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## 'Sine and Cosine for Connected Angles' printed from http://nrich.maths.org/

The diagram shows a nine-point pegboard. We'll take the board
radius as one (in other words it's a unit circle).

Looking at triangle $OAM$, why is $AM$ equal to
$\sin40^\circ$?

Looking at triangle $OAB$, why is $AB$ equal to
$2\sin20^\circ$?

Looking at triangle $ABM$, why is $AM$ also equal to
$AB\cos20^\circ$ which works out as $2\sin20^\circ$
$\cos20^\circ$

So it looks like the same $AM$ length value can be calculated by
using $\sin40^\circ$

and also calculated by using $2\sin20^\circ\cos 20^\circ$

Now switch to a $10$ point pegboard and find the two ways to
calculate the $AM$ length on that board.

How about on a $12$ point board? What general result is
emerging?

Draw a diagram and use it to provide reasoning which accounts for
that general result.

For printable sets of circle templates for use with this activity,
please see

Printable Resources page.

Many thanks to Geoff Faux who introduced us
to the merits of the 9 pin circular geo-board.
The boards, moulded in crystal clear ABS
that can be used on an OHP (185 cm in diameter), together with a
teacher's guide, are available from Geoff at Education
Initiatives