cos(5x)-cos(x)=sin(3x)

Solution

cos (5 x) - cos (x) = sin (3 x)

x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}, x = \frac{7 π}{12} + πn, x = \frac{11 π}{12} + πn

Solution steps

cos (5 x) - cos (x) = sin (3 x)

Subtract

sin (3 x)

from both sides

cos (5 x) - cos (x) - sin (3 x) = 0

Rewrite using trig identities

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

Factor

- sin (3 x) - 2 sin (2 x) sin (3 x) : - sin (3 x) (2 sin (2 x) + 1)

- sin (3 x) (2 sin (2 x) + 1) = 0

Solving each part separately

sin (3 x) = 0 or 2 sin (2 x) + 1 = 0

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

2 sin (2 x) + 1 = 0 : x = \frac{7 π}{12} + πn, x = \frac{11 π}{12} + πn

Combine all the solutions

x = \frac{2 πn}{3}, x = \frac{π}{3} + \frac{2 πn}{3}, x = \frac{7 π}{12} + πn, x = \frac{11 π}{12} + πn

1 = cos (\frac{( π ( t - 22 ) )}{183})

cot (x) = - csc^{2} (x)

3 cos^{2} (θ) + 4 cos (θ) - 4 = 0

sin (x) = - cos^{2} (x) - 1

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