Special Trig Values: Multiples of π/4

With a strong understanding of the unit circle definition of the trigonometric functions, trig values for multiples of $\frac{\pi}{4}$ are easy to derive.

Consider the unit circle pictured below:

Think back to the unit circle definitions of $\sin \theta$ and $\cos \theta$.

So, a point $(x,y)$ on the unit circle corresponds to $(\sin\theta,\cos\theta)$.

Since our angles are all multiples of $45^\circ,$ we actually get that $\left|\sin\dfrac{\pi}{4}k\right| = \left|\cos \dfrac{\pi}{4}k\right|$. In particular, we have $$\left|\sin\dfrac{\pi}{4}k\right| = \left|\cos \dfrac{\pi}{4}k\right| = \frac{1}{\sqrt2} = \frac{\sqrt2}{2}.$$ The signs are be determined by the quadrant that the angle is in, as normal.