In our previous post, we show a
compass-and-straightedge construction of a
regular pentagon based on the following trigonometric formula $$\cos{\frac{\pi}{5}} = \frac{1 + \sqrt{5}}{4}.$$
We derive this formula of $\cos{\frac{\pi}{5}}$ by observing that $\cos{\frac{2 \pi}{5}} = -\cos{\frac{3 \pi}{5}}$ and then applying the trigonometric formulas for double angle and triple angle:
$$\cos{2 x} = 2 \cos^2{x} - 1,$$ $$\cos{3 x} = 4 \cos^3{x} - 3 \cos{x}$$ to set up a cubic equation for $\cos{\frac{\pi}{5}}$.
It seems a good occasion now for us to learn about trigonometric multiple-angle formulas. In this post, we will show how to derive formulas for
$\sin{nx}$,
$\cos{nx}$,
$\tan{nx}$ and
$\cot{nx}$ using
de Moivre's identity of the
complex numbers.