SOHCAHTOA: Everything You Need to Know for GCSE Maths
Page Contents
SOHCAHTOA: How to find missing sides and missing angles in right-angled triangles
This blog contains everything you need to know about SOHCAHTOA for GCSE Maths. We will explain what SOHCAHTOA means, why it is essential in GCSE exams, how it is used in right-angled triangles, and how to apply it confidently to find missing sides and angles.
SOHCAHTOA is a memory aid used to remember the three trigonometric ratios in a right-angled triangle. These ratios allow us to calculate unknown sides or angles using basic trigonometry. In GCSE exams (Edexcel, AQA and OCR), SOHCAHTOA regularly appears in calculator and non-calculator papers, often in geometry, worded problems, and multi-step questions involving bearings, 3D shapes, or real-life contexts.
SOHCAHTOA applies only to right-angled triangles.
The Three Trigonometric Ratios
\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}
\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}
\tan \theta = \frac{\text{opposite}}{\text{adjacent}}
To decide which ratio to use for each trig problem, we can memorise the three trigonometric ratios using the mnemonic SOH CAH TOA. The letters O, A, and H represent the opposite, adjacent sides and the hypotenuse.
These formulas allow students to solve problems where one side and one acute angle are known. Here at Think Academy, we also provide structured SOHCAHTOA worksheets based on GCSE-style exam questions, designed to develop fluency in recognising which ratio to use and presenting clear working for full marks. Come to try our online GCSE maths course for FREE, and get all the GCSE maths worksheets.
How to Identify the Sides of a Right-Angled Triangle
Before using SOHCAHTOA, you must correctly label the triangle.
Step 1: Identify the Hypotenuse
The hypotenuse is always:
- The longest side
- The side directly opposite the right angle
It does not depend on the given angle — it is fixed.
Step 2: Identify Opposite and Adjacent
These depend on the angle you are using.
- Opposite: the side directly across from the chosen angle
- Adjacent: the side next to the chosen angle (but not the hypotenuse)
A common GCSE mistake is labelling adjacent incorrectly by including the hypotenuse. The hypotenuse is never adjacent.
Using SOHCAHTOA to Find Unknown Sides
Method
- Label the triangle correctly.
- Choose the correct ratio.
- Substitute known values.
- Rearrange if needed.
- Round appropriately (usually 3 significant figures).
Worked Example 1
Angle = 30°, Hypotenuse = 12 cm
\sin 30^\circ = \frac{\text{opposite}}{12}
\text{opposite} = 12 \sin 30^\circ
\text{opposite} = 6 \text{ cm}
Worked Example 2
Angle = 40°, Adjacent = 7 cm
\cos 40^\circ = \frac{7}{\text{hypotenuse}}
\text{hypotenuse} = \frac{7}{\cos 40^\circ}
\text{hypotenuse} \approx 9.14 \text{ cm}
Worked Example 3
Angle = 50°, Opposite = 8 cm
\tan 50^\circ = \frac{8}{\text{adjacent}}
\text{adjacent} = \frac{8}{\tan 50^\circ}
\text{adjacent} \approx 6.71 \text{ cm}
Using SOHCAHTOA to Find Unknown Angles
Method
- Label the triangle.
- Choose the correct ratio.
- Use inverse functions.
- Ensure calculator is in degree mode.
- Round appropriately (usually 1 decimal place).
Similarly use SHIFT + \cos for \cos^{-1} and SHIFT + \tan for \tan^{-1} .
Worked Example 1
Opposite = 5 cm, Hypotenuse = 13 cm
\sin \theta = \frac{5}{13}
\theta = \sin^{-1}\left(\frac{5}{13}\right)
\theta \approx 22.6^\circ
Worked Example 2
Opposite = 7 cm, Adjacent = 4 cm
\tan \theta = \frac{7}{4}
\theta = \tan^{-1}\left(\frac{7}{4}\right)
\theta \approx 60.3^\circ
Common Mistakes in SOHCAHTOA
- Forgetting that SOHCAHTOA only applies to right-angled triangles
- Misidentifying the hypotenuse
- Choosing the wrong ratio (e.g., using sine instead of cosine)
- Using inverse functions when finding a side
- Leaving calculator in radians mode
- Rounding too early and losing accuracy
- Not showing working clearly (losing method marks)
Summary
SOHCAHTOA is one of the most important topics in GCSE Maths. It provides a systematic way to solve right-angled triangle problems involving missing sides and angles. Once students can confidently identify triangle sides and select the correct ratio, most GCSE trigonometry questions become straightforward.
If you would like structured practice, try our SOHCAHTOA worksheets based on GCSE exam-style questions. You can also come and try our online GCSE Maths course for FREE, where you’ll gain access to step-by-step explanations, exam-focused practice, and downloadable GCSE Maths worksheets to strengthen your understanding of SOHCAHTOA and beyond.
See how Think Academy helps achieve top grades in GCSE exams, and book a trial for free:
Read more:
Sine Rule: One of the Key GCSE Maths Topics You Must Know
Cosine Rule: Top GCSE Maths Topics to Master
