What is the general solution for 8sin(θ)cos(θ)tan(θ)=csc(θ) ?

The general solution for 8sin(θ)cos(θ)tan(θ)=csc(θ) is θ= pi/6+2pin,θ=(5pi)/6+2pin

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8sin(θ)cos(θ)tan(θ)=csc(θ)

8 sin (θ) cos (θ) tan (θ) = csc (θ)

θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn

Solution steps

8 sin (θ) cos (θ) tan (θ) = csc (θ)

Subtract

csc (θ)

from both sides

8 sin (θ) cos (θ) tan (θ) - csc (θ) = 0

Express with sin, cos

\frac{- 1 + 8 sin ^{3} ( θ )}{sin ( θ )} = 0

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

- 1 + 8 sin^{3} (θ) = 0

Solve by substitution

sin (θ) = \frac{1}{2}, sin (θ) = - \frac{1}{4} + i \frac{3}{4}, sin (θ) = - \frac{1}{4} - i \frac{3}{4}

sin (θ) = \frac{1}{2} : θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn

sin (θ) = - \frac{1}{4} + i \frac{3}{4} :

No Solution

sin (θ) = - \frac{1}{4} - i \frac{3}{4} :

No Solution

Combine all the solutions

θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn

\frac{cot ( θ )}{sec ( θ )} = sin (θ)

16 sin (2 x) = 0

cos (2 α) + sin (α) = 0

sin (9 0^{\circ} - x^{\circ}) = \frac{4}{5}

sin (x) = sin (8 0^{\circ})

What is the general solution for 8sin(θ)cos(θ)tan(θ)=csc(θ) ?
The general solution for 8sin(θ)cos(θ)tan(θ)=csc(θ) is θ= pi/6+2pin,θ=(5pi)/6+2pin