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Sine and Cosine

Stage: 4 Challenge Level: Challenge Level:1

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?
The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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