Exact values of sine and cosine

If you know the sine of any angle, you can find its cosine from the Pythagorean theorem. And if you know the sine of an angle you can find the sine of any multiple of that angle using the identity for the sine of a sum.

You can find that sin 30° = 1/2 from a simple argument with triangles, and you can find the sin 45° = √2/2 by splitting a square on a diagonal and using the Pythagorean theorem. From these two values you can bootstrap your way to the sine and cosine of any multiple of 30° or 45°.

The utility of knowing exact sines and cosines declines rapidly beyond this point, but we can keep going just for fun.

Since 15 = 45 − 30, you can use the sum-angle formula to find

\sin 15^\circ = \frac{\sqrt{2} }{4}\left(\sqrt{3} - 1\right)

and from there every multiple of 15 degrees.

Because pentagons are constructible, you find the sine of 36 degrees (derivation).

\sin 36^\circ = \sqrt{\frac{1}{8}\left(5 - \sqrt{5}\right)}

And since 6 = 36 − 30, you can find the sine of 6°.

If you know the sine of an angle, you can find the sine of half that angle using the half-angle identity. So from the sine of 6 degrees you can find the sine of 3 degrees, and hence every multiple of 3 degrees.

\sin 3^\circ = \frac{1}{4} \sqrt{8 - \sqrt{10 - 2 \sqrt{5}} - \sqrt{3} \left(1 + \sqrt{5}\right)}

You could keep playing this game to find the sine of 1.5 degrees etc.

Gauss proved that a 17-sided regular polygon is constructible with a straight edge and compass, and so from his construction you can find the sine of 180/17 degrees just as the constructibility of a pentagon gave us the sine of 180/5 degrees.

More generally, the Gauss-Wantzel theorem says that a regular n-gon is constructible if and only if n is a power of 2 times a product of distinct Fermat primes. There are five known Fermat primes: 3, 5, 17, 257, and 65537. There are 32 numbers that are the product of (known) Fermat primes [1], if we count 1 as the empty product. So we can find the sine of 180/n degrees exactly if n is a power of 2 times one of these Fermat prime products. And from there we can bootstrap to all their multiples and sums.

More trigonometry posts

[1] See the next post for a fractal pattern hiding inside these numbers.