[Leejohnston]

Trigonometry Formula Sheet — SOH CAH TOA & More

A clear, simple, GCSE-friendly trigonometry reference sheet. 
Use this for triangles, angles, physics problems, and exam revision.

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1. The Trig Ratios (SOH CAH TOA)

Right-angled triangles only.

SOH → Sin = Opposite / Hypotenuse 
CAH → Cos = Adjacent / Hypotenuse 
TOA → Tan = Opposite / Adjacent

Example: 
sin(θ) = opposite / hypotenuse 
cos(θ) = adjacent / hypotenuse 
tan(θ) = opposite / adjacent

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2. Finding Missing Sides

If you know an angle + one side:

• Opposite side = sin(θ) × hypotenuse 
• Adjacent side = cos(θ) × hypotenuse 
• Opposite side = tan(θ) × adjacent 
• Adjacent side = opposite ÷ tan(θ)

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3. Finding Missing Angles

Use inverse trigonometry:

θ = sin⁻¹(opposite / hypotenuse) 
θ = cos⁻¹(adjacent / hypotenuse) 
θ = tan⁻¹(opposite / adjacent)

These buttons on calculators are usually: 
• SHIFT + sin 
• SHIFT + cos 
• SHIFT + tan

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4. Pythagoras’ Theorem

For right-angled triangles:

a² + b² = c² 
(c is the hypotenuse)

Examples: 
• 3² + 4² = 5² 
• 9 + 16 = 25 

Used when you know two sides and need the third.

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5. Exact Trig Values (Need-to-Know)

These come up in higher-tier questions:

sin(30°) = 1/2 
cos(60°) = 1/2 
sin(45°) = √2/2 
cos(45°) = √2/2 
tan(45°) = 1

Memorising these saves time in exams.

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6. Trig on the Unit Circle (Simple Version)

Useful for higher-level maths:

• sin = y-value 
• cos = x-value 
• tan = sin / cos 

Angles measured from the positive x-axis.

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7. Common Mistakes

❌ Using the wrong ratio (SOH/CAH/TOA) 
✔ Always label opposite, adjacent, hypotenuse first

❌ Forgetting to use SHIFT for angle finding 
✔ Use sin⁻¹, cos⁻¹, tan⁻¹

❌ Using Pythagoras on non-right-angled triangles 
✔ Only works for 90° triangles

❌ Mixing up sides in diagrams 
✔ Draw a quick sketch — it helps every time

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Summary

Know these essentials:
• SOH CAH TOA 
• Inverse trig to find angles 
• Pythagoras for side lengths 
• Exact values for 30°, 45°, 60°

Master these and almost every trigonometry question becomes easy.