Trigonometry/The sine of 15 degrees

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We have

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

If θ = 15º then

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

so after some manipulation (left as an exercise),

\displaystyle \sin(15^\circ) = \frac{\sqrt{6}-\sqrt{2}}{4} = \cos(75^\circ)
\displaystyle \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 3º and its multiples.