Any time you tilt a ladder, aim a telescope, or check how steep a hill is, you're doing trigonometry in your head. Here's the simple machinery underneath.
Trigonometry is the branch of mathematics that connects angles to side lengths. Its three main characters — sine, cosine and tangent — are just ratios that compare the sides of a right‑angled triangle. Once you see them as ratios, everything from wave graphs to GPS calculations starts to make sense.
sin θ = Opposite / Hypotenuse
Compares the side opposite the angle to the longest side.
cos θ = Adjacent / Hypotenuse
Compares the side next to the angle to the longest side.
tan θ = Opposite / Adjacent
Compares the two shorter sides — the slope of the angle.
The Right Triangle Setup
SIDES & ANGLESEvery right triangle has one 90° corner and two acute angles. Pick one of those acute angles and call it θ (theta). The three sides get special names relative to θ:
- Hypotenuse — the longest side, always opposite the 90° angle.
- Opposite — the side across from θ.
- Adjacent — the side next to θ that isn't the hypotenuse.
These labels flip if you choose the other acute angle, which is why it's important to always mark which angle you're working from.
In a right triangle, the hypotenuse is always the same side no matter which acute angle you pick — but “opposite” and “adjacent” swap when you switch angles.
SOHCAHTOA: The Three Ratios
MEMORY AIDThe word SOHCAHTOA packs all three definitions into one string. Each group of three letters tells you which sides to divide:
Because the hypotenuse is always the longest side, sine and cosine are always between 0 and 1 (for acute angles). Tangent can be larger than 1 — it blows up as the angle approaches 90°.
Worked example
A right triangle has an angle θ = 30°, and the hypotenuse is 10 cm. How long is the opposite side?
- We know the angle and hypotenuse, and we want the opposite side — that's the sine ratio.
- Write it: sin 30° = opposite / 10
- Multiply both sides by 10: opposite = 10 × sin 30°
- Since sin 30° = 0.5: opposite = 5 cm
The Unit Circle: Beyond Right Triangles
ANGLES EVERYWHERERight triangles only handle angles between 0° and 90°. To talk about any angle — 120°, 200°, even negative angles — mathematicians use the unit circle.
Picture a circle of radius 1 centred at the origin. Start at the point (1, 0) and rotate anticlockwise by an angle θ. The coordinates of the point you land on are (cos θ, sin θ). That's the whole idea: cosine is the horizontal coordinate, sine is the vertical coordinate.
This extends the definitions naturally to all angles and makes the repeating wave patterns of sine and cosine visible — they oscillate between −1 and 1 forever.
The unit circle turns trigonometry from a triangle‑only tool into a language for rotation, waves, and anything that repeats — sound, light, alternating current, planetary orbits.
Key Values to Know
These five angles show up constantly. Memorising them (or knowing how to derive them from the unit circle) pays off immediately.
| Angle θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 0.5 | ≈ 0.866 | ≈ 0.577 |
| 45° | ≈ 0.707 | ≈ 0.707 | 1 |
| 60° | ≈ 0.866 | 0.5 | ≈ 1.732 |
| 90° | 1 | 0 | undefined |
Try It: Interactive Triangle Explorer
Adjust the angle and the hypotenuse length below. The triangle redraws, and sine, cosine and tangent update in real time — along with the actual side lengths. This is trigonometry as it happens.
Notice how the opposite side grows faster than the adjacent side as the angle increases — that's tangent climbing.
Sine, Cosine & Tangent Side by Side
Here's the full picture in one table — what each ratio compares, its range for acute angles, and what it's useful for.
| Ratio | Formula | Range (0°–90°) | Use it when you know… |
|---|---|---|---|
| Sine | opposite / hypotenuse | 0 → 1 | An angle and the hypotenuse, and want the opposite side |
| Cosine | adjacent / hypotenuse | 0 → 1 | An angle and the hypotenuse, and want the adjacent side |
| Tangent | opposite / adjacent | 0 → ∞ | An angle and one short side, and want the other |
Where Trigonometry Shows Up in Real Life
Trigonometry is not an abstract exercise — it's the maths of anything involving angles and distances.
- Surveying & navigation: finding the height of a mountain or the distance to a ship from a known baseline.
- Architecture & engineering: calculating roof slopes, ramp angles, and the forces in a bridge truss.
- Computer graphics & games: rotating objects in 2D and 3D, and simulating light and shadows.
- Sound & electricity: sine waves describe alternating current, sound waves, and musical tones.
- Astronomy: measuring distances to stars using parallax angles and the sine rule.
Tangent is just sine divided by cosine: tan θ = sin θ / cos θ. And the Pythagorean identity sin²θ + cos²θ = 1 falls straight out of the unit circle's radius‑1 definition.