Trigonometry SOHCAHTOA –

Trigonometry using SOHCAHTOA is a core GCSE Maths topic that builds directly on Pythagoras' theorem. While Pythagoras lets you find a missing side when you know two sides, SOHCAHTOA lets you work with angles — finding a missing side when you know one side and an angle, or finding a missing angle when you know two sides. It appears on both Foundation and Higher papers across AQA, Edexcel and OCR, and the marks on offer are significant. This guide explains the three ratios, gives you a clear method for deciding which to use, provides fully worked examples at both tiers, and finishes with practice questions. For more detail, read our trigonometry blog post, and for a formula reference see our GCSE Maths formulas guide.

What Is SOHCAHTOA?

SOHCAHTOA is a mnemonic for the three trigonometric ratios in a right-angled triangle:

$$\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \quad (\text{SOH})$$

$$\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \quad (\text{CAH})$$

$$\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}} \quad (\text{TOA})$$

Labelling the Sides

Before using any ratio, label the three sides relative to the angle you are working with:

• Hypotenuse (H): the longest side, opposite the right angle.
• Opposite (O): the side directly across from the angle θ.
• Adjacent (A): the side next to the angle θ (that is not the hypotenuse).

Choosing the Right Ratio

Once you have labelled the sides, look at which two sides are involved (the one you know and the one you want). Choose the ratio that uses those two sides:

• O and H → use sin
• A and H → use cos
• O and A → use tan

Step-by-Step Method

Finding a Missing Side

1. Label the sides O, A, H relative to the given angle.
2. Identify which two sides are involved (one known, one unknown).
3. Choose the correct ratio (sin, cos or tan).
4. Substitute the known values.
5. Rearrange to isolate the unknown side.
6. Calculate using your calculator (make sure it is in degree mode).

Finding a Missing Angle

1. Label the sides O, A, H.
2. Identify which two sides are known.
3. Choose the correct ratio.
4. Substitute both known sides.
5. Use the inverse function (sin⁻¹, cos⁻¹ or tan⁻¹) to find the angle.
6. Round as instructed.

Worked Example 1 — Foundation Level

In a right-angled triangle, the angle is 35° and the hypotenuse is 10 cm. Find the length of the side opposite the 35° angle. Give your answer to 1 decimal place.

Step 1: Label: H = 10, O = ?, angle = 35°.

Step 2: O and H → use sin.

Step 3: sin 35° = O/10.

Step 4: O = 10 × sin 35° = 10 × 0.5736 = 5.7 cm (1 d.p.).

Finding an Angle

A right-angled triangle has an opposite side of 7 cm and an adjacent side of 4 cm. Find the angle.

Step 1: O = 7, A = 4 → use tan.

Step 2: tan θ = 7/4 = 1.75.

Step 3: θ = tan⁻¹(1.75) = 60.3° (1 d.p.).

Worked Example 2 — Higher Level

A right-angled triangle has an angle of 52° and the side adjacent to this angle is 8 cm. Find the hypotenuse. Give your answer to 3 significant figures.

Step 1: A = 8, H = ?, angle = 52°.

Step 2: A and H → use cos.

Step 3: cos 52° = 8/H.

Step 4: H = 8 ÷ cos 52° = 8 ÷ 0.61566 = 12.99 = 13.0 cm (3 s.f.).

Multi-Step Problem

A vertical flagpole is supported by a wire attached to the top of the pole and anchored 6 m from the base. The wire makes an angle of 70° with the ground. Find the height of the flagpole and the length of the wire.

Height (opposite): tan 70° = h/6, so h = 6 × tan 70° = 6 × 2.7475 = 16.5 m (1 d.p.).

Wire (hypotenuse): cos 70° = 6/w, so w = 6 ÷ cos 70° = 6 ÷ 0.3420 = 17.5 m (1 d.p.).

Common Mistakes

• Mislabelling the sides. The opposite and adjacent depend on which angle you are working with. If you switch to a different angle, the labels change.
• Calculator in radian mode. At GCSE, angles are in degrees. Check your calculator shows "D" or "DEG", not "R" or "RAD".
• Using the wrong ratio. If you mix up sin and cos, the answer will be wrong. Double-check your side labels.
• Forgetting to use the inverse function for angles. If tan θ = 1.75, the angle is tan⁻¹(1.75), not just 1.75.
• Rounding too early. Keep the full value from your calculator until the final step.
• Applying SOHCAHTOA to non-right-angled triangles. SOHCAHTOA only works when there is a 90° angle. For other triangles, use the sine or cosine rule.

Exam Tips

• Label O, A, H on the diagram before doing anything else. This takes five seconds and prevents the most common errors.
• Write the ratio and formula — examiners award a method mark for stating, for example, sin 35° = O/H.
• Show rearrangement clearly. If sin θ = O/H, then O = H × sin θ. Write this step.
• For elevation and depression problems, draw a clear right-angled triangle and mark the angle.
• Exact values (Higher): know that sin 30° = 0.5, cos 60° = 0.5, tan 45° = 1, sin 45° = √2/2, sin 60° = √3/2, and so on.
• Pythagoras or trig? If you have two sides and want the third, use Pythagoras. If an angle is involved (given or needed), use trigonometry.

Practice Questions

Question 1 (Foundation) In a right-angled triangle, the hypotenuse is 13 cm and the side opposite angle θ is 5 cm. Find angle θ.

Question 2 (Foundation) Find the adjacent side when the angle is 40° and the hypotenuse is 15 cm.

Question 3 (Higher) A ramp rises 2.5 m over a horizontal distance of 8 m. Find the angle the ramp makes with the horizontal.

Question 4 (Higher) From the top of a 40 m cliff, the angle of depression to a boat is 28°. How far is the boat from the base of the cliff?

Question 5 (Higher) Find the exact value of sin 60° × cos 30° + cos 60° × sin 30°.

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Related Topics

• Pythagoras' Theorem — used alongside trig in right-angled triangle problems
• Trigonometry: Sine and Cosine Rules — for non-right-angled triangles (Higher)
• Bearings — often combined with trigonometry
• Angles in Polygons — understanding angle relationships
• Exact Trigonometric Values — required for Higher tier

Summary

SOHCAHTOA provides three ratios — sin, cos and tan — for working with right-angled triangles. Label the sides O, A, H relative to your angle, choose the ratio that involves the known and unknown sides, substitute, and solve. For finding a side, rearrange and calculate. For finding an angle, use the inverse function (sin⁻¹, cos⁻¹, tan⁻¹). Always check your calculator is in degree mode, show your working clearly, and remember that SOHCAHTOA only applies to right-angled triangles. This is a high-value topic that rewards careful labelling and systematic method.