sin(3x)+sin(x)=cos(x)

Solution

sin (3 x) + sin (x) = cos (x)

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

Solution steps

sin (3 x) + sin (x) = cos (x)

Subtract

cos (x)

from both sides

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

Rewrite using trig identities

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

Factor

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

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

Solving each part separately

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

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

2 sin (2 x) - 1 = 0 : x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{π}{12} + πn, x = \frac{5 π}{12} + πn

sin^{2} (θ) - 2 = sin (θ), - π \leq x \leq 2 π

sin (2 x) = cos (\frac{x}{2})

5 - 7 sin (x) - 2 cos^{2} (x) = 0

2 + cos (θ) = 0

\frac{1 + sin ( x )}{sin ( x )} = 1 + cos (x)