Sine, cosine & tangent

Three ratios. One right triangle.

The idea

Cosine is adjacent divided by hypotenuse. Tangent is opposite divided by adjacent. On the unit circle the hypotenuse is 1, so cosine becomes the x-coordinate and sine becomes the y-coordinate. Tangent equals sine divided by cosine.

Work through an example

At 45°, the two legs of the right triangle are equal. On the unit circle, both coordinates are √2/2, and tangent is 1. At 90°, cosine is 0, which makes tangent undefined.

What to watch for

Opposite and adjacent depend on which acute angle you choose. The hypotenuse is always the side opposite the right angle. For angles beyond 90°, the unit circle extends these definitions using signed coordinates.