Pythagoras' theorem and trigonometry

These two topics appear on every OCR J560 paper. Higher-tier extensions include 3D Pythagoras, the sine rule, cosine rule and area = ½ab sin C (these are G22–G23, but the basics from G20 are foundational).

Pythagoras' theorem

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

Finding the hypotenuse: c = √(a² + b²). Example: a=3, b=4 → c = √(9+16) = √25 = 5.

Finding a shorter side: a = √(c² − b²). Example: c=13, b=5 → a = √(169−25) = √144 = 12.

Trigonometric ratios (SOHCAHTOA)

In a right-angled triangle:

• Sin θ = Opposite / Hypotenuse
• Cos θ = Adjacent / Hypotenuse
• Tan θ = Opposite / Adjacent

Memory: SOH CAH TOA.

Finding a side

Example: Find x if the hypotenuse is 10 cm and the angle is 35°. x is opposite: sin 35° = x/10 → x = 10 sin 35° ≈ 5.74 cm.

Finding an angle

Example: opposite = 7, hypotenuse = 11. sin θ = 7/11 → θ = sin⁻¹(7/11) ≈ 39.5°.

3D Pythagoras

To find a diagonal through a cuboid (length l, width w, height h):

1. Find the base diagonal: d = √(l² + w²).
2. Apply Pythagoras again: diagonal = √(d² + h²) = √(l² + w² + h²).

Example: cuboid 3 × 4 × 12. d = √(9+16) = 5; diagonal = √(25+144) = √169 = 13 cm.

Angles of elevation and depression

• Elevation: angle measured upwards from horizontal.
• Depression: angle measured downwards from horizontal.

These always form right-angled triangles with the horizontal.

Common OCR exam mistakes

1. Applying Pythagoras to non-right-angled triangles — check for the right angle before using a² + b² = c².
2. SOHCAHTOA: identifying the correct sides relative to the marked angle, not any angle.
3. Calculator mode: always ensure calculator is in degree mode (not radians) for GCSE.
4. 3D: finding the wrong diagonal — the space diagonal uses all three dimensions.