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Sine and Cosine

Age 14 to 16 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 the last part of the film, you may also find the Pause button useful.

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?