Trigonometry/The sine of 15 degrees

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.