4sin(x)cos(x)+2sin(x)-2cos(x)=1

Solution

4 sin (x) cos (x) + 2 sin (x) - 2 cos (x) = 1

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Solution steps

4 sin (x) cos (x) + 2 sin (x) - 2 cos (x) = 1

Subtract

1

from both sides

4 sin (x) cos (x) + 2 sin (x) - 2 cos (x) - 1 = 0

Factor

4 sin (x) cos (x) + 2 sin (x) - 2 cos (x) - 1 : (2 cos (x) + 1) (2 sin (x) - 1)

(2 cos (x) + 1) (2 sin (x) - 1) = 0

Solving each part separately

2 cos (x) + 1 = 0 or 2 sin (x) - 1 = 0

2 cos (x) + 1 = 0 : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

2 sin (x) - 1 = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Combine all the solutions

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

sin (x) cos^{2} (x) = sin (x)

3 sin (x) cos (x) = cos^{2} (x)

2 cos^{2} (4 x) + 1 = 0

1 + cos (2 θ) = - 3 sin (θ)

2 cos (θ) + sin (2 θ) = 0