IGCSE Trigonometry: Complete SOHCAHTOA Guide for Right-Angled Triangles

IGCSE trigonometry is one of the most important topics in Cambridge IGCSE Mathematics, and SOHCAHTOA is the foundation for solving right-angled triangle problems. Whether you’re preparing for Paper 2 or Paper 4, mastering trigonometry formulas is essential for achieving top grades.

This comprehensive IGCSE trigonometry guide covers everything you need to know about SOHCAHTOA, including sine, cosine, and tangent formulas, step-by-step worked examples, common exam questions, and expert tips from Tutopiya’s IGCSE maths tutors. We’ll also show you how to avoid the most common mistakes that cost students valuable marks.

🎯 What you’ll learn: By the end of this guide, you’ll know how to identify the correct trigonometry formula, solve right-angled triangle problems confidently, find missing angles and sides, and structure your working to earn full method marks in IGCSE exams.

Already studying with Tutopiya? Practice these skills with our dedicated IGCSE Trigonometry practice deck featuring exam-style questions and instant feedback.

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Why IGCSE Trigonometry and SOHCAHTOA Matter

IGCSE trigonometry appears in both Paper 2 and Paper 4 of the Cambridge IGCSE Mathematics exam. Here’s why it’s so important:

• High frequency topic: SOHCAHTOA questions appear in almost every IGCSE maths paper
• Multi-topic integration: Trigonometry connects algebra, geometry, and real-world applications
• Exam weight: Typically worth 8-12 marks per paper, making it crucial for grade boundaries
• Common applications: Used in angles of elevation, depression, 3D problems, and practical scenarios
• Foundation for advanced topics: Essential for understanding sine rule, cosine rule, and trigonometric graphs

Key insight from examiners: Students often lose marks not because they don’t understand trigonometry, but because they make simple errors in labeling sides or rounding too early. This guide will help you avoid these pitfalls.

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Identify the Sides First (The Golden Pre-Step)

1. Hypotenuse (H) – always opposite the right angle.
2. Opposite (O) – opposite the angle you’re measuring.
3. Adjacent (A) – the remaining side touching the angle.

Without this 3-second audit, it’s easy to substitute the wrong values and blow the question.

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SOHCAHTOA Formula Vault

Ratio	Teen-friendly Formula	Use When	Calculator Tip
Sine	sin θ = opposite ÷ hypotenuse	You know the opposite side and hypotenuse	SHIFT → sin⁻¹ to find an angle
Cosine	cos θ = adjacent ÷ hypotenuse	You know the adjacent side and hypotenuse	Keep calculator in degree mode
Tangent	tan θ = opposite ÷ adjacent	You know the opposite and adjacent sides	Great for slope/gradient problems

Memory hook: “Some Old Hens / Can’t Always Hide / Their Old Age” — more vivid than the classic chant.

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Step-by-Step Playbook (Guaranteed Method Marks)

1. Sketch + label the triangle, even if one is provided.
2. Circle the target: Find θ or Find length x.
3. Choose the ratio based on the two sides you know.
4. Write the formula before substituting numbers.
5. Solve algebraically, delaying rounding until the final step.
6. State answer with units (cm, m, degrees).

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Worked Example (Paper 4 Style)

A drone cable is secured to the top of a mast 18 m high. The cable forms a $62^\circ$ angle with the ground. Calculate the length of the cable.

1. Known sides: angle + Opposite (18 m). Need Hypotenuse.
2. Use sine: sin 62° = 18 ÷ H.
3. Rearrange: H = 18 ÷ sin 62° = 20.37 m.
4. Final answer (3 s.f.): 20.4 m.

Examiner win: writing the rearranged equation gets you the method mark even if you mis-key the calculator.

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Reverse Trigonometry: Finding an Angle

When the unknown is the angle, you need to use the inverse trigonometric functions:

θ = sin⁻¹(O/H)
θ = cos⁻¹(A/H)
θ = tan⁻¹(O/A)

Calculator steps:

1. Calculate the ratio (e.g., 5 ÷ 12 = 0.4167)
2. Press SHIFT then the appropriate function (sin⁻¹, cos⁻¹, or tan⁻¹)
3. Enter the ratio value
4. Press = to get the angle in degrees

Important checks:

• Angles in right-angled triangles are always between 0° and 90°
• If your answer is negative or greater than 90°, you’ve made an error
• Always round to the required precision (usually 1 decimal place or nearest degree)

Worked Example: Finding an Angle

A 5 m ladder leans against a wall. The foot of the ladder is 3 m from the wall. Calculate the angle the ladder makes with the ground.

Solution:

1. Label the triangle: We have adjacent (3 m) and hypotenuse (5 m)
2. Use cosine: cos θ = 3 ÷ 5 = 0.6
3. Find the angle: θ = cos⁻¹(0.6) = 53.1° (1 d.p.)

Exact Trigonometric Values (IGCSE Extended)

For IGCSE Extended Mathematics, you should memorize these exact values:

Angle	sin	cos	tan
30°	1/2	√3/2	1/√3
45°	1/√2	1/√2	1
60°	√3/2	1/2	√3

Memory tip: For 30° and 60°, the values swap between sine and cosine. For 45°, sine and cosine are equal.

