Give any two values from a right-angled triangle and this solves the rest. The useful part is that it tells you which of sin, cos or tan to use and why, which is the bit people actually get stuck on.
The three ratios are just definitions of how the sides of a right-angled triangle relate to one of its angles.
| Ratio | Formula | Use it when you have |
|---|---|---|
| SOH — sine | sin θ = opposite ÷ hypotenuse | Opposite and hypotenuse |
| CAH — cosine | cos θ = adjacent ÷ hypotenuse | Adjacent and hypotenuse |
| TOA — tangent | tan θ = opposite ÷ adjacent | Opposite and adjacent |
The choosing method is simple once you see it. Label the two sides you know or want, then look at which pair they are. If the hypotenuse is not involved at all, it is tan. If it is involved along with the opposite, it is sin. If it is involved along with the adjacent, it is cos.
When you know two sides and want the angle, you use the inverse functions, written sin⁻¹, cos⁻¹ and tan⁻¹, and sometimes called arcsin, arccos and arctan. They do the opposite job: you feed in a ratio and get back the angle that produces it.
So if sin θ = 0.5, then θ = sin⁻¹(0.5) = 30°. On most calculators this is the shift or second-function key above sin. The inverse is not the same as 1 ÷ sin, which is a genuinely different thing despite the notation looking similar.