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

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

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tan(x)=-2sin(x)

tan (x) = - 2 sin (x)

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

Solution steps

tan (x) = - 2 sin (x)

Subtract

- 2 sin (x)

from both sides

tan (x) + 2 sin (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 cos (θ) = 1

cos (x) = sin (4 x)

0 = 2 + 4 sin (x)

sin (x) = \frac{1}{5}

cos (x) + cos (x) tan (x) = 0

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