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# Sine and Cosine

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?## You may also like

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Age 14 to 16

Challenge Level

- Problem
- Teachers' Resources

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?

The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design...

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

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.