Question

if cos x cos (pi/7)+sin x sin(pi/7)=-(sqrt 2)/2,then x can equal

Answer 1

cos x cos (pi/7)+sin x sin(pi/7) this is cos(x - pi/7)
so
cos(x-pi/7) = -(sqrt2)/2
this means
x-pi/7 = 3pi/4 or x-pi/7 = 5pi/4

now solve for x....

Answer 2

cos(x)cos(pi/7) + sin(x)sin(pi/7) = cos( x - pi/7)

so \[\cos(x - \pi/7) = - \sqrt{2}/2\]

the angles are in the 2nd (180 - x) and 3rd (180 + x) quadrants since cos is negative


\[\cos^{-1}(\sqrt{2}/2) = \pi/4\]

then
2rd quadrant

\[(x - \pi/7) = 3\pi/4 \]

\[x = 25\pi/28\]

3rd quadrant

\[(x - \pi/7) = 5\pi/4\]

\[x = 39\pi/28\]