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AQASpec G20-G22Foundation & Higher~35 min

Pythagoras & Trigonometry

Mathematics · Topic revision workspace

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Topic overview

What you need to know

Pythagoras & Trigonometry is a key part of GCSE Mathematics. Build fluency with the method, then apply it to unfamiliar and multi-step problems. In a right-angled triangle: a² + b² = c², where c is the hypotenuse (the longest side, opposite the right angle). The longest side of a right-angled triangle, opposite the right angle. Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent. Used to find angles and sides in right-angled triangles.

Exam tip

Write every stage of your method. Method marks can still be earned when the final answer is wrong.

Revision notes

Core ideas

Secure these ideas first. Say each definition in your own words, then connect it to the topic overview.

  • Pythagoras' Theorem: In a right-angled triangle: a² + b² = c², where c is the hypotenuse (the longest side, opposite the right angle).
  • Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.
  • SOH CAH TOA: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent. Used to find angles and sides in right-angled triangles.
  • Sine (sin θ): The ratio of the opposite side to the hypotenuse in a right-angled triangle. sin θ = O/H.

Revision notes

Apply it in the exam

The exam will rarely ask for an isolated definition. Practise selecting the right idea and using it as part of a complete explanation or method.

  • Cosine (cos θ): The ratio of the adjacent side to the hypotenuse in a right-angled triangle. cos θ = A/H.
  • Tangent (tan θ): The ratio of the opposite side to the adjacent side in a right-angled triangle. tan θ = O/A.
  • Sine Rule: a/sin A = b/sin B = c/sin C. Used in non-right-angled triangles when you know a side-angle pair.
  • Cosine Rule: a² = b² + c² − 2bc cos A. Used when you know two sides and the included angle (or all three sides).

Core knowledge

Key facts for Pythagoras & Trigonometry

1

Pythagoras' Theorem

In a right-angled triangle: a² + b² = c², where c is the hypotenuse (the longest side, opposite the right angle).

2

Hypotenuse

The longest side of a right-angled triangle, opposite the right angle.

3

SOH CAH TOA

Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent. Used to find angles and sides in right-angled triangles.

4

Sine (sin θ)

The ratio of the opposite side to the hypotenuse in a right-angled triangle. sin θ = O/H.

5

Cosine (cos θ)

The ratio of the adjacent side to the hypotenuse in a right-angled triangle. cos θ = A/H.

6

Tangent (tan θ)

The ratio of the opposite side to the adjacent side in a right-angled triangle. tan θ = O/A.

7

Sine Rule

a/sin A = b/sin B = c/sin C. Used in non-right-angled triangles when you know a side-angle pair.

8

Cosine Rule

a² = b² + c² − 2bc cos A. Used when you know two sides and the included angle (or all three sides).

Active recall

Close the notes and answer these

  1. 1.Without looking, explain pythagoras' theorem and give one example or consequence.
  2. 2.Without looking, explain hypotenuse and give one example or consequence.
  3. 3.Without looking, explain soh cah toa and give one example or consequence.
  4. 4.Without looking, explain sine (sin θ) and give one example or consequence.

Content reviewed 23 July 2026 against the current linked specification.

Finished learning?

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