Trigonometric Ratio of Submultiple of an Angle - Practice Questions & MCQ

Trigonometric Ratio of Submultiple of an Angle
1. Trigonometric Ratio of $\theta$ in terms of $\theta / 2$

$
\begin{aligned}
\sin (\theta) & =2 \sin \frac{\theta}{2} \cos \frac{\theta}{2} \\
& =\frac{2 \tan \frac{\theta}{2}}{1+\tan ^2 \frac{\theta}{2}} \\
\cos (\theta) & =\cos ^2 \frac{\theta}{2}-\sin ^2 \frac{\theta}{2} \\
& =1-2 \sin ^2 \frac{\theta}{2} \\
& =2 \cos ^2 \frac{\theta}{2}-1 \\
& =\frac{1-\tan ^2 \frac{\theta}{2}}{1+\tan ^2 \frac{\theta}{2}} \\
\tan (\theta) & =\frac{2 \tan ^{\frac{\theta}{2}}}{1-\tan ^2 \frac{\theta}{2}}
\end{aligned}
$
All the above trigonometric ratios can be derived by replacing ' $\theta$ ' with ' $\theta / 2$ ' in the double angle formulas.

2. Trigonometric Ratio of $\theta$ in terms of $\theta / 3$
1. $\sin \mathrm{A}=3 \sin \frac{\mathrm{~A}}{3}-4 \sin ^3 \frac{\mathrm{~A}}{3}$
2. $\cos \mathrm{A}=4 \cos ^3 \frac{\mathrm{~A}}{3}-3 \cos \frac{A}{3}$
3. $\tan \mathrm{A}=\frac{3 \tan \frac{\mathrm{~A}}{3}-\tan ^3 \frac{\mathrm{~A}}{3}}{1-3 \tan ^2 \frac{\mathrm{~A}}{3}}$

The above trigonometric ratio of angle ' $A$ ' in terms of ' $A / 3$ ' can be derived by replacing ' $A$ ' with ' $A / 3$ ' in the triple angle formulas