Right-Angled Trigonometry (SOHCAHTOA)
Trigonometry connects the angles of a right-angled triangle to the lengths of its sides. Almost every mark lost in this topic comes from one place: labelling the sides wrongly. Get the labels right and the rest is substitution.
1. Labelling the sides — the step that decides everything
Every right-angled triangle has three sides, and two of the three labels depend on which angle you are using.
| Side | How to find it |
|---|---|
| Hypotenuse (H) | The longest side, always opposite the right angle. Never changes. |
| Opposite (O) | The side directly across from the angle you are working with |
| Adjacent (A) | The side next to the angle, that is not the hypotenuse |
Opposite and adjacent swap when the angle changes. They are not fixed properties of the triangle. If a question has two parts using two different angles, you must re-label for each part. Confusion between opposite and adjacent was by far the most common error in real lessons — it appeared more often than every calculator and rounding mistake combined.
A reliable routine:
- Mark the right angle
- Mark the angle you’re using (given or wanted)
- The side opposite the right angle is H
- The side opposite your marked angle is O
- The remaining side is A
Label the triangle on the paper before you choose a ratio. It takes five seconds and prevents the error that costs the most marks in this topic.
The “base” of a triangle is not automatically the adjacent side. Adjacent is defined relative to the angle, not to how the triangle is drawn on the page. A triangle rotated on the page still labels the same way.
2. SOHCAHTOA
SOH — sin θ = O / H CAH — cos θ = A / H TOA — tan θ = O / A
Choosing the ratio: identify which two sides are involved — the one you know and the one you want — then pick the ratio containing exactly those two.
| You know / want | Use |
|---|---|
| Opposite and Hypotenuse | sin |
| Adjacent and Hypotenuse | cos |
| Opposite and Adjacent | tan |
tan is opposite over adjacent, never over the hypotenuse. This specific mix-up appeared repeatedly. tan is the only ratio with no hypotenuse in it — if the hypotenuse is involved, it is sin or cos.
These ratios only work in right-angled triangles. If there is no right angle, you need the sine rule or cosine rule instead. Equally, don’t reach for the sine rule when a right angle is sitting there — SOHCAHTOA is quicker and less error-prone.
3. Finding a missing side
Example: find x, given a 35° angle and a hypotenuse of 12 cm, where x is opposite the angle.
- Label: x is O, 12 is H
- O and H → sin
- sin 35° = x / 12
- x = 12 × sin 35°
- x = 6.88 cm (3 s.f.)
When the unknown is on the bottom:
Find x, given a 40° angle, where the side adjacent is x and the opposite is 9 cm.
- O and A → tan
- tan 40° = 9 / x
- Multiply both sides by x: x × tan 40° = 9
- x = 9 / tan 40° = 10.7 cm (3 s.f.)
If the unknown is in the denominator, it does not simply move across as a multiplication. Rearrange properly: x = 9 ÷ tan 40°, not 9 × tan 40°. Getting this backwards produces an answer that is wrong but looks reasonable.
A quick sanity check: the hypotenuse must be the longest side. If your “hypotenuse” comes out shorter than another side, something is wrong.
4. Finding a missing angle
When you know two sides and want the angle, use the inverse functions: sin⁻¹, cos⁻¹, tan⁻¹ (on a calculator, usually SHIFT then sin/cos/tan).
Example: opposite = 5 cm, adjacent = 8 cm. Find θ.
- O and A → tan
- tan θ = 5/8 = 0.625
- θ = tan⁻¹(0.625)
- θ = 32.0° (3 s.f.)
You cannot find an angle with Pythagoras. Pythagoras relates the three sides only. Finding an angle needs a trigonometric ratio — this was a recurring misconception.
Use the inverse function, not the ordinary one. tan 0.625 is a completely different quantity from tan⁻¹(0.625).
5. Angles of elevation and depression
The angle of elevation is measured upward from the horizontal. The angle of depression is measured downward from the horizontal.
Both are measured from the horizontal, never from the vertical.
Method: always draw a diagram. Mark the horizontal, the line of sight, the right angle, and the angle given. Most of these questions become a routine SOHCAHTOA problem the moment the diagram is drawn.
