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

The general solution for -tan(2x)+2sin(x)=0 is x=2pin,x=pi+2pin,x=(4pin)/3 ,x=(2pi)/3+(4pin)/3

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

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

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

Solution steps

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

Express with sin, cos

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

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

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

Rewrite using trig identities

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

Factor

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

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

Solving each part separately

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

cos (2 x) - cos (x) = 0 : x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}

Combine all the solutions

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

10 tan (a) + 8 = 5 tan (a) + 3

cos (x) csc (x) = - 2 cos (x)

cos (2 x) + cos (4 x) = 0

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

sin^{2} (x) - 3 cos (x) = 3

What is the general solution for -tan(2x)+2sin(x)=0 ?
The general solution for -tan(2x)+2sin(x)=0 is x=2pin,x=pi+2pin,x=(4pin)/3 ,x=(2pi)/3+(4pin)/3