Trigonometry connects the angles of a right triangle to the lengths of its sides. That link is surprisingly powerful: it lets you find a height you can't reach, a distance you can't walk, or an angle you can't measure directly. It underpins navigation, engineering, physics and computer graphics. This lesson introduces the three core ratios that make it work.
Trigonometry starts with a right triangle (one 90° angle). Pick one of the other angles and call it θ (theta). Relative to θ, the three sides get names: the hypotenuse is the longest side, always opposite the right angle; the opposite side is across from θ; and the adjacent side is next to θ (but not the hypotenuse).
Which side is 'opposite' and 'adjacent' depends on which angle you chose as θ — the hypotenuse never changes, but the other two swap if you switch angles.
For a given angle, the ratio of two sides is always the same, no matter how big the triangle is. There are three key ratios, remembered as SOH-CAH-TOA: sin θ = Opposite ÷ Hypotenuse, cos θ = Adjacent ÷ Hypotenuse, and tan θ = Opposite ÷ Adjacent.
So for the triangle with opposite 3, adjacent 4 and hypotenuse 5: sin θ = 3/5 = 0.6, cos θ = 4/5 = 0.8, and tan θ = 3/4 = 0.75. Your calculator's sin, cos and tan buttons give these ratios for any angle.
These ratios let you find sides and angles you can't measure. If you know an angle and one side, you can find another side by rearranging the right ratio. If you know two sides, you can find the angle using the inverse buttons on a calculator (sin⁻¹, cos⁻¹, tan⁻¹).
This is how surveyors find the height of a mountain from a distance and an angle, and how a game engine works out where to draw things on screen. Three simple ratios open the door to a huge amount of real-world maths.
Try these. Pick an answer to see whether it's right and why.