Sine, cosine, tangent

Right triangles hide everywhere: in the ladder against the wall, in the roof gable, in the ramp. If you know one angle and one side, you can work out the remaining sides without measuring. The key is three fixed ratios: sine, cosine and tangent. Here you get to know them. The topic Solving triangles then shows how to rearrange them and work out missing lengths.

The sides around the right angle

The longest side lies opposite the right angle and is called the hypotenuse. The other two sides are the legs. Which leg is which depends on the angle you are looking at: the opposite leg lies across from that angle, the adjacent leg sits next to it.

The three ratios

For an angle $\alpha$ the three ratios are always defined the same way, no matter how big the triangle is. Only the shape, that is the angle, decides the value. Sine compares the opposite leg with the hypotenuse, cosine the adjacent leg with the hypotenuse, and tangent the opposite leg with the adjacent leg.

A memory aid

In English many people use SOH CAH TOA: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent. If you prefer a sentence: sine reaches for the opposite leg, cosine for the adjacent leg, and both divide by the hypotenuse. Only tangent stays with the two legs.

The angle drives the ratio

Because the hypotenuse is the longest side, sine and cosine in a right triangle always sit between 0 and 1. As the angle grows from 0° towards 90°, sine gets bigger and cosine gets smaller. At $\alpha = 45°$ the opposite and adjacent legs are equally long, which is why $\tan 45° = 1$ there. Tangent knows no such limit: once the opposite leg is longer than the adjacent one, it grows beyond 1. You will meet such fixed values more closely in the unit circle.