Pythagoras' theorem and right-angled triangles

• Pythagoras' theorem: a² + b² = c², where c is the hypotenuse.
• sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, tan θ = opposite ÷ adjacent.
• Use them to find sides and acute angles, and to solve problems in two dimensions, including bearings.
• The shortest distance from a point to a line is the perpendicular distance.
• Angles of elevation and depression are measured from the horizontal.
• Give angles in degrees correct to 1 decimal place, and lengths correct to 3 significant figures.

Square, add or subtract, square root

The hypotenuse is the longest side, opposite the right angle. To find the hypotenuse, add the squares of the other two sides. To find a shorter side, subtract.

SOH CAH TOA

Label the sides from the angle θ: opposite is across from θ, adjacent is next to θ (not the hypotenuse), hypotenuse is opposite the right angle. Choose the ratio that uses the side you know and the side you want. Check your calculator is in degrees.

The inverse functions

When you know two sides, find the ratio and then use sin⁻¹, cos⁻¹ or tan⁻¹. Write angles correct to 1 decimal place.

Find the right-angled triangle

Sketch the situation, mark the right angle and the angle you know or want, and label every length. Some problems need Pythagoras and trigonometry one after the other. Bearings questions give right-angled triangles when you draw north and east lines.