Special Angles and Function Values

Some angles are special because their trig values come out as finite radical expressions — clean numbers like $\frac{1}{2}$, $\frac{\sqrt{2}}{2}$ and $\frac{\sqrt{3}}{2}$ rather than endless decimals. These are the angles worth committing to memory, and laying them around the unit circle is the standard way to do it. This builds directly on trigonometric-functions-properties-and-inverses, where the functions are defined as coordinates on that circle.

The unit circle

Each point on the unit circle is $(\cos \theta ,\sin \theta )$, and $\tan \theta =\sin \theta /\cos \theta$ is the slope of the radius. The special angles are the multiples of $30^{\circ }$ ($\frac{\pi }{6}$) and $45^{\circ }$ ($\frac{\pi }{4}$), which split each quadrant evenly.

The four quadrantal angles sit on the axes:

Angle	Radians	$(\cos \theta ,\sin \theta )$	$\tan \theta$
$0^{\circ }/360^{\circ }$	$0/2\pi$	$(1,0)$	$0$
$90^{\circ }$	$\frac{\pi }{2}$	$(0,1)$	$\infty$
$180^{\circ }$	$\pi$	$(-1,0)$	$0$
$270^{\circ }$	$\frac{3\pi }{2}$	$(0,-1)$	$\infty$

First-quadrant values

Memorise the first quadrant and the rest follow by symmetry. A handy pattern: writing each sine as $\frac{\sqrt{n}}{2}$ for $n=0,1,2,3,4$ makes the progression obvious.

Angle	Radians	$\sin \theta$	$\cos \theta$	$\tan \theta$
$0^{\circ }$	$0$	$0$	$1$	$0$
$30^{\circ }$	$\frac{\pi }{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{3}$
$45^{\circ }$	$\frac{\pi }{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$
$60^{\circ }$	$\frac{\pi }{3}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
$90^{\circ }$	$\frac{\pi }{2}$	$1$	$0$	$\infty$

Reading $\sin$ down and $\cos$ up, the two columns are mirror images — a direct consequence of the co-function relationship $\cos \theta =\sin (90^{\circ }-\theta )$.

Extending by symmetry

The other three quadrants are reflections of the first, so only the signs change — the magnitudes stay identical to the matching reference angle:

Quadrant	Angle range	$\cos$	$\sin$	$\tan$
I	$0^{\circ }$–$90^{\circ }$	$+$	$+$	$+$
II	$90^{\circ }$–$180^{\circ }$	$-$	$+$	$-$
III	$180^{\circ }$–$270^{\circ }$	$-$	$-$	$+$
IV	$270^{\circ }$–$360^{\circ }$	$+$	$-$	$-$

So, for instance, $120^{\circ }$ has reference angle $60^{\circ }$ in quadrant II, giving $\left(-\frac{1}{2},\frac{\sqrt{3}}{2}\right)$; and $225^{\circ }$ has reference angle $45^{\circ }$ in quadrant III, giving $\left(-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right)$. The sign rule is just the even/odd and quadrant symmetry discussed in trigonometric-functions-properties-and-inverses.

Draft note: a future reference-angles note could expand the symmetry trick into a general method for any angle.