Trigonometry

(from Greek trigonon 'triangle' and metron 'measure': the mathematics of angles and side ratios)

What is trigonometry?

Trigonometry studies the relationship between the angles of a triangle and the lengths of its sides. It starts with right triangles — triangles with one 90-degree angle — and the three basic ratios: sine, cosine, and tangent.

The three basic ratios

In a right triangle, pick one of the two non-right angles and call it θ (theta). Relative to that angle, one of the other two sides is called "opposite" (across from θ) and the other is "adjacent" (next to θ); the longest side, opposite the right angle, is always the "hypotenuse." The three ratios are:

Sine of θ = opposite ÷ hypotenuse

Cosine of θ = adjacent ÷ hypotenuse

Tangent of θ = opposite ÷ adjacent

A common way to remember this is "SOH-CAH-TOA": Sine-Opposite-Hypotenuse, Cosine-Adjacent-Hypotenuse, Tangent-Opposite-Adjacent.

Why the ratio stays the same no matter the triangle's size

Here's a fact worth actually proving, not just accepting: for a fixed angle θ, the ratio opposite ÷ hypotenuse is always the same number, no matter how big or small the right triangle is.

Take two right triangles that both contain the same angle θ. Since both are right triangles, and both share the angle θ, their third angles must also match (because a triangle's angles always add to 180 degrees, and two of the three angles — the right angle and θ — are already fixed and equal in both triangles). Triangles with all three matching angles are similar, meaning one is just a scaled-up or scaled-down copy of the other — every side of the larger triangle is the same multiple of the matching side in the smaller one. Since both the opposite side and the hypotenuse get scaled by that same multiple, their ratio stays exactly the same. That's why "sine of 30 degrees" can be a single, fixed number, rather than depending on which particular triangle you happened to draw.

A worked example

Suppose a right triangle has an angle of 30 degrees, with the side opposite that angle measuring 5, and the hypotenuse measuring 10. Then sine of 30 degrees = 5 ÷ 10 = 0.5. This ratio, 0.5, would come out exactly the same for any right triangle with a 30-degree angle, no matter its overall size — which is exactly the fact proven above.

The Pythagorean identity

Since sine and cosine both come from the same right triangle, they're connected. Starting from the Pythagorean theorem, (opposite)² + (adjacent)² = (hypotenuse)², divide every term by (hypotenuse)²:

(opposite ÷ hypotenuse)² + (adjacent ÷ hypotenuse)² = 1

The left side is exactly sin(θ)² + cos(θ)², usually written sin²θ + cos²θ. So sin²θ + cos²θ = 1 — true for every angle θ, because it comes directly from the Pythagorean theorem itself.

Why trigonometry matters

Anywhere you need to connect an angle to a distance — measuring the height of a tall building from its shadow, finding a ship's position from two bearings, or describing anything that repeats in a regular cycle, like sound waves or the seasons — trigonometry is the tool that makes the connection.