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

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

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2tan(θ)+sin(θ)cos(θ)=0

2 tan (θ) + sin (θ) cos (θ) = 0

θ = 2 πn, θ = π + 2 πn

Solution steps

2 tan (θ) + sin (θ) cos (θ) = 0

Express with sin, cos

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

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

2 sin (θ) + cos^{2} (θ) sin (θ) = 0

Factor

2 sin (θ) + cos^{2} (θ) sin (θ) : sin (θ) (cos^{2} (θ) + 2)

sin (θ) (cos^{2} (θ) + 2) = 0

Solving each part separately

sin (θ) = 0 or cos^{2} (θ) + 2 = 0

sin (θ) = 0 : θ = 2 πn, θ = π + 2 πn

cos^{2} (θ) + 2 = 0 :

No Solution

Combine all the solutions

θ = 2 πn, θ = π + 2 πn

sec (x) = \frac{25}{7}

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

tan (θ) = \frac{- 3}{3}

csc (2 θ) = \frac{1}{2} (sec (θ)) (cos (θ))

sec^{2} (x) + 2 sec (x) - 8 = 0

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