Question

If $\sqrt{3}\tan \mathrm{\theta }=3\sin \mathrm{\theta }$, find the value of sin θ.

Answer 1

$$\begin{gathered} \mathrm{Given}:\hspace{0.17em}\sqrt{3}\tan \theta =3\sin \theta \\ \sqrt{3}\tan \theta =3\sin \theta \\ \Rightarrow \sqrt{3}\frac{\sin \theta }{\cos \theta }=3\sin \theta \\ \Rightarrow \frac{\sqrt{3}\sin \theta }{\cos \theta }-3\sin \theta =0 \\ \Rightarrow \frac{\sqrt{3}\sin \theta -3\sin \theta \cos \theta }{\cos \theta }=0 \\ \Rightarrow \sqrt{3}\sin \theta -3\sin \theta \cos \theta =0 \\ \Rightarrow \sqrt{3}\sin \theta \left(1-\sqrt{3}\cos \theta \right)=0 \\ \Rightarrow \sin \theta =0\mathrm{or}1-\sqrt{3}\cos \theta =0 \\ \Rightarrow \sin \theta =0\mathrm{or}\cos \theta =\frac{1}{\sqrt{3}} \\ \Rightarrow \sin \theta =0\mathrm{or}{\cos }^2\theta =\frac{1}{3} \\ \Rightarrow \sin \theta =0\mathrm{or}1-{\cos }^2\theta =1-\frac{1}{3} \\ \Rightarrow \sin \theta =0\mathrm{or}{\sin }^2\theta =\frac{2}{3} \\ \Rightarrow \sin \theta =0\mathrm{or}\sin \theta =\sqrt{\frac{2}{3}} \\ \mathrm{Hence},\mathrm{the}\mathrm{value}\mathrm{of}\sin \theta \mathrm{is}0\mathrm{or}\sqrt{\frac{2}{3}}. \end{gathered}$$