SOH-CAH-TOA: Sine, Cosine and Tangent Explained

Right-triangle trigonometry · 5 minute read

Trigonometry starts with one idea: in a right triangle, the ratios between side lengths depend only on the angles. Three ratios cover almost every beginner problem.

Name the sides first

Pick one acute angle, called θ. The hypotenuse is the longest side, opposite the right angle. The opposite side faces θ, and the adjacent side touches θ (but is not the hypotenuse). If you switch to the other acute angle, opposite and adjacent swap.

The three ratios

sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent

The memory phrase SOH-CAH-TOA stands for Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

Worked example

A right triangle has opposite side 5, adjacent side 12, and hypotenuse 13. Find sin θ, cos θ and tan θ.

sin θ = 5/13 ≈ 0.385
cos θ = 12/13 ≈ 0.923
tan θ = 5/12 ≈ 0.417

To find the angle itself, use the inverse function: θ = tan⁻¹(5/12) ≈ 22.6°.

Common mistakes

Calculator in the wrong mode. Use degrees when the problem gives degrees.
Labelling opposite and adjacent from the wrong angle.
Using sine or cosine when you know only the two legs; tangent is the right choice then.
Rounding too early. Keep extra digits until the final answer.

Practice it

Try the right-triangle question bank, with step-by-step solutions for every problem.

Practice right-triangle trigonometry