### Cosines Rule

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

### Squ-areas

Three squares are drawn on the sides of a triangle ABC. Their areas are respectively 18 000, 20 000 and 26 000 square centimetres. If the outer vertices of the squares are joined, three more triangular areas are enclosed. What is the area of this convex hexagon?

### Far Horizon

An observer is on top of a lighthouse. How far from the foot of the lighthouse is the horizon that the observer can see?

# Sine and Cosine

##### Stage: 4 Challenge Level:

This problem is a follow on to the problems Round and Round and Round and Where Is the Dot?
Use your calculator to find decimal values for the following:

$\sin 50 ^{\circ}$, $\cos 40 ^{\circ}$
$\sin 70 ^{\circ}$, $\cos 20 ^{\circ}$
$\sin 15^{\circ}$, $\cos 75^{\circ}$

What do you notice and why does that happen?

Look at the film below - does that fit with your description?

Look especially at Stage 3 of the film, you may also find the Pause button useful.

This text is usually replaced by the Flash movie.

The film suggests a way to understand Sine and Cosine ratios (or lengths, if the hypotenuse has length one), for angles beyond the $0 ^{\circ}$ to $90^{\circ}$ range, in other words beyond angles which occur in right-angled triangles.

Which of these statements do you think are true?:

$\sin 150 ^{\circ}= \sin 30^{\circ}$ (notice that 180 - 30 = 150)
$\sin 150 ^{\circ}= \sin 330 ^{\circ}$
$\sin 150 ^{\circ}= \sin 210^{\circ}$
$\sin 30^{\circ}= \sin 330 ^{\circ}$
$\cos 30 ^{\circ}= \cos 330 ^{\circ}$
$\cos 50^{\circ}= \cos 130 ^{\circ}$
$\sin 150 ^{\circ}= \cos 30 ^{\circ}$
$\sin 150 ^{\circ}= \cos 60 ^{\circ}$
$\sin 300 ^{\circ}= \cos 30 ^{\circ}$

You could use your calculator to check.

What other relationships can you find?

Can you make some general statements?