac{\pi }{7}~~~~+2\pi n}`