The angle of depression from the top equals the angle of elevation from the bottom — they are alternate angles between two parallel horizontals. Questions often give you one and expect you to use the other.
Sketch even when a diagram is provided, if the provided one is cluttered. Tutors repeatedly advised redrawing the single triangle you actually need.
6. Multi-step and 3D problems
Harder questions chain two triangles together — the answer from the first becomes an input to the second.
Approach:
- Identify the triangle that has enough information to solve on its own
- Solve it completely
- Carry that result into the next triangle
State which triangle you are working in. Writing “In triangle ABC:” before each stage makes your method legible to the examiner and protects your method marks when an early value is wrong.
Carry full accuracy between stages. Round only the final answer. Rounding an intermediate value to 3 s.f. and feeding it into the next step introduces errors large enough to lose accuracy marks — keep the value on your calculator, or store it in memory.
7. Exact values worth knowing
Useful for non-calculator papers and for questions asking for an exact answer:
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
sin 60° is √3/2 ≈ 0.866, not 0.8. When a question asks for an exact value, give the surd — a decimal is not exact.
8. Calculator care
Check your calculator is in DEGREE mode. If it is in radians, every answer in the topic will be wrong while looking perfectly plausible. Look for D or DEG in the display. This was flagged by tutors more than once, and it is the single most damaging setting error in the subject.
- Use brackets for anything more complicated than a single value
- Check the order of operations — some calculators need the number before the function, others after
- Take the inverse functions from SHIFT, and confirm you pressed it
Rounding:
Give 3 significant figures unless the question says otherwise. If it specifies decimal places, follow the question. And always give the units — cm, m, degrees.
9. Mistakes that cost marks
Mixing up opposite and adjacent — the dominant error in this topic.
Not re-labelling when the question moves to a different angle.
Using tan as opposite over hypotenuse.
Using SOHCAHTOA in a triangle with no right angle.
Using the sine rule in a right-angled triangle when SOHCAHTOA was quicker.
Trying to find an angle with Pythagoras.
Forgetting the inverse function when finding an angle.
Mishandling the unknown when it is in the denominator.
Leaving the calculator in radian mode.
Rounding intermediate values instead of carrying full accuracy.
Measuring elevation or depression from the vertical.
Omitting units, or giving more or fewer figures than asked.
Frequently asked questions
What does SOHCAHTOA stand for? sin = Opposite/Hypotenuse, cos = Adjacent/Hypotenuse, tan = Opposite/Adjacent.
How do I know which side is the adjacent? It is the side next to your angle that is not the hypotenuse. It changes if you use a different angle.
Which ratio should I use? Identify the two sides involved — the one you know and the one you want — and choose the ratio containing exactly those two.
How do I find an angle? Form the ratio from two known sides, then apply the inverse function (sin⁻¹, cos⁻¹, tan⁻¹).
Can I use SOHCAHTOA in any triangle? No — only right-angled triangles. Otherwise use the sine or cosine rule.
Can I find an angle using Pythagoras? No. Pythagoras only relates the three sides.
What is the angle of elevation? The angle measured upward from the horizontal to the line of sight. Depression is measured downward from the horizontal.
Why is my answer completely wrong? Check your calculator is in degree mode, and check your side labels.
How should I round? 3 significant figures unless the question specifies otherwise — and keep full accuracy in intermediate steps.
What is the exact value of sin 60°? √3/2.
Quick revision checklist
- I can identify the hypotenuse instantly
- I label opposite and adjacent relative to the angle in use
- I re-label when the question changes angle
- I know SOHCAHTOA and can pick the ratio from the two sides involved
- I know tan never involves the hypotenuse
- I only use these ratios in right-angled triangles
- I can find a missing side, including when the unknown is on the bottom
- I can find a missing angle using the inverse functions
- I never try to find an angle with Pythagoras
- I can draw and use angles of elevation and depression
- I can chain two triangles and state which one I’m in
- I carry full accuracy and round only at the end
- I know the exact values for 30°, 45° and 60°
- I check my calculator is in degree mode
- I give 3 significant figures and the correct units
These notes cover right-angled trigonometry in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.
Finished this topic?
Saved on this device — no account needed.