# Solve for: 6cos(θ)^2sin(θ)-3cos(θ)^2=0

## Expression: $6{\cos\left({θ}\right)}^{2}\sin\left({θ}\right)-3{\cos\left({θ}\right)}^{2}=0$

Use ${\cos\left({t}\right)}^{2}=1-{\sin\left({t}\right)}^{2}$ to expand the expression

$6\left( 1-{\sin\left({θ}\right)}^{2} \right) \times \sin\left({θ}\right)-3{\cos\left({θ}\right)}^{2}=0$

Use ${\cos\left({t}\right)}^{2}=1-{\sin\left({t}\right)}^{2}$ to expand the expression

$6\left( 1-{\sin\left({θ}\right)}^{2} \right) \times \sin\left({θ}\right)-3\left( 1-{\sin\left({θ}\right)}^{2} \right)=0$

Factor out $3\left( 1-{\sin\left({θ}\right)}^{2} \right)$ from the expression

$3\left( 1-{\sin\left({θ}\right)}^{2} \right) \times \left( 2\sin\left({θ}\right)-1 \right)=0$

Divide both sides of the equation by $3$

$\left( 1-{\sin\left({θ}\right)}^{2} \right) \times \left( 2\sin\left({θ}\right)-1 \right)=0$

When the product of factors equals $0$, at least one factor is $0$

$\begin{array} { l }1-{\sin\left({θ}\right)}^{2}=0,\\2\sin\left({θ}\right)-1=0\end{array}$

Solve the equation for $θ$

$\begin{array} { l }\begin{array} { l }θ=\frac{ π }{ 2 }+kπ,& k \in ℤ\end{array},\\2\sin\left({θ}\right)-1=0\end{array}$

Solve the equation for $θ$

$\begin{array} { l }\begin{array} { l }θ=\frac{ π }{ 2 }+kπ,& k \in ℤ\end{array},\\\begin{array} { l }θ=\frac{ π }{ 6 }+2kπ,& k \in ℤ\end{array},\\\begin{array} { l }θ=\frac{ 5π }{ 6 }+2kπ,& k \in ℤ\end{array}\end{array}$

Find the union

$\begin{array} { l }\begin{array} { l }θ=\frac{ π }{ 6 }+\frac{ 2kπ }{ 3 },& k \in ℤ\end{array},\\\begin{array} { l }θ=\frac{ π }{ 2 }+kπ,& k \in ℤ\end{array}\end{array}$

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