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3D Trigonometry: Applying SOHCAHTOA in 3D Problems

3D trigonometry questions combine Pythagoras’ theorem with SOHCAHTOA. Here’s the systematic approach:

Step-by-Step Method for 3D Problems

1. Identify the right-angled triangle in the base (plan view)
2. Use Pythagoras to find the diagonal/base length
3. Create a new right-angled triangle using the vertical height and the diagonal you just found
4. Apply SOHCAHTOA to find the required angle or side
5. Quote intermediate values to 3 significant figures before proceeding

Worked Example: 3D Trigonometry

A rectangular box has dimensions 6 cm × 8 cm × 10 cm (length × width × height). Calculate the angle between the longest diagonal and the base.

Solution:

1. Find the diagonal of the base: d = √(6² + 8²) = √(36 + 64) = √100 = 10 cm
2. Create a right-angled triangle with:
   • Adjacent (base diagonal) = 10 cm
   • Opposite (height) = 10 cm
   • Hypotenuse = longest diagonal
3. Find the angle: tan θ = 10 ÷ 10 = 1, so θ = tan⁻¹(1) = 45°

Answer: The angle is 45°.

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Common Examiner Traps (and How to Dodge Them)

• Mixing up adjacent/opposite when the angle moves. Re-label every time.
• Premature rounding (e.g., using 7.4 instead of 7.39). Keep full calculator precision.
• Wrong calculator mode (RAD vs DEG). Double-check the display.
• Forgetting context – include direction/units, e.g., “The ladder is 3.6 m long.”

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IGCSE Trigonometry Practice Questions

Test your understanding with these exam-style IGCSE trigonometry questions:

Question 1: Finding a Side Length

A kite string is 42 m long and makes a 38° angle with the ground. Calculate the vertical height of the kite to 3 significant figures.

Solution:

• Identify: We have the hypotenuse (42 m) and need the opposite side
• Use: sin 38° = opposite ÷ 42
• Calculate: opposite = 42 × sin 38° = 25.9 m (3 s.f.)

Question 2: Finding an Angle

A ramp rises 1.2 m over a horizontal distance of 3.6 m. Calculate the incline angle to one decimal place.

Solution:

• Identify: We have opposite (1.2 m) and adjacent (3.6 m)
• Use: tan θ = 1.2 ÷ 3.6 = 0.333...
• Calculate: θ = tan⁻¹(0.333...) = 18.4° (1 d.p.)

Question 3: Angle of Depression

A security camera mounted at 5 m height observes a doorway 11 m away horizontally. Calculate the angle of depression to the nearest degree.

Solution:

• Identify: We have opposite (5 m) and adjacent (11 m)
• Use: tan θ = 5 ÷ 11 = 0.4545...
• Calculate: θ = tan⁻¹(0.4545...) = 24.6° ≈ 25° (nearest degree)

Question 4: Real-World Application (Paper 4 Style)

A ladder 6.5 m long leans against a wall. The foot of the ladder is 2.1 m from the base of the wall. Calculate: a) The height the ladder reaches up the wall b) The angle the ladder makes with the ground

Solution: a) Using Pythagoras: height = √(6.5² - 2.1²) = √(42.25 - 4.41) = √37.84 = 6.15 m (3 s.f.) b) Using trigonometry: cos θ = 2.1 ÷ 6.5 = 0.323..., so θ = cos⁻¹(0.323) = 71.1° (1 d.p.)

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Frequently Asked Questions About IGCSE Trigonometry

What is SOHCAHTOA in IGCSE trigonometry?

SOHCAHTOA is a mnemonic device that helps you remember the three main trigonometry formulas for right-angled triangles:

• SOH: Sine = Opposite ÷ Hypotenuse
• CAH: Cosine = Adjacent ÷ Hypotenuse
• TOA: Tangent = Opposite ÷ Adjacent

How do I know which trigonometry formula to use?

Identify which two sides you know (or the angle and one side), then choose the formula that uses those sides:

• Know opposite and hypotenuse? Use sine
• Know adjacent and hypotenuse? Use cosine
• Know opposite and adjacent? Use tangent

What’s the difference between finding a side and finding an angle?

• Finding a side: Use the formula directly (e.g., sin 30° = opposite ÷ 10)
• Finding an angle: Use the inverse function (e.g., θ = sin⁻¹(opposite ÷ hypotenuse))

Do I need to memorize exact trig values for IGCSE?

For Cambridge IGCSE Extended Mathematics, you should know the exact values for 30°, 45°, and 60° angles. However, for most SOHCAHTOA questions, you’ll use your calculator.

How do I avoid common IGCSE trigonometry mistakes?

1. Always label the sides (Opposite, Adjacent, Hypotenuse) before starting
2. Check your calculator is in degree mode (not radians)
3. Don’t round intermediate values - keep full calculator precision
4. Always include units in your final answer
5. Check your answer makes sense (angles in right-angled triangles are < 90°)

Is trigonometry tested in both IGCSE papers?

Yes, IGCSE trigonometry appears in both Paper 2 (calculator) and Paper 4 (extended). Paper 2 typically has shorter questions, while Paper 4 includes multi-step problems and real-world applications.

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Related IGCSE Maths Resources

Strengthen your IGCSE Mathematics preparation with these comprehensive guides:

• IGCSE Maths Revision Notes, Syllabus and Preparation Tips - Complete syllabus overview, topic breakdown, and revision strategies
• IGCSE Math Study Tips: Science-based Study Tools - Evidence-based techniques to master IGCSE Mathematics
• IGCSE Past Papers Guide - Access free IGCSE past papers and exam resources

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