Trigonometry/The sine of 15 degrees

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

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

so after some manipulation (left as an exercise),

$\displaystyle \sin(15^O) = \frac{\sqrt{3}-1}{2\sqrt{2}} = \cos(75^O)$

$\displaystyle \cos(15^O) = \frac{\sqrt{3}+1}{2\sqrt{2}} = \sin(75^O)$

These results may be combined with those from the previous section to find the sines and cosines of 3º and its multiples.

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