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). Opposite and adjacent swap if you switch which acute angle you are using. The diagrams below show the same idea in four orientations.

Opposite, adjacent, hypotenuse

The hypotenuse is always opposite the right angle. Opposite and adjacent depend on which acute angle is θ.

Tap each orientation. Watch how opposite and adjacent move with θ, while the hypotenuse stays across from the right angle.

Tap each orientation. Watch how opposite and adjacent move with θ, while the hypotenuse stays across from the right angle.

Practise naming the sides until you can do it without looking at the labels.

SOH CAH TOA: sin = \frac{Opp}{Hyp}, cos = \frac{Adj}{Hyp}, tan = \frac{Opp}{Adj} .

Exact values for common angles
Exact values for common angles

Memorise this table. You will use it without a calculator. Notice sin and cos swap between 30 ° and 60 °. At 90 °, tan is — (not defined).

Right triangle — labelled for θ

Opposite 3.0Adjacent 4.0

θ ≈ 36.9° · sin θ ≈ 0.600 · cos θ ≈ 0.800 · tan θ ≈ 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) = \frac{Opposite}{Hypotenuse} 2 sin (θ) = \frac{5}{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) = \frac{Opposite}{Adjacent} 2 tan (θ) = \frac{8}{6} \approx 1.333 Answer: tan (θ) \approx 1.333

Practice

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