2sqrt(3)cos(θ)sin(θ)-cos(θ)=2cos(θ)

Solution

23 cos (θ) sin (θ) - cos (θ) = 2 cos (θ)

θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn, θ = \frac{π}{3} + 2 πn, θ = \frac{2 π}{3} + 2 πn

Solution steps

23 cos (θ) sin (θ) - cos (θ) = 2 cos (θ)

Subtract

2 cos (θ)

from both sides

23 cos (θ) sin (θ) - 3 cos (θ) = 0

Factor

23 cos (θ) sin (θ) - 3 cos (θ) : cos (θ) (23 sin (θ) - 3)

cos (θ) (23 sin (θ) - 3) = 0

Solving each part separately

cos (θ) = 0 or 23 sin (θ) - 3 = 0

cos (θ) = 0 : θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn

23 sin (θ) - 3 = 0 : θ = \frac{π}{3} + 2 πn, θ = \frac{2 π}{3} + 2 πn

Combine all the solutions

θ = \frac{π}{2} + 2 πn, θ = \frac{3 π}{2} + 2 πn, θ = \frac{π}{3} + 2 πn, θ = \frac{2 π}{3} + 2 πn

cos (2 x) = 2

cos (x) - cos (7 x) = sin (4 x)

cos (\frac{θ}{2}) = 1

csc^{2} (θ) + csc (θ) - 2 = 0

2 sin (x) = cos (x)