24 min read

The Three Ratios

Sine, cosine, and tangent connect an angle in a right-angled triangle to the ratio of two of its sides.

Naming the Sides

Relative to a chosen angle (not the right angle) in a right-angled triangle, the three sides get special names: Hypotenuse : the longest side, opposite the right angle (always the same, whichever angle you choose). Opposite : the side directly across from the chosen angle. Adjacent : the remaining side, next to the chosen angle (but not the hypotenuse).

SOH CAH TOA: Sin=Opp/Hyp, Cos=Adj/Hyp, Tan=Opp/Adj.

Right triangle

Opposite 3.0Adjacent 4.0

sin ≈ 0.600 · cos ≈ 0.800 · tan ≈ 0.750

Change a value and watch:

Change the opposite and adjacent sides and watch sin, cos, and tan update.

The hypotenuse is calculated automatically using Pythagoras' theorem.

Example 1 — working out a ratio

A right-angled triangle has Opposite = 5, Hypotenuse = 13 (relative to angle θ). Find sin(θ). 1 sin(θ) = Opposite \div Hypotenuse 2 sin(θ) = 5 \div 13 \approx 0.385 Answer: sin(θ) \approx 0.385

Example 2 — using tan

A right-angled triangle has Opposite = 8, Adjacent = 6 (relative to angle θ). Find tan(θ). 1 tan(θ) = Opposite \div Adjacent 2 tan(θ) = 8 \div 6 \approx 1.333 Answer: tan(θ) \approx 1.333

Practice

For angle θ with Adjacent = 7 and Hypotenuse = 25, what is cos(θ)?