[tex]\bf 3sin^2(\theta )+16sin(\theta )-35=0\impliedby
\begin{array}{llll}
\textit{notice this is just a quadratic }\\\\ax^2+bx+c=0
\end{array}
\\\\\\\
[3sin(\theta )-5][sin(\theta )+7]=0\\\\
-------------------------------\\\\
3sin(\theta )-5=0\implies 3sin(\theta )=5\implies sin(\theta )=\cfrac{5}{3}
\\\\\\
\measuredangle \theta =sin^{-1}\left( \frac{5}{3} \right)[/tex]
now, before continuing on the first root, recall that the sine has a range of -1 ⩽ sin(θ) ⩽1, however 5/3 is greater than 1, and therefore is out of range, meaning, there's no such angle, so we've got zilch there.
now, let's check the second root,
[tex]\bf sin(\theta )+7=0\implies sin(\theta )=-7\implies \measuredangle \theta =sin^{-1}(-7)[/tex]
well, hell -7 is smaller than -1, so is also outside the range for sine, and so, the second root gave also zilch, so, in short, there's no such angle.
