Geometry · GCSE Maths
SOHCAHTOA
GCSE Maths SOHCAHTOA: label opposite, adjacent and hypotenuse from the angle, choose sine, cosine or tangent, and rearrange carefully when the unknown is on the denominator.
Label from the angle first, then pick the ratio that contains the two sides you know or want. Unknown on the bottom means divide, not multiply.
The important bits
What you need to know
- 1
SOHCAHTOA only applies to right-angled triangles. The right angle must be marked before you name a side as hypotenuse.
- 2
Hypotenuse is opposite the right angle and is the longest side. From the acute angle you are using, opposite does not touch that angle; adjacent does, other than the hypotenuse.
- 3
sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent. Tick the two sides you have or want; the ratio with two ticks is the one to use.
- 4
To find a side, write the full equation first: sin 32° = x / 11, then multiply both sides by 11. Do not start with “sin 32 × 11” as a slogan if you cannot also handle the unknown on the bottom.
- 5
If the unknown is on the denominator, multiply both sides by that unknown, then divide by the trigonometric value: cos 40° = 7 / x becomes x = 7 / cos 40°.
- 6
To find an angle, use the inverse: θ = tan⁻¹(opp/adj). The calculator must be in degrees. Inverse is for angles; ordinary sin, cos, tan are for sides.
- 7
Exact values: sin 30° = 1/2, cos 30° = √3/2, tan 45° = 1, sin 60° = √3/2, cos 60° = 1/2, sin 45° = cos 45° = √2/2. Use them on non-calculator papers.
- 8
Angles of elevation and depression are from the horizontal. Draw the horizontal and the right angle before you label opposite and adjacent.
Quotations worth analysing
Short evidence. Real method.
“sin θ = opp / hyp”
Write the ratio from the labelled triangle, not from a side you merely want to find. The hypotenuse is never opposite the acute angle you are working with.
“x = 7 / cos 40°”
This is the method mark that multiplications miss. cos 40° = 7 / x is not x = 7 cos 40°. Treat the ratio as an equation.
“Label opposite, adjacent and hypotenuse”
Most wrong trig answers are labelling errors. Circle the angle, mark H, O and A, then choose the ratio. Calculators cannot fix a mislabelled triangle.
Go deeper
Label, tick, then write an equation
Circle the angle you are allowed to use. H is fixed: it faces the right angle. O faces your circled angle. A is the remaining side that touches the circled angle. Tick the two of H, O, A that are involved. Two ticks on S means sine; on C, cosine; on T, tangent. Then write a complete equation, for example tan 50° = 8 / x. That equation is algebra. Multiply both sides by x: x tan 50° = 8. Divide by tan 50°: x = 8 / tan 50°. A size check catches the flip: 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 unknown was treated as if it were on top. Keep four decimal places if you write the trig value down, then round the side to 1 d.p. unless the question says otherwise.
Go deeper
Unknown on the denominator is still balance method
Students remember “multiply by the trig” from the cases where the unknown is on top, then apply it always. When sin θ = opp / hyp and hyp is unknown, multiplying by sin θ makes the hypotenuse smaller than the opposite, which is impossible. The balance method does not care about trigonometry: if a = b / x, then x = b / a. So if sin 35° = 10 / x, x = 10 / sin 35°. If cos 35° = x / 10, x = 10 cos 35°. Write the two cases as a pair in revision until the difference is boring. Elevation and depression questions are the same ratios after you have drawn the horizontal. The extra marks sit in extracting the correct angle, not in a new flavour of sine.
Go deeper
Exact values are labelled triangles in disguise
sin 30° = 1/2 because a 30–60–90 triangle has sides 1, √3, 2. The side opposite 30° is half the hypotenuse. cos 30° is adjacent over hypotenuse, √3/2. tan 45° = 1 because an isosceles right-angled triangle has equal opposite and adjacent. On a non-calculator paper, “sin 30° = x / 10” is x = 5, not a decimal hunt. If the question asks for an exact value, leave √2 and √3 in the answer. Mixing 0.5 with 0.87 on a non-calculator paper is how an exact-value question loses its last mark. Still label O, A, H; the exact values do not excuse a cosine used on opposite and hypotenuse.
See the idea in action
In a right-angled triangle, angle θ = 35°, the hypotenuse is 12 cm, and the side opposite θ is unknown. Find that side to 1 d.p. Step 1: Label H = 12 cm, O = x, A unused. Opposite and hypotenuse mean sine. Step 2: sin 35° = x / 12. Step 3: Multiply both sides by 12: x = 12 sin 35°. Step 4: x = 12 × 0.5736… = 6.883… = 6.9 cm to 1 d.p. Check: 35° < 45°, so O should be less than A and clearly less than 12. If instead A = 12 and O = x with the same angle, that would be tan 35° = x / 12, x = 12 tan 35° ≈ 8.4 cm.
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
Multiplying by sin, cos or tan when the unknown is on the denominator, instead of dividing.
- 03
Using an angle that is not the one from which opposite and adjacent were labelled, or leaving the calculator in radians.
cos 40° = 7 / x. What is x?
Ax = 7 / cos 40°
Bx = 7 cos 40°
Cx = cos 40° / 7
Dx = 7 sin 40°
Show the answer
x = 7 / cos 40°. Multiply both sides by x, then divide by cos 40°. x = 7 cos 40° would be the rearrangement of cos 40° = x / 7, the unknown-on-top case. x = cos 40° / 7 inverts the 7 wrongly.
Quick questions
If this is the bit you searched
How do I know whether to use sin, cos or tan?
Label O, A and H from the angle. Use the ratio that contains the two sides you know or want. If H is not involved, use tan.
What do I do if the unknown is on the bottom of the fraction?
Write the equation, multiply both sides by the unknown, then divide by the trigonometric value. For cos θ = 7 / x, x = 7 / cos θ.
When do I use sin⁻¹ rather than sin?
Use sin, cos or tan to find a side. Use the inverse to find an angle once you know the two sides in the ratio.
Can I use SOHCAHTOA if the triangle is not right-angled?
No. Use the sine rule or cosine rule instead. Dropping a perpendicular to create a right-angled triangle is allowed if you then apply SOHCAHTOA to that new triangle.