Pythagoras and trigonometry: from 2D to 3D
These tools are the workhorses of geometry. Pythagoras finds missing sides in right-angled triangles; trigonometric ratios (sin, cos, tan) find missing angles or sides when an angle is involved.
Pythagoras' theorem
In a right-angled triangle with hypotenuse c and legs a, b: a² + b² = c².
Worked example: legs 6 and 8 → c² = 36 + 64 = 100 → c = 10.
Trigonometric ratios (SOH CAH TOA)
For a right-angled triangle with angle θ, opposite (O), adjacent A and hypotenuse (H):
- sin θ = O / H.
- cos θ = A / H.
- tan θ = O / A.
Worked example: θ = 30°, hypotenuse = 10. Find opposite.
- sin 30° = O/10 → O = 10 × 0.5 = 5.
Finding angles
Use inverse functions: sin⁻¹, cos⁻¹, tan⁻¹.
Worked example: in a triangle, opposite = 4, adjacent = 3 (right angle elsewhere). Find θ.
- tan θ = 4/3.
- θ = tan⁻¹(4/3) ≈ 53.1°.
Pythagoras in 3D
For diagonals of cuboids and 3D distances, apply Pythagoras twice (or use extended formula):
For a cuboid with dimensions a × b × c, the space diagonal d: d² = a² + b² + c².
Worked example: cuboid 3 × 4 × 12.
- d² = 9 + 16 + 144 = 169 → d = 13.
Trigonometry in 3D
Often involves identifying the right-angled triangle within a 3D figure, then applying SOH CAH TOA.
Worked example: angle between space diagonal and base of cuboid 3 × 4 × 12.
- Base diagonal = √(9 + 16) = 5.
- Space diagonal = 13.
- tan θ = 12/5 → θ = 67.4°.
⚠Common mistakes
- Mixing up O, A, H — H is always opposite the right angle.
- Using degrees mode incorrectly — calculator must be in DEG for GCSE.
- Calculator setting — sin⁻¹ vs sin (use INV/2nd key).
- Pythagoras with non-right-angled triangles — only works on right-angled. Use sine/cosine rule otherwise.
- Not labelling sides — write O, A, H on diagram first.
➜Try this— Quick check
Right-angled triangle with hypotenuse 5, opposite 3. Find θ.
- sin θ = 3/5 = 0.6 → θ = sin⁻¹(0.6) ≈ 36.87°.
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