Geometry · GCSE Maths
Trigonometry
Use SOHCAHTOA on right-angled triangles: label opposite, adjacent and hypotenuse from the angle, then choose sine, cosine or tangent.
SOH CAH TOA. Label opposite, adjacent and hypotenuse from the angle you are using — never from a side you merely like the look of.
The important bits
What you need to know
- 1
In a right-angled triangle, the hypotenuse is the longest side, opposite the right angle. It is never opposite an acute angle you are working with.
- 2
From the chosen acute angle, opposite is the side that does not touch the angle. Adjacent is the side that does touch it, other than the hypotenuse.
- 3
SOH: sin θ = opposite / hypotenuse. CAH: cos θ = adjacent / hypotenuse. TOA: tan θ = opposite / adjacent.
- 4
To find a side, write the ratio, substitute the two known values, and solve the equation with the balance method.
- 5
To find an angle, use the inverse: θ = sin⁻¹(opp/hyp), and likewise for cos⁻¹ and tan⁻¹. The calculator must be in degrees.
- 6
Exact values at GCSE: sin 30° = 1/2, cos 30° = √3/2, tan 45° = 1, sin 60° = √3/2, sin 45° = √2/2. Learn the 30–60–90 and 45–45–90 triangles.
- 7
The Sine Rule and Cosine Rule (higher) apply when the triangle is not right-angled. Do not force SOHCAHTOA onto an angle that is not next to a right angle.
- 8
Angles of elevation and depression are measured from the horizontal. Draw them; do not invent a vertical angle because the diagram is sketchy.
Go deeper
Label from the angle, then choose the ratio
Most trigonometry errors are labelling errors, not calculator errors. Circle the angle you are allowed to use. The hypotenuse is fixed: it faces the right angle. The opposite faces your circled angle. The adjacent is the remaining side that touches the circled angle. Only then say SOHCAHTOA. If you know opposite and hypotenuse, sine is the ratio that contains both. If you know adjacent and hypotenuse, cosine. If the hypotenuse is not involved, tangent. Write the full equation, for example sin 32° = x / 11, before you rearrange. Rearrangement is ordinary algebra: multiply both sides by 11. Students who write “sin 32 × 11” without the equation still often get the number, but they cannot recover when the unknown is on the bottom, as in cos 40° = 7 / x, which is x = 7 / cos 40°, not 7 × cos 40°.
Go deeper
Unknown on the denominator needs a balance
SOHCAHTOA produces an equation. Treat it with the same respect as 3x = 12. If tan 50° = 8 / x, multiply both sides by x: x tan 50° = 8, then divide by tan 50°: x = 8 / tan 50°. The physical picture is a side adjacent to 50° when the opposite is 8. If you multiply by tan 50° you have described a different triangle. A quick size check helps: 50° is more than 45°, so opposite should be longer than adjacent, so x should be less than 8. If the calculator offers 9.5, the rearrangement has flipped. Keep four decimal places in the trigonometric value if you write it down, then round the final side to a sensible degree of accuracy, usually one decimal place unless the question says otherwise.
Go deeper
Elevation, depression and the horizontal
An angle of elevation is the angle you look up from the horizontal. An angle of depression is the angle you look down from the horizontal. They are equal when they are alternate angles on parallel horizontals, which is why a depression from a cliff often becomes an elevation at the boat. Draw the horizontal. Mark the right angle where a vertical height meets the ground or the sea. Only then is the triangle a SOHCAHTOA triangle. Bearings questions add a north line: a bearing is an angle measured clockwise from north, written in three figures. The trigonometry has not changed; the extra marks sit in extracting the correct angle from the bearing diagram. If you cannot name opposite and adjacent in a sentence, do not press sin yet.
See the idea in action
A right-angled triangle has an angle of 35° with opposite side unknown and hypotenuse 12 cm. Opposite and hypotenuse mean sine: sin 35° = x / 12, so x = 12 sin 35° ≈ 6.9 cm to 1 d.p. Check: 35° is less than 45°, so the opposite should be shorter than the adjacent and clearly shorter than 12.
Exam technique
Turn knowledge into marks
Write SOHCAHTOA in the margin and tick the two sides you know or want. The ratio you tick twice is the one to use. Then write the equation before you rearrange.
Common mistakes
Do not give these marks away
- 01
Labelling the hypotenuse as opposite because it is the side you are trying to find.
- 02
Using sin, cos or tan with an angle that is not the one from which the sides were labelled.
- 03
Multiplying by the trigonometric value when the unknown is on the denominator, instead of dividing.
In a right-angled triangle, sin 30° = opposite / hypotenuse. If the hypotenuse is 10 cm, the opposite side is
A5 cm
B5√3 cm
C10√3 cm
D20 cm
Show the answer
5 cm. sin 30° = 1/2, so opposite = 10 × 1/2 = 5 cm. 5√3 cm would be the opposite in a 60° triangle with hypotenuse 10, or the adjacent in this 30° triangle.
Quick questions
If this is the bit you searched
How do I remember SOHCAHTOA?
Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent. Label the triangle from the angle first; the mnemonic only chooses the ratio.
When do I use sin⁻¹ rather than sin?
Use sine (or cos, or tan) when you are finding a side. Use the inverse when you already know the two sides in the ratio and you need the angle.
Why did my calculator give a wrong trig value?
It is probably in radians. GCSE work is in degrees. Check the D/R mode. Also check you used the inverse key only when finding an angle.