Sheet № 109 · Foundation + Higher · AQA · Edexcel · OCR
SOHCAHTOA Finding Missing Sides –
Finding a missing side using SOHCAHTOA is one of the most important trigonometry skills in GCSE Maths. You need one angle and one side of a right-angled triangle, and the correct trig ratio does the rest.
§Key definitions
Question:
In a right-angled triangle, the angle is 40° and the hypotenuse is 12 cm. Find the side opposite the 40° angle. Give your answer to 1 decimal place.
Answer:
O = 7.7 cm (1 d.p.).
Q1 (Foundation):
The angle is 55° and the adjacent side is 10 cm. Find the opposite side to 1 d.p.
Q2 (Foundation):
The angle is 30° and the hypotenuse is 20 cm. Find the adjacent side.
Q3 (Higher):
The angle is 48° and the opposite side is 9 cm. Find the adjacent side to 1 d.p.
§Formulas to memorise
sin θ = Opposite ÷ Hypotenuse (SOH)
cos θ = Adjacent ÷ Hypotenuse (CAH)
tan θ = Opposite ÷ Adjacent (TOA)
O = 12 × sin 40°
O = 12 × 0.6428
H = 15 ÷ sin 62°
H = 15 ÷ 0.8829
The ramp is the hypotenuse (H = 8 m). The vertical rise is opposite the 25° angle (O = ?).
O = 8 × sin 25°
O = 8 × 0.4226
Worked example
In a right-angled triangle, the angle is 40° and the hypotenuse is 12 cm. Find the side opposite the 40° angle. Give your answer to 1 decimal place.
Working: Label: H = 12, O = ?, angle = 40°. O and H are involved, so use sin (SOH). sin 40° = O ÷ 12 O = 12 × sin 40° O = 12 × 0.6428
⚠ Common mistakes
- ✗Mislabelling O and A. The opposite and adjacent depend entirely on which angle you are working with. If you switch to a different angle, the labels swap.
- ✗Multiplying when you should divide (or vice versa). If the unknown is on the bottom of the fraction, you need to divide. If it is on the top, you multiply. Use a formula triangle if this helps.
- ✗Calculator in radian mode. GCSE questions use degrees. Check your calculator display shows D or DEG, not R or RAD. An answer like 0.12 instead of 7.7 is a sign your calculator is in radians.
✦ Exam tips
- →Write the ratio name (e.g. "sin 40° = O/H") before substituting — this earns a method mark even if your final answer has a rounding error.
- →If the unknown side is the hypotenuse, you will always divide by the trig value.
- →If the unknown side is O or A, you will always multiply.
- →Show every line of working: label, ratio, substitution, calculation, answer.