Trigonometry can sound intimidating at first, but SOH CAH TOA is actually one of the simplest and most useful formulas in GCSE Maths. Once you understand it, many exam questions become much easier and quicker to solve.

This guide explains SOH CAH TOA step by step, using plain English, clear diagrams (described) and worked examples so you can feel confident using it in exams.

What does SOH CAH TOA mean?

SOH CAH TOA is a memory aid that helps you remember three trigonometry formulas used in right-angled triangles.

It stands for:

• SOH → Sin = Opposite ÷ Hypotenuse
• CAH → Cos = Adjacent ÷ Hypotenuse
• TOA → Tan = Opposite ÷ Adjacent

These formulas help you find missing sides or angles in a right-angled triangle.

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What is a right-angled triangle?

A right-angled triangle is a triangle where one angle is exactly 90°.

Important parts of the triangle:

• Hypotenuse → the longest side, opposite the right angle
• Opposite → the side opposite the angle you are working with
• Adjacent → the side next to the angle (not the hypotenuse)

When do I use SOH CAH TOA?

You use SOH CAH TOA when:

• The triangle is right-angled
• You are given one side and one angle
• You need to find a missing side or angle

How do I know which formula to use?

Ask yourself two questions:

1. Which side do I know? (Opposite, Adjacent, or Hypotenuse)
2. Which side do I want to find?

Then choose:

• Sin if it involves Opposite and Hypotenuse
• Cos if it involves Adjacent and Hypotenuse
• Tan if it involves Opposite and Adjacent

Why Do We Put the “Missing Side” in the Formula?

Think of SOH CAH TOA like a treasure map. If you want to find the hidden treasure (the missing side), the map has to show where the treasure is, otherwise you would wander forever! It is the same in maths: the side you want to find has to be in the formula so you can use the angle and the other side to work it out. If it was not there, there would be nothing to solve. So when we pick sin, cos, or tan, we choose the one that includes the side we know and the side we want, then rearrange the equation to uncover the answer, like solving a fun little mystery.

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Worked Example 1: Finding a side using SIN

Question:
A right-angled triangle has an angle of 30°.
The hypotenuse is 10 cm.
Find the opposite side.

Step 1: Choose the formula

We are using:

• Opposite
• Hypotenuse

So we use SIN.

Sin = Opposite ÷ Hypotenuse

Step 2: Write the calculation

$$\sin (30\mathrm{°})=\frac{\text{Opposite}}{10}$$​

Step 3: Rearrange

$$\text{Opposite}=10\times \sin (30\mathrm{°})$$

Step 4: Calculate

$$\sin (30\mathrm{°})=0.5$$$$\text{Opposite}=10\times 0.5=5.00$$

✅ Answer:

The opposite side is 5.00 cm

Worked Example 2: Finding a side using COS

Question:
A right-angled triangle has an angle of 60°.
The hypotenuse is 12 cm.
Find the adjacent side.

Step 1: Choose the formula

We are using:

• Adjacent
• Hypotenuse

So we use COS.

Cos = Adjacent ÷ Hypotenuse

Step 2: Write the calculation

$$\cos (60\mathrm{°})=\frac{\text{Adjacent}}{12}$$

Step 3: Rearrange

$$\text{Adjacent}=12\times \cos (60\mathrm{°})$$

Step 4: Calculate

$$\cos (60\mathrm{°})=0.5$$$$\text{Adjacent}=12\times 0.5=6.00$$

✅ Answer:

The adjacent side is 6.00 cm

Worked Example 3: Finding a side using TAN

Question:
A right-angled triangle has an angle of 45°.
The adjacent side is 8 cm.
Find the opposite side.

Step 1: Choose the formula

We are using:

• Opposite
• Adjacent

So we use TAN.

Tan = Opposite ÷ Adjacent

Step 2: Write the calculation

$$\tan (45\mathrm{°})=\frac{\text{Opposite}}{8}$$

Step 3: Rearrange

$$\text{Opposite}=8\times \tan (45\mathrm{°})$$

Step 4: Calculate

$$\tan (45\mathrm{°})=1$$$$\text{Opposite}=8\times 1=8.00$$

✅ Answer:

The opposite side is 8.00 cm

Worked Example 4: Finding a side with SIN (rounded answer)

Question:
A right-angled triangle has an angle of 37°.
The hypotenuse is 12 cm.
Find the opposite side, rounded to 2 decimal places.

Step 1: Choose the formula

We are using:

• Opposite
• Hypotenuse

So we use SIN.

Sin = Opposite ÷ Hypotenuse

Step 2: Write the calculation

$$\sin (37\mathrm{°})=\frac{\text{Opposite}}{12}$$

Step 3: Rearrange

$$\text{Opposite}=12\times \sin (37\mathrm{°})$$

Step 4: Calculate using a calculator

Make sure your calculator is in Degrees (DEG).

$$\sin (37\mathrm{°})\approx 0.6018$$$$\text{Opposite}=12\times 0.6018=7.2216$$

Rounded to 2 decimal places:$$\text{Opposite}=7.22\text{ cm}$$

✅ Answer:

The opposite side is 7.22 cm

Common mistakes students make with SOH CAH TOA

• Forgetting to check the triangle is right-angled
• Choosing the wrong side as opposite or adjacent
• Not putting the calculator in degrees
• Forgetting to round answers correctly

How do I remember SOH CAH TOA?

Popular memory tricks include:

• “Some Old Horses Can Always Trot On”
• Writing it on your exam paper as soon as the test starts
• Practicing one example of SIN, COS and TAN every revision session

Why is SOH CAH TOA important for GCSE Maths?

SOH CAH TOA:

• Comes up in GCSE exams
• Is often easy marks if you know the steps
• Is essential for higher-tier questions
• Builds foundations for A-Level Maths and Physics

Final tip from Apollo Scholars

If you ever feel stuck:

1. Draw the triangle
2. Label the sides
3. Write out SOH CAH TOA
4. Take it one step at a time

With practice, trigonometry becomes one of the most reliable topics to score marks in GCSE Maths.