What is the general solution for -2/3 sin(x)+1/3 sin(2x)=0 ?

The general solution for -2/3 sin(x)+1/3 sin(2x)=0 is x=2pin,x=pi+2pin

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

- \frac{2}{3} sin (x) + \frac{1}{3} sin (2 x) = 0

x = 2 πn, x = π + 2 πn

Solution steps

- \frac{2}{3} sin (x) + \frac{1}{3} sin (2 x) = 0

Simplify

- \frac{2}{3} sin (x) + \frac{1}{3} sin (2 x) : \frac{- 2 sin ( x ) + sin ( 2 x )}{3}

\frac{- 2 sin ( x ) + sin ( 2 x )}{3} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

Rewrite using trig identities

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

Factor

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

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

Solving each part separately

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

cos (x) - 1 = 0 : x = 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn

cos (θ) = 0.84

cos (θ) = 0.88

cos (θ) = 0.76

cos (θ) = 0.25

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

What is the general solution for -2/3 sin(x)+1/3 sin(2x)=0 ?
The general solution for -2/3 sin(x)+1/3 sin(2x)=0 is x=2pin,x=pi+2pin