How to Solve: Pythagoras’ Theorem & Trigonometry (SOH CAH TOA, 3D) – GCSE/A-Level Maths Guide

How to Solve: Pythagoras’ Theorem & Trigonometry (SOH CAH TOA, 3D) – GCSE/A-Level Maths Guide

Introduction

Mastering Pythagoras and SOH CAH TOA unlocks 10-15% of your GCSE Maths exam marks—and real-life problems like measuring roofs, navigation, or even gaming physics. One question could be the difference between a Grade 5 and a Grade 7.

What You Need To Know First

1. Right-angled triangles: A triangle with one 90° angle.
2. Labelling sides: Opposite, adjacent, and hypotenuse (longest side).
3. Basic algebra: Rearranging equations to solve for unknowns.

Key Vocabulary

Formulas To Know

1. Pythagoras’ Theorem

Formula: a² + b² = c² - a and b = shorter sides (legs). - c = hypotenuse (longest side). MEMORISE THIS – Not given on exam sheets.

2. SOH CAH TOA (Trigonometry Ratios)

Formulas: - Sine (SOH): sin(θ) = Opposite / Hypotenuse - Cosine (CAH): cos(θ) = Adjacent / Hypotenuse - Tangent (TOA): tan(θ) = Opposite / Adjacent MEMORISE THIS – Not given on exam sheets.

3. 3D Pythagoras (Space Diagonal)

Formula: d = √(a² + b² + c²) - d = space diagonal (longest diagonal in a cuboid). - a, b, c = length, width, height. MEMORISE THIS – Sometimes given, but not always.

Step-by-Step Method

For 2D Pythagoras:

1. Identify the right angle – Look for the 90° symbol.
2. Label the sides – Hypotenuse (c) is opposite the right angle. The other two sides are a and b.
3. Write the formula – a² + b² = c².
4. Substitute the known values – Plug in the numbers.
5. Solve for the unknown – Square, add/subtract, then square root.
6. Check units – If lengths are in cm, answer must be in cm.

For SOH CAH TOA:

1. Label the triangle – Mark the angle (θ), opposite, adjacent, and hypotenuse.
2. Choose the correct ratio – SOH, CAH, or TOA based on the sides you have.
3. Write the formula – e.g., sin(θ) = Opposite / Hypotenuse.
4. Substitute the known values – Plug in the numbers.
5. Rearrange to solve – Use inverse sine (sin⁻¹) if finding an angle.
6. Check your calculator mode – Must be in degrees (DEG) for GCSE/A-Level.

For 3D Pythagoras:

1. Draw the shape – Label all given lengths (e.g., length, width, height).
2. Find the 2D diagonal first – Use Pythagoras on the base (e.g., √(a² + b²)).
3. Use the 2D diagonal as one side – Now apply Pythagoras again with the height (c).
4. Write the full formula – d = √(a² + b² + c²).
5. Calculate and simplify – Square, add, then square root.

Worked Examples

Example 1 – Basic Pythagoras

Question: A right-angled triangle has sides 3 cm and 4 cm. Find the hypotenuse.

Solution: 1. Label sides: a = 3, b = 4, c = ?. 2. Write formula: 3² + 4² = c². 3. Substitute: 9 + 16 = c². 4. Add: 25 = c². 5. Square root: c = √25 = 5 cm.

What we did and why: We used Pythagoras because it’s a right-angled triangle. We squared the two shorter sides, added them, and took the square root to find the hypotenuse.

Example 2 – Medium SOH CAH TOA

Question: In a right-angled triangle, the hypotenuse is 10 cm, and the angle is 30°. Find the opposite side.

Solution: 1. Label sides: Opposite = ?, Hypotenuse = 10, Angle = 30°. 2. Choose ratio: SOH (sin(θ) = Opposite / Hypotenuse). 3. Write formula: sin(30°) = Opposite / 10. 4. Substitute: 0.5 = Opposite / 10 (since sin(30°) = 0.5). 5. Rearrange: Opposite = 10 × 0.5 = 5 cm.

What we did and why: We used SOH because we had the hypotenuse and needed the opposite side. We substituted the known values and solved for the unknown.

Example 3 – Exam-Style 3D Pythagoras

Question: A cuboid has length 3 cm, width 4 cm, and height 12 cm. Find the space diagonal.

Solution: 1. Draw the cuboid and label sides: a = 3, b = 4, c = 12. 2. Find the 2D diagonal of the base: √(3² + 4²) = √(9 + 16) = √25 = 5 cm. 3. Use the 2D diagonal as one side: d = √(5² + 12²). 4. Calculate: d = √(25 + 144) = √169 = 13 cm.

What we did and why: We used 3D Pythagoras because we needed the longest diagonal in a 3D shape. We first found the 2D diagonal, then used it with the height to find the space diagonal.

Common Mistakes

Exam Traps

1-Minute Recap

"Listen up—this is your last-minute cheat sheet for Pythagoras and trig!

1. Pythagoras: a² + b² = c². Hypotenuse is always c. Square, add, square root.
2. SOH CAH TOA: Label your triangle first. SOH = Opposite/Hypotenuse, CAH = Adjacent/Hypotenuse, TOA = Opposite/Adjacent. Use inverse (sin⁻¹, cos⁻¹, tan⁻¹) to find angles.
3. 3D Pythagoras: Find the 2D diagonal first, then use it with the height. Formula: d = √(a² + b² + c²).
4. Common traps: Check calculator mode (DEG!), label sides correctly, and don’t forget units.
5. Exam tip: If stuck, draw the triangle and write down what you know. Most marks come from setting it up right!

You’ve got this—go smash that exam!