Question

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

Answer 1

Main formula cos (a - b) =cosacosb + sina sinb
arccos(-x) = pi -arccosx

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

so cos(x-pi/7) = -(sqrt 2)/2,  x -pi/7 = arc cos[-(sqrt 2)/2 ]= pi - arc cos[(sqrt 2)/2 ]

arc cos[(sqrt 2)/2 ] = Pi/4 + 2kpi, and then x -pi/7 = pi - pi/4 +2kpi,
 implies x= -pi/7 + pi - pi/4 = -pi/7 + 3pi/4 = -19pi/28 + 2kpi