What is the general solution for sin(x)= 1/2 tan(x) ?

The general solution for sin(x)= 1/2 tan(x) is x=2pin,x=pi+2pin,x= pi/3+2pin,x=(5pi)/3+2pin

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sin(x)= 1/2 tan(x)

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

x = 2 πn, x = π + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Solution steps

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

Subtract

\frac{1}{2} tan (x)

from both sides

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

Simplify

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

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

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

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

Express with sin, cos

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

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

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

Factor

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

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

Solving each part separately

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

2 cos (x) - 1 = 0 : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

8 (1 - cos (θ)) = sin^{2} (θ)

x, cos (x) = sin (x)

cos (2 θ) - cos (θ) = - 1

- 6 cos (θ) + 4 = 4

4 cos (x) = - sin^{2} (x) + 4

What is the general solution for sin(x)= 1/2 tan(x) ?
The general solution for sin(x)= 1/2 tan(x) is x=2pin,x=pi+2pin,x= pi/3+2pin,x=(5pi)/3+2pin