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}

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

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

cos (4 x) = \frac{- 1}{2}

2 cos (2 x) + sin (x) = 1

2 cos^{2} (x) - 3 cos (x) + 1 = 0, 0 \leq x \leq 2 π

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