Solve finding all solutions in 0 2 pi 2cos x 2sin x square

Solve, finding all solutions in [0, 2 pi). 2cos x - 2sin x = squareroot 6 x = (Simplify your answer. Type your answer in radians. Type an exact answer, using pi as needed. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as needed.)

Solution

2 cos x - 2 sin x = sqrt 6

2 cos x = sqrt 6 + 2 sin x

squaring both sides

( 2cos x)^2 = ( sqrt 6 + 2 sin x )^2

4 cos^2 x - ( sqrt 6 + 2 sin x )^2 = 0

cos^2 x = 1- sin^2 x

4 ( 1- sin^2 x) - ( sqrt 6 + 2 sin x )^2 = 0

- 2 - 8 sin^2 x - 4 sin x sqrt 6 = 0

let sin x = y

-2 - 8y^2 - 4 sqrt 6 y = 0

on solving for y

we get

y = - ( sqrt 2 + sqrt 6 )/4

y = - ( sqrt 6 - sqrt 2 ) / 4

y = sin x

so sin x = - ( sqrt 2 + sqrt 6 )/4

sin x = - ( sqrt 6 - sqrt 2 ) / 4

solutions are

x = 2pi n + sin^-1 ( 1/4 ( sqrt 2 - sqrt 6)) ,    2pi - .26

x = 2pi n + sin^-1 ( 1/4 ( -sqrt 2 - sqrt 6)) , 2pi - 1.31