Pythagoras and trigonometry

Maths · Geometry and measures. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

Both tools work only in right-angled triangles. Use Pythagoras' theorem when the question has three sides and no angle. Use the trigonometric ratios sin, cos and tan when an angle is given or asked for.

Label the sides before you start. The hypotenuse is the longest side, opposite the right angle. The opposite side is across from the angle you are using, and the adjacent side is next to it. Then choose the ratio that contains the two sides in the question.

In 3D, find a right-angled triangle inside the solid, draw it flat, and work as normal. You may need two steps. On the non-calculator paper you must know the exact values for 0°, 30°, 45°, 60° and 90°.

Rules to remember

Pythagoras: a² + b² = c², where c is the hypotenuse. For a shorter side, subtract: a² = c² − b².
sin θ = opposite ÷ hypotenuse; cos θ = adjacent ÷ hypotenuse; tan θ = opposite ÷ adjacent (SOH CAH TOA).
To find an angle use the inverse function, e.g. θ = tan⁻¹(opposite ÷ adjacent).
Diagonal of a cuboid: d² = length² + width² + height².
sin 0° = 0, sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2, sin 90° = 1.
cos 0° = 1, cos 30° = √3/2, cos 45° = √2/2, cos 60° = 1/2, cos 90° = 0.
tan 0° = 0, tan 30° = 1/√3 (the same as √3/3), tan 45° = 1, tan 60° = √3.

Common mistake

Adding the squares when finding a shorter side. If the hypotenuse is already known, subtract the squares, and check that your answer is shorter than the hypotenuse.

Worked example

A point on level ground is 20 m from the base of a vertical tower. The angle of elevation of the top of the tower from this point is 60°. Find the height of the tower.

The height is opposite the 60° angle and the 20 m is adjacent, so use tan.
tan 60° = height ÷ 20, so height = 20 × tan 60°.
tan 60° = √3, so height = 20√3 = 20 × 1.732... = 34.6 m (3 significant figures).

Answer: 20√3 m, which is 34.6 m to 3 significant figures

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