Question

If 1/3 sin theta=1/4 cos theta,then what is the value of sin 4 theta?

Answer 1

$$\begin{gathered} \mathrm{It}\mathrm{is}\mathrm{given}\mathrm{that}, \\ \frac{1}{3}\mathrm{sin\theta }=\frac{1}{4}\mathrm{cos\theta } \\ \mathrm{This}\mathrm{implies}\mathrm{that}, \\ \frac{\mathrm{sin\theta }}{\mathrm{cos\theta }}=\frac{3}{4}\mathrm{or}\mathrm{tan\theta }=\frac{3}{4} \\ \mathrm{It}\mathrm{is}\mathrm{required}\mathrm{to}\mathrm{find}\mathrm{the}\mathrm{value}\mathrm{of}\sin 4\mathrm{\theta }. \\ \mathrm{Rewrite}\sin 4\mathrm{\theta }\mathrm{as}\sin 2\left(2\mathrm{\theta }\right)\mathrm{and}\mathrm{use}\mathrm{the}\mathrm{double}\mathrm{angle} \\ \mathrm{formula},\sin 2\mathrm{x}=2\mathrm{sinxcosx}\mathrm{to}\mathrm{get}, \\ \sin 4\mathrm{\theta }=\sin 2\left(2\mathrm{\theta }\right) \\ =2\sin \left(2\mathrm{\theta }\right)\cos \left(2\mathrm{\theta }\right) \\ \mathrm{Use}\mathrm{the}\mathrm{following}\mathrm{formulas}. \\ \sin \left(2\mathrm{\theta }\right)=\frac{2\mathrm{tan\theta }}{1+{\tan }^2\mathrm{\theta }}\mathrm{and}\cos \left(2\mathrm{\theta }\right)=\frac{1-{\tan }^2\mathrm{\theta }}{1+{\tan }^2\mathrm{\theta }} \\ \mathrm{This}\mathrm{further}\mathrm{implies}\mathrm{that}, \\ \sin 4\mathrm{\theta }=2\frac{2\mathrm{tan\theta }}{1+{\tan }^2\mathrm{\theta }}\frac{1-{\tan }^2\mathrm{\theta }}{1+{\tan }^2\mathrm{\theta }} \\ \mathrm{Substitute}\mathrm{the}\mathrm{values}\mathrm{tan\theta }=\frac{3}{4}\mathrm{to}\mathrm{get}, \\ \sin 4\mathrm{\theta }=2\frac{2\left(\frac{3}{4}\right)}{1+{\left(\frac{3}{4}\right)}^2}\frac{1-{\left(\frac{3}{4}\right)}^2}{1+{\left(\frac{3}{4}\right)}^2} \\ =2\frac{\frac{3}{2}}{1+{\displaystyle \frac{9}{16}}}\frac{1-\frac{9}{16}}{1+\frac{9}{16}} \\ =2\frac{\frac{3}{2}}{{\displaystyle \frac{25}{16}}}\frac{\frac{7}{16}}{\frac{25}{16}} \\ =2\times \frac{3}{\cancel{2}}\times \frac{{\displaystyle \stackrel{8}{\cancel{16}}}}{{\displaystyle 25}}\times \frac{7}{\cancel{16}}\times \frac{\cancel{16}}{{\displaystyle 25}} \\ =\frac{2\times 3\times \times 8\times 7}{25\times 25} \\ =\frac{336}{625} \\ =0.5376 \end{gathered}$$