8cos^2(x)sin(x)+4cos^2(x)=0

Solution

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

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Solution steps

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

Factor

8 cos^{2} (x) sin (x) + 4 cos^{2} (x) : 4 cos^{2} (x) (2 sin (x) + 1)

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

Solving each part separately

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

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

2 sin (x) + 1 = 0 : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

- 6 sin (6 x) = 0

tan (x) = \frac{7}{6}

- 3 arcsin (2 x) = π

3 tan (3 x) = 3

\frac{1}{2} + cos (θ) = 0