Review question

Ref: R7021

## Suggestion

$ABCDE$ is a regular pentagon. By projecting the broken line $AED$ on the line $AB$, or otherwise, show that $\cos\dfrac{\pi}{5}-\cos\dfrac{2\pi}{5}=\dfrac{1}{2}.$

Let’s assume that the side lengths of the pentagon are $1$.

Can we draw a large, clear diagram of the pentagon $ABCDE$, with $AB$ horizontally at the bottom?

What are the angles inside a regular pentagon, in radians?

If we now drop perpendiculars from $D$ and $E$ to the horizontal, can we start to find lengths and angles?

Hence, or otherwise, show that $\cos\dfrac{\pi}{5} = \dfrac{\sqrt{5}+1}{4}$.

Using the first part, are there any trigonometrical identities we could use? To get this just in terms of $\cos\dfrac{\pi}{5}$?

Show further that $\cos\dfrac{3\pi}{5} = -\dfrac{\sqrt{5}-1}{4}$.

What’s the sum of $\dfrac{3\pi}{5}$ and $\dfrac{2\pi}{5}$? How does this connect $\cos\dfrac{3\pi}{5}$ and $\cos\dfrac{2\pi}{5}$?