Special Trig Values: Multiples of π/12

We'll use degrees instead of radians here.

The multiples of $15$ degrees are particularly nice because they are half of 30.

Let's find $\sin 15$ first.

There are two methods: 1) use the half-angle formula or 2) use the angle bisector theorem on a 30-60-90 triangle.

You can do the second method on your own; We'll use the half angle formula here

We know $\sin x = \pm \sqrt{\frac{(1 - \cos x)}{2}}$

$\cos 30 = \frac{\sqrt{3}}{2}$, so we plug that in to get $$\begin{align*} \sin 15 &=\pm\sqrt{\frac{1-\frac{\sqrt{3}}{2}}{2}} \\ &= \pm \sqrt{ \frac{2-\sqrt{3}}{4}} \\ &= \pm \dfrac{\sqrt{6}-\sqrt{2}}{4} \end{align*}$$

Sin 15 is positive, so we obtain $\sin 15 = \frac{\sqrt{6}-\sqrt{2}}{4}.$ Using the half angle formula for cosines gives $\cos 15 =\frac{\sqrt{6}+\sqrt{2}}{4}.$

These values also give the values for $\sin 75$ and $\cos 75$-- $$\begin{align*} \sin 15 &= \cos 75 = \frac{\sqrt{6}-\sqrt{2}}{4} \\ \cos 15 &= \sin 75= \frac{\sqrt{6}+\sqrt{2}}{4}\end{align*}$$ Since the values for sin/cosine of 30, 45, and 60 are easily calculated, we can found all the values for sin/cosine of multiples of 15.