Review question

# Why is $\cos \theta + \cos (\theta + 2\pi/3) + \cos(\theta + 4\pi/3)=0$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9782

## Suggestion

1. Prove that $\cos(A+B)=\cos A\cos B - \sin A\sin B,$ where the angles $A$, $B$ and $A+B$ are all acute.

There is a standard geometric proof of this formula using the unit circle. If you haven’t come across it before, you may like to have a look at the diagram hidden below.

What lengths can you calculate, in terms of $\alpha$ and $\beta$?

1. By projecting the sides of an equilateral triangle onto a certain line, prove that $\cos \theta + \cos \left(\theta + \frac{2\pi}{3}\right) + \cos\left(\theta+\frac{4\pi}{3}\right)=0$ and find the value of the expression $\sin \theta + \sin \left(\theta + \frac{2\pi}{3}\right) + \sin\left(\theta+\frac{4\pi}{3}\right).$

Could we draw an equilateral triangle at an angle $\theta$ to the horizontal? What angles do we know? What horizontal lengths can we deduce?

To find the value of the $\sin$ sum, what vertical lengths can we deduce?