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.

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.