### Coke Machine

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design. Coins inserted into the machine slide down a chute into the machine and a drink is duly released. How many more revolutions does the foreign coin make over the 50 pence piece going down the chute? N.B. A 50 pence piece is a 7 sided polygon ABCDEFG with rounded edges, obtained by replacing AB with arc centred at E and radius EA; replacing BC with arc centred at F radius FB ...etc..

### At a Glance

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

### Six Discs

Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?

# Cosines Rule

##### Stage: 4 Challenge Level:

Why do this problem?
The first part is an easy application of the cosine rule. The second part requires some algebraic skills and gives practice in proving a formula.

Possible approach
Use as practice. The problem does not require learners to decide on the method to use.

Key questions
Look at the formula to be proved. Write down the cosine formula for the triangles in the diagram using the same notation. Can you see how to manipulate the formulas you have established to give the formula to be proved?