Use de Moivre's theorem to find an expression for $\tan 3\theta$ in terms of $\tan \theta$

Question

Use de Moivre's theorem to find an expression for $\tan 3\theta$ in terms of $\tan \theta$ only. By using a suitable value of $\theta$ in your result, find the exact values of $\tan \frac{\pi}{12}$ and $\tan \frac{5\pi}{12}$. Show that $\frac{\tan \frac{5\pi}{12}}{\tan \frac{\pi}{12}} = 7 + 4\sqrt{3}$.

Accepted Answer

(a) $\tan3\theta=\dfrac{3\tan\theta-\tan^3\theta}{1-3\tan^2\theta}$, where $\cos\theta\cos3\theta\ne0$. (b) $\tan\dfrac\pi{12}=2-\sqrt3$, $\tan\dfrac{5\pi}{12}=2+\sqrt3$. (c) $7+4\sqrt3$ (shown).

Tangent of multiple angles

Tangent of multiple angles

Solution:

Solution locked