Pearson Edexcel International GCSE in Mathematics (Specification A) · 4MA1 · Free · no account needed

Right-Angled Trigonometry

In one sentence. Sine, cosine and tangent: finding sides and angles in right triangles, from ladders to angles of elevation — plus the exact values worth knowing.

Trigonometry — one of 4 modules in this strand, and one of 53 in Pearson Edexcel International GCSE Mathematics. Both tiers.

This module ends with Claudius Ptolemy. Learn Right-Angled Trigonometry, teach it, then defend it against Claudius Ptolemy.

Where this sits in the specification

Specification 4MA1, 4.8 Trigonometry. Incandio is aligned to this specification; it is not published by, endorsed by or affiliated with Pearson, and the sub-topic numbers are given so you can check every module against your own copy.

  • 4.8 — Trigonometry

1 · Understand it

What it is

In a right triangle, the ratios between sides depend only on the angle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent (SOH-CAH-TOA). Given an angle and a side, the ratios find the other sides; given two sides, the INVERSE functions (sin⁻¹ etc.) find the angle.

Why it works

All right triangles with the same acute angle are similar — same shape, different size — so their side RATIOS are identical. sin 30° = 0.5 means: in every right triangle with a 30° angle, the opposite side is exactly half the hypotenuse. The trig functions are just these universal ratios, tabulated.

When a question is asking for it

One angle + one side known → label the sides relative to the angle (opposite/adjacent/hypotenuse), pick the ratio naming the two sides involved, solve. Two sides known, angle wanted → same labelling, then the inverse function. Angles of elevation/depression and ladder problems are this in costume.

How to hold it in your head

Everything flows from LABELLING: the hypotenuse is opposite the right angle; the 'opposite' faces your angle; the 'adjacent' touches it. Label first, choose the ratio second, calculate third. Most trig errors are labelling errors in disguise.

2 · The mistakes that cost marks

  • Labelling 'opposite' relative to the wrong angle. Opposite and adjacent swap when you switch angles; the hypotenuse never moves.
  • Forgetting the inverse when finding an angle — reporting sin θ = 0.911 as '0.911°'. The angle needs sin⁻¹.
  • Calculator in radians. Check the D/DEG symbol before any trig work.
  • Rounding mid-calculation: keep full accuracy until the final line, or angles drift by whole degrees.

3 · Worked examples

Laid out the way an examiner wants working shown — every step is a mark.

A ladder 4.5 m long leans against a wall, reaching 4.1 m up. What angle does it make with the ground?

  1. Relative to the ground angle: opposite = 4.1 (the wall height), hypotenuse = 4.5 (the ladder).
  2. sin θ = 4.1/4.5 = 0.9111…
  3. θ = sin⁻¹(0.9111…) = 65.7° (1 d.p.).

Answer: 65.7°

From 50 m away from a tower's base, the angle of elevation of its top is 32°. Find the tower's height.

  1. Opposite = height h; adjacent = 50. Tangent names both: tan 32° = h/50.
  2. h = 50 × tan 32° = 31.2 m (1 d.p.).

Answer: 31.2 m

4 · Prove it — the five questions

The five questions climb Grade 6 → 9+ and are marked inside Incandio on your own device, against authored accepted forms, with a diagnosis of the wrong answer you actually gave. The answers stay in the app so that the practice is worth doing; the questions themselves are here.

  1. Grade 6 — In a right-angled triangle, the hypotenuse is 12 cm and one angle is 30°. Work out the length of the side OPPOSITE the 30° angle. (answer in cm)
  2. Grade 7 — In a right-angled triangle, the hypotenuse is 20 cm and one angle is 35°. Work out the side ADJACENT to the 35° angle, correct to 1 decimal place. (answer in cm)
  3. Grade 8 — A 4.5 m ladder leans against a vertical wall, reaching 4.1 m up the wall. Work out the angle between the ladder and the GROUND, correct to 1 decimal place. (answer in °)
  4. Grade 9 — From a point 50 m from the base of a tower, the angle of elevation of the top is 32°. Work out the height of the tower, correct to 1 decimal place. (answer in m)
  5. 9+ — Using exact values, work out tan 60° × sin 30°. Give your answer exactly. (Type it like sqrt3/2.)

Think you understand it?

Reading is the start. On Incandio a topic counts as mastered only once you have taught it and defended it — the two things a revision guide cannot ask of you.

Incandio this topic → Prove it. Teach it. Defend it. What does “Incandio it” mean?

  1. Prove it — five questions on Right-Angled Trigonometry, marked on the spot, that tell you what was missing rather than just what was wrong.
  2. Teach it — Explain how Right-Angled Trigonometry works — and why — to Ember, an AI pupil who does not have it yet. The method matters more than the vocabulary.
  3. Defend it — Argue Right-Angled Trigonometry against Claudius Ptolemy — three rounds, marked against descriptors you can read before you begin.
  4. Then the exam — read all 10 authored exam-style questions on this unit, each with its command word and its tariff, then have them marked on Incandio with the mistake you actually made named.

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