What is the general solution for 5sin^2(x)=9tan(x) ?

The general solution for 5sin^2(x)=9tan(x) is x=2pin,x=pi+2pin

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5sin^2(x)=9tan(x)

5 sin^{2} (x) = 9 tan (x)

x = 2 πn, x = π + 2 πn

Solution steps

5 sin^{2} (x) = 9 tan (x)

Subtract

9 tan (x)

from both sides

5 sin^{2} (x) - 9 tan (x) = 0

Express with sin, cos

\frac{- 9 sin ( x ) + 5 cos ( x ) sin ^{2} ( x )}{cos ( x )} = 0

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

- 9 sin (x) + 5 cos (x) sin^{2} (x) = 0

Factor

- 9 sin (x) + 5 cos (x) sin^{2} (x) : sin (x) (- 9 + 5 sin (x) cos (x))

sin (x) (- 9 + 5 sin (x) cos (x)) = 0

Solving each part separately

sin (x) = 0 or - 9 + 5 sin (x) cos (x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

- 9 + 5 sin (x) cos (x) = 0 :

No Solution

Combine all the solutions

x = 2 πn, x = π + 2 πn

cos (θ) = \frac{5}{4}

cos (x) = sin (2 x), 0 \leq x \leq 2 π

tan (θ) (sin (θ) - 1) = 0

sin (θ) - 4 = - 5

sin (θ) - 4 = - 3

What is the general solution for 5sin^2(x)=9tan(x) ?
The general solution for 5sin^2(x)=9tan(x) is x=2pin,x=pi+2pin