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

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

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

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

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

Solution steps

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

Subtract

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

from both sides

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

Simplify

sin (x) + \frac{tan ( x )}{2} : \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{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Combine all the solutions

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

2 sin (3 θ) - 1 = 0, 0 \leq θ \leq 2 π

sin (θ) = - \frac{5}{7}

cos (x) sin (x) = sin (x)

sec^{2} (x) - 1 = 1

sec (θ) = 8

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