Question

find all values of theta between 0⁰ and 360⁰ which satisfy the equation 6sin²theta + Sin theta - 1 =0​

Answer 1

`\underset{ax^2+bx+c=0}{\stackrel{\textit{notice this is a quadratic}}{6sin^2(\theta )+sin(\theta )-1=0}}\implies [3sin(\theta )-1][2sin(\theta )+1]=0 \\\\[-0.35em] ~\dotfill\\\\ 3sin(\theta )-1=0\implies 3sin(\theta )=1\implies sin(\theta )=\cfrac{1}{3} \\\\\\ \theta =sin^(-1)\left( \cfrac{1}{3}\right)\implies \theta \approx \begin{cases} \stackrel{\textit{I Quadrant}}{19.47^o}\\\\ \stackrel{\textit{II Quadrant}}{160.53^o} \end{cases} \\\\[-0.35em] ~\dotfill`

`2sin(\theta )+1=0\implies 2sin(\theta )=-1\implies sin(\theta )=\cfrac{-1}{2} \\\\\\ \theta =sin^(-1)\left( \cfrac{-1}{2} \right)\implies \theta = \begin{cases} \stackrel{\textit{III Quadrant}}{210^o}\\\\ \stackrel{\textit{IV Quadrant}}{330^o} \end{cases}`