What is the general solution for tan(θ+pi)+2sin(θ+pi)=0 ?

The general solution for tan(θ+pi)+2sin(θ+pi)=0 is θ=2pin,θ=pi+2pin,θ= pi/3+2pin,θ=(5pi)/3+2pin

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tan(θ+pi)+2sin(θ+pi)=0

tan (θ + π) + 2 sin (θ + π) = 0

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

Solution steps

tan (θ + π) + 2 sin (θ + π) = 0

Rewrite using trig identities

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

Simplify

\frac{sin ( θ )}{cos ( θ )} - 2 sin (θ) : \frac{sin ( θ ) - 2 sin ( θ ) cos ( θ )}{cos ( θ )}

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

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

sin (θ) - 2 sin (θ) cos (θ) = 0

Factor

sin (θ) - 2 sin (θ) cos (θ) : - sin (θ) (2 cos (θ) - 1)

- sin (θ) (2 cos (θ) - 1) = 0

Solving each part separately

sin (θ) = 0 or 2 cos (θ) - 1 = 0

sin (θ) = 0 : θ = 2 πn, θ = π + 2 πn

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

Combine all the solutions

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

sin (θ) = \frac{2}{3}, \frac{π}{2} < θ < π, cos (2 θ) = x

tan (3 x) = 2

tan (θ) = 3 - π

sin (x) = \frac{15}{39}

tan (a) = \frac{40}{12.5}

What is the general solution for tan(θ+pi)+2sin(θ+pi)=0 ?
The general solution for tan(θ+pi)+2sin(θ+pi)=0 is θ=2pin,θ=pi+2pin,θ= pi/3+2pin,θ=(5pi)/3+2pin