Sine, cosine and tangent: the three ratios

The three trigonometric ratios

In a right-angled triangle, given an acute angle, three ratios between the lengths of the sides are defined as follows:

• sin(angle) = opposite / hypotenuse
• cos(angle) = adjacent / hypotenuse
• tan(angle) = opposite / adjacent

A mnemonic device

These formulas are often remembered using the acronym SOH-CAH-TOA:

• SOH: Sine = Opposite / Hypotenuse
• CAH: Cosine = Adjacent / Hypotenuse
• TOA: Tangent = Opposite / Adjacent

Numerical example

Let’s take a right-angled triangle with hypotenuse = 5 cm, side opposite angle x = 3 cm, and adjacent side = 4 cm (a 3-4-5 triangle).

• sin(x) = 3/5 = 0.6
• cos(x) = 4/5 = 0.8
• tan(x) = 3/4 = 0.75

Common pitfalls

1. Do not confuse the side opposite the angle with the side adjacent to it: these depend on the angle chosen.
2. The result of a sine or cosine is always between 0 and 1, as the hypotenuse is the longest side. If you get 1.5, there’s a mistake!
3. The tangent, on the other hand, can easily exceed 1.