We have

If then

so after some manipulation (left as an exercise),

These results may be combined with those from the previous section to find the sines and cosines of and its multiples.

The latest reviewed version was checked on *22 August 2013*. There is 1 pending change awaiting review.

We have

- $\sin(\theta )={\sqrt {\frac {1-\cos(2\theta )}{2}}}$
- $\cos(\theta )={\sqrt {\frac {1+\cos(2\theta )}{2}}}$

If $\theta =15^{\circ }$ then

- $\cos(2\theta )=\cos(30^{\circ })={\frac {\sqrt {3}}{2}}$

so after some manipulation (left as an exercise),

- $\sin(15^{\circ })={\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}=\cos(75^{\circ })$
- $\cos(15^{\circ })={\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}=\sin(75^{\circ })$

These results may be combined with those from the previous section to find the sines and cosines of $=3^{\circ }$ and its multiples.