Exact Values of Sine, Cosine and Tangent

In a right angled triangle, pick one acute angle and call it theta. The hypotenuse is the longest side, opposite the right angle. The opposite side sits across from theta, and the adjacent side is the remaining leg. For that chosen angle, sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. The special angles are 0, 30, 45, 60, and 90 degrees. Here is the full table. sin: 0, 1/2, root 2 over 2, root 3 over 2, 1. cos: 1, root 3 over 2, root 2 over 2, 1/2, 0. tan: 0, root 3 over 3, 1, root 3, undefined.

The 30 and 60 values come from a halved equilateral triangle. Cut an equilateral triangle of side 2 in half. The hypotenuse is the old side, so it is 2. The cut splits the base in half, so the shortest leg is 1. Pythagoras gives the long leg: long leg squared plus 1 squared equals 2 squared, so the long leg is root 3.

Now read the ratios. For 30 degrees, opposite is 1 and hypotenuse is 2, so sin 30 = 1/2. For 60 degrees, opposite is root 3 and hypotenuse is 2, so sin 60 = root 3 over 2. Cosine uses the adjacent side, so cos 30 = root 3 over 2 and cos 60 = 1/2. Tangent is opposite over adjacent, so tan 30 = root 3 over 3 and tan 60 = root 3.

Cut a square in half instead and you get a 45-45-90 triangle with equal legs of 1 and hypotenuse root 2, so sin 45 = cos 45 = root 2 over 2 and tan 45 = 1. Leave answers with roots in surd form, like root 3 over 2, because they never come out neat as decimals.

The right-triangle definitions need acute angles, so the endpoints 0 and 90 degrees need a small extension. Measure the angle from the positive x-axis and use a unit circle of radius 1 centred at the vertex. A ray at angle theta meets the circle at (cos theta, sin theta). At 0 degrees the point is (1, 0), so cos 0 = 1 and sin 0 = 0. At 90 degrees the point is (0, 1), so cos 90 = 0 and sin 90 = 1. Tangent is y over x where x is not zero, so tan 90 asks for 1 over 0, which is undefined.