Sheet № 49 · Foundation + Higher · AQA · Edexcel · OCR
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
§Key definitions
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.).
Height (opposite):
tan 70° = h/6, so h = 6 × tan 70° = 6 × 2.7475 = 16.5 m (1 d.p.).
§Formulas to memorise
\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})
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).
O and H → use sin
A and H → use cos
O and A → use tan
Worked example
See example below.
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.
⚠ 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.
✦ 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.