sin(8x)+sin(4x)-0.5sin(6x)=0

Solution

sin (8 x) + sin (4 x) - 0.5 sin (6 x) = 0

x = \frac{πn}{3}, x = \frac{π}{6} + \frac{πn}{3}, x = \frac{1.31811 \dots}{2} + πn, x = π - \frac{1.31811 \dots}{2} + πn

Solution steps

sin (8 x) + sin (4 x) - 0.5 sin (6 x) = 0

Rewrite using trig identities

- 0.5 sin (6 x) + 2 cos (2 x) sin (6 x) = 0

Factor

- 0.5 sin (6 x) + 2 cos (2 x) sin (6 x) : sin (6 x) (2 cos (2 x) - 0.5)

sin (6 x) (2 cos (2 x) - 0.5) = 0

Solving each part separately

sin (6 x) = 0 or 2 cos (2 x) - 0.5 = 0

sin (6 x) = 0 : x = \frac{πn}{3}, x = \frac{π}{6} + \frac{πn}{3}

2 cos (2 x) - 0.5 = 0 : x = \frac{arccos ( 0.25 )}{2} + πn, x = π - \frac{arccos ( 0.25 )}{2} + πn

Combine all the solutions

x = \frac{πn}{3}, x = \frac{π}{6} + \frac{πn}{3}, x = \frac{arccos ( 0.25 )}{2} + πn, x = π - \frac{arccos ( 0.25 )}{2} + πn

Show solutions in decimal form

x = \frac{πn}{3}, x = \frac{π}{6} + \frac{πn}{3}, x = \frac{1.31811 \dots}{2} + πn, x = π - \frac{1.31811 \dots}{2} + πn

4 sin^{2} (θ) + 4 cos (θ) = 0

cos^{2} (x) = \frac{( 8 k - 9 )}{9}

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

4 sin^{4} (x) + 3 sin^{2} (x) - 1 = 0

3 tan (θ) - 3 = 0