sin(3x)-sin(x)=cos(2x)

Solution

sin (3 x) - sin (x) = cos (2 x)

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

Solution steps

sin (3 x) - sin (x) = cos (2 x)

Subtract

cos (2 x)

from both sides

sin (3 x) - sin (x) - cos (2 x) = 0

Rewrite using trig identities

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

Factor

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

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

2 tan^{2} (x) + 3 sec (x) = 0

tan (x) = - \frac{3}{4}

tan (x) = - \frac{3}{2}

cos (2 θ) + 4 sin^{2} (θ) = 2

- 4 cos (4 x) = 0