Pythagoras and trigonometry

Pythagoras' theorem

In any right-angled triangle: a² + b² = c² where c is the hypotenuse.

Finding the hypotenuse: c = √(a² + b²). Finding a shorter side: a = √(c² − b²).

Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25.

3D Pythagoras (Edexcel Higher): the space diagonal of a cuboid l × w × h is d = √(l² + w² + h²). Often tested as a two-step Pythagoras in a 3D context (e.g. diagonal of the base, then use that as a leg).

SOHCAHTOA

Label sides relative to the angle θ (NOT relative to the right angle):

• sin θ = Opposite / Hypotenuse
• cos θ = Adjacent / Hypotenuse
• tan θ = Opposite / Adjacent

Finding a side: choose the correct ratio; rearrange algebraically. Example: cos 40° = adj/15 → adj = 15 cos 40°.

Finding an angle: use inverse trig. Example: tan θ = 5/12 → θ = tan⁻¹(5/12) ≈ 22.6°.

Exact trigonometric values (Edexcel Paper 1 requirement)

Edexcel tests these without a calculator on Paper 1:

These come from the 30-60-90 triangle (sides 1, √3, 2) and the 45-45-90 triangle (sides 1, 1, √2).

Angles of elevation and depression

Appear in Edexcel context problems involving buildings, cliffs, flagpoles.

• Elevation: angle above horizontal.
• Depression: angle below horizontal.

Both are measured from a horizontal line.

Trigonometry in 3D (Higher)

A common Edexcel Higher question: find the angle between a line and a plane (e.g. the angle a diagonal makes with the base of a cuboid). Strategy: identify the right-angled triangle, label sides, apply SOHCAHTOA.