Question

If root 3 tan theta=3 sin theta.find the value of sin theta

Answer 1

$\underline{\underline{\mathfrak{\Large{Solution : }}}}$



$\underline{\textsf{Given , }} \\ \\ \sf \implies \sqrt{3} \: tan \: \theta \: = \: 3 \: sin \: \theta$


$\underline{\mathsf{To \: Find \: \longrightarrow sin \: \theta }}$



$\sf \implies \sqrt{3} \: tan \: \theta \: = \: 3 \: sin \: \theta \\ \\ \sf \implies \sqrt{3} \: \dfrac{\cancel{sin \: \theta}}{cos \: \theta} \: = \: 3 \: \cancel{sin \: \theta } \:\quad \left\{ tan \: \theta \: = \: \dfrac{sin \: \theta}{cos \: \theta} \right\} \\ \\ \sf \implies \dfrac{\sqrt{3}}{3} \: = \: cos \: \theta$



$\underline{\textsf{By Trigonometric Identity , }} \\ \\ \sf \implies sin \: \theta \: = \: \sqrt{1 \: - \: cos^2 \theta} \\ \\ \sf \implies sin \: \theta \: = \: \sqrt{1 \: - \: \left(\dfrac{\sqrt{3}}{3} \right)^2 } \\ \\ \sf \implies sin \: \theta \: = \: \sqrt{1 \: - \: \dfrac{3}{9}} \\ \\ \sf \implies sin \: \theta \: = \: \sqrt{1 \: - \: \dfrac{1}{3}} \\ \\ \sf \implies sin \: \theta \: = \: \sqrt{\dfrac{3 \: - \: 1}{3}} \\ \\ \sf \: \: \therefore \: \: sin \: \theta \: = \: \sqrt{\dfrac{2}{3}}$

Answer 2

$\underline{\underline{\mathfrak{\Large{Solution : }}}} \\\\\\</p><p></p><p> </p><p> </p><p> </p><p> </p><p></p><p></p><p></p><p>\begin{lgathered}\underline{\textsf{Given , }} \\ \\ \sf \implies \sqrt{3} \: tan \: \theta \: = \: 3 \: sin \: \theta\end{lgathered} \\\\\\</p><p></p><p></p><p>\underline{\mathsf{To \: Find \: \longrightarrow sin \: \theta }} \\\\\\</p><p></p><p> </p><p> </p><p></p><p></p><p></p><p>\begin{lgathered}\sf \implies \sqrt{3} \: tan \: \theta \: = \: 3 \: sin \: \theta \\ \\ \sf \implies \sqrt{3} \: \dfrac{\cancel{sin \: \theta}}{cos \: \theta} \: = \: 3 \: \cancel{sin \: \theta } \:\quad \left\{ tan \: \theta \: = \: \dfrac{sin \: \theta}{cos \: \theta} \right\} \\ \\ \sf \implies \dfrac{\sqrt{3}}{3} \: = \: cos \: \theta\end{lgathered} \\\\\\</p><p></p><p></p><p></p><p></p><p>\begin{lgathered}\underline{\textsf{By Trigonometric Identity , }} \\ \\ \sf \implies sin \: \theta \: = \: \sqrt{1 \: - \: cos^2 \theta} \\ \\ \sf \implies sin \: \theta \: = \: \sqrt{1 \: - \: \left(\dfrac{\sqrt{3}}{3} \right)^2 } \\ \\ \sf \implies sin \: \theta \: = \: \sqrt{1 \: - \: \dfrac{3}{9}} \\ \\ \sf \implies sin \: \theta \: = \: \sqrt{1 \: - \: \dfrac{1}{3}} \\ \\ \sf \implies sin \: \theta \: = \: \sqrt{\dfrac{3 \: - \: 1}{3}} \\ \\ \sf \: \: \therefore \: \: sin \: \theta \: = \: \sqrt{\dfrac{2}{3}}\end{lgathered}$