Exact Values

For most angles a calculator gives a messy decimal, but a handful of special angles — 0^\circ, 30^\circ, 45^\circ, 60^\circ and 90^\circ — have neat exact values you can write down without a calculator. They all come from just two special triangles.

The 45-45-90 triangle is a right isosceles triangle: two legs of length 1 and a hypotenuse of \sqrt2.
The 30-60-90 triangle is half an equilateral triangle: sides 1, \sqrt3 and 2.

Reading the side ratios (SOH-CAH-TOA) straight off these triangles:

\sin 45^\circ = \cos 45^\circ = \tfrac{1}{\sqrt2} = \tfrac{\sqrt2}{2}

\sin 30^\circ = \cos 60^\circ = \tfrac12, \qquad \sin 60^\circ = \cos 30^\circ = \tfrac{\sqrt3}{2}

\tan 30^\circ = \tfrac{1}{\sqrt3}, \qquad \tan 45^\circ = 1, \qquad \tan 60^\circ = \sqrt3

The exact trig values at the special angles:

\sin: \sin 0^\circ = 0, \sin 30^\circ = \tfrac12, \sin 45^\circ = \tfrac{\sqrt2}{2}, \sin 60^\circ = \tfrac{\sqrt3}{2}, \sin 90^\circ = 1;
\cos: \cos 0^\circ = 1, \cos 30^\circ = \tfrac{\sqrt3}{2}, \cos 45^\circ = \tfrac{\sqrt2}{2}, \cos 60^\circ = \tfrac12, \cos 90^\circ = 0;
\tan: \tan 0^\circ = 0, \tan 30^\circ = \tfrac{1}{\sqrt3}, \tan 45^\circ = 1, \tan 60^\circ = \sqrt3, \tan 90^\circ is undefined.

Where the values come from

Step through the two triangles. Every exact value above is just one side divided by another, read straight off these pictures.