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SOH-CAH-TOA: sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, and tan θ = opposite ÷ adjacent. Notes, worked examples and 11 questions answered.
Trigonometry Review is part of Sine and cosine rules in scalene triangles in GCSE AQA Higher Further Maths.
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SOH-CAH-TOA: sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, and tan θ = opposite ÷ adjacent. Notes, worked examples and 11 questions answered.
Use this lesson to stay inside Sine and cosine rules in scalene triangles while connecting the detail back to the wider Geometry topic.
Read the core explanations from Trigonometry Review before testing yourself from memory.
Architects calculate roof angles, surveyors measure land, and engineers design bridges — all using the same three ratios you are about to review. Trigonometry turns a triangle with one known angle and one known side into a fully solved shape.
The key insight is this: in any right-angled triangle, once you fix one of the acute angles, the ratios of the sides are always the same, no matter how big or small the triangle is. That fixed relationship is what sine, cosine, and tangent capture.
Before choosing a ratio, you must label the three sides correctly relative to the angle you are working with — call it $\theta$. The side opposite the right angle is always the hypotenuse (H). The side directly across from $\theta$ is the opposite (O). The side touching $\theta$ that is not the hypotenuse is the adjacent (A).
A common trap: students label the adjacent and opposite correctly for one angle, then forget to re-label when the question switches to the other acute angle. Always re-label from scratch for each new $\theta$.
The mnemonic SOH-CAH-TOA encodes all three ratios. Each chunk tells you which two sides form the ratio for that function.
Choosing the right ratio is a two-step decision: first identify which two sides are involved (known and unknown), then pick the ratio that connects exactly those two sides. If the hypotenuse is not involved at all, tangent is your ratio.
When a side is unknown, write the chosen ratio as a fraction, substitute the known values, then rearrange to isolate the unknown. The rearrangement follows one of two patterns: multiply across if the unknown is on top, or divide if the unknown is on the bottom.
Worked example: Finding the Opposite Side Using Sine
Triangle $ABC$ has a right angle at $C$. Angle $A = 35^\circ$ and the hypotenuse $AB = 12$ cm. Calculate the length of $BC$ (the side opposite angle $A$). Give your answer to 1 decimal place.
Answer: $BC = 6.9$ cm (to 1 decimal place)
Worked example: Finding the Adjacent Side Using Cosine
In right-angled triangle $PQR$, the right angle is at $R$. Angle $P = 52^\circ$ and the hypotenuse $PQ = 9$ cm. Find the length of $PR$ (adjacent to angle $P$). Give your answer to 1 decimal place.
Answer: $PR = 5.5$ cm (to 1 decimal place)
Worked example: Finding the Hypotenuse When It Is in the Denominator
A right-angled triangle has an angle of $40^\circ$. The side opposite this angle is $7$ cm. Find the hypotenuse. Give your answer to 2 decimal places.
Answer: $H = 10.89$ cm (to 2 decimal places)
When both relevant sides are known and the angle is unknown, you reverse the process using an inverse trig function. On your calculator, these are labelled $\sin^{-1}$, $\cos^{-1}$, and $\tan^{-1}$ — sometimes accessed via a SHIFT or 2nd key.
The inverse function undoes the ratio: if $\sin\theta = 0.5$, then $\theta = \sin^{-1}(0.5) = 30^\circ$. You are asking the calculator: "which angle produces this ratio?"
Worked example: Finding an Unknown Angle Using Tangent
A right-angled triangle has an opposite side of $8$ cm and an adjacent side of $5$ cm. Find the angle $\theta$ between the adjacent side and the hypotenuse. Give your answer to 1 decimal place.
Answer: $\theta = 58.0^\circ$ (to 1 decimal place)
Now that all three ratios and both directions (finding sides and finding angles) have been covered, here are four problems to consolidate the method. For each one, start by labelling the sides before picking a ratio.
Trigonometry questions are reliable mark-earners once the method is secure — but examiners know exactly where students drop marks. The guidance below targets the most frequent errors seen in GCSE mark schemes.
One final check worth building into every answer: does the size of your answer make sense? The hypotenuse must always be the longest side. An angle must sit between $0^\circ$ and $90^\circ$. If either condition is violated, revisit the labelling before moving on.
Published by Revise Right. Use the official specification reference to check the requirements for your course. Content updated: 2026-03-14.
Move to the related lessons or back to the topic page when you need the wider sequence before exam-style practice.
SOH-CAH-TOA is the mnemonic for the three trigonometric ratios in a right-angled triangle: sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, and tan θ = opposite ÷ adjacent. Pick the ratio that connects the two sides involved; if the hypotenuse is not involved, use tangent.
Relative to the angle θ, the hypotenuse is the side opposite the right angle, the opposite is the side directly across from θ, and the adjacent is the side touching θ that is not the hypotenuse. Re-label from scratch when the question switches to the other acute angle.
Tan is used when the two sides involved are the opposite and the adjacent, so the hypotenuse is not involved at all. SOH-CAH-TOA gives tan θ = opposite ÷ adjacent.
Choose the ratio, substitute the known values, then rearrange. If angle A = 35° and the hypotenuse is 12 cm, sin 35° = BC ÷ 12, so BC = 12 × sin 35° ≈ 6.9 cm. Multiply across when the unknown is on top of the fraction.
When the hypotenuse is the unknown in the denominator, multiply both sides by it, then divide. For an angle of 40° with opposite side 7 cm, sin 40° = 7 ÷ H, so H = 7 ÷ sin 40° ≈ 10.89 cm, which is the longest side, as it must be.
A missing angle is found by working out the ratio of the two known sides, then using the inverse function, such as tan⁻¹, on a calculator. For an opposite side of 8 cm and adjacent side of 5 cm, tan θ = 8 ÷ 5 = 1.6, so θ = tan⁻¹(1.6) ≈ 58.0°.
An inverse trig function, sin⁻¹, cos⁻¹ or tan⁻¹, undoes the ratio to find the angle that produces it. For example, if sin θ = 0.5, then θ = sin⁻¹(0.5) = 30°; on a calculator these are often accessed via a SHIFT or 2nd key.
The opposite and adjacent sides depend on which angle θ is being used, so they must be labelled again from scratch when a question switches to the other acute angle. Only the hypotenuse, opposite the right angle, stays the same.
A trigonometry answer is checked by making sure the hypotenuse is the longest side and any angle in a right-angled triangle lies between 0° and 90°. For example, 7 ÷ sin 40° ≈ 10.89 cm is longer than the 7 cm side, as a hypotenuse must be.
The opposite side is found using sine when the hypotenuse is known. With a hypotenuse of 12 cm and angle 35°, sin 35° = BC ÷ 12, so BC = 12 × sin 35° ≈ 6.9 cm to 1 decimal place.
In any right-angled triangle, once one of the acute angles is fixed, the ratios of the sides are always the same no matter how big or small the triangle is. Sine, cosine and tangent capture that fixed relationship.