ત્રિકોણમિતીય પ્રતિવિધેયો — ગણિત ધોરણ 12 સાયન્સ — Question

$\sin \left( {3{{\sin }^{ - 1}}\frac{1}{3}} \right)$

ધારો કે, ${\sin ^{ - 1}}\frac{1}{3} = \theta ;\theta \in \left[ { - \frac{\pi }{2},\frac{\pi }{2}} \right].$ $\therefore \sin \theta = \frac{1}{3}$

$\therefore$ ${\sin ^{ - 1}}\left( {3{{\sin }^{ - 1}}\frac{1}{3}} \right) = \sin 3\theta $

$ = 3\sin \theta - 4{\sin ^3}\theta $

$ = 3\left( {\frac{1}{3}} \right) - 4{\left( {\frac{1}{3}} \right)^3}$

$ = 1 - \frac{4}{{27}} = \frac{{27 - 4}}{{27}} = \frac{{23}}{{27}}$