Trigonometry SOHCAHTOA Explained: KS3 and GCSE Maths

SOHCAHTOA is a memory aid for the three trigonometric ratios in a right-angled triangle: sin = Opposite ÷ Hypotenuse, cos = Adjacent ÷ Hypotenuse, tan = Opposite ÷ Adjacent. Choose the ratio that connects the angle you know (or want) to the two sides involved, then rearrange to find the unknown.

What do the letters in SOHCAHTOA mean?

Letters | Ratio | Formula
SOH | Sine | sin θ = Opposite ÷ Hypotenuse
CAH | Cosine | cos θ = Adjacent ÷ Hypotenuse
TOA | Tangent | tan θ = Opposite ÷ Adjacent

The angle θ (theta) is always one of the two non-right angles in the triangle.

How do you label the sides of a right-angled triangle?

Before using SOHCAHTOA, label the three sides relative to the angle you are working with (not the right angle):

Hypotenuse (H): The longest side — always opposite the right angle.

Opposite (O): The side directly across from the marked angle θ.

Adjacent (A): The side next to θ that is not the hypotenuse.

Relabelling for each question is important: the "opposite" and "adjacent" sides swap if you switch to a different angle.

How do you find a missing side?

Step-by-step:

Label the sides H, O, A relative to the given angle.

Identify which two sides are involved (the known side and the unknown side).

Pick the ratio (SOH, CAH, or TOA) that links those two sides.

Write the formula, substitute known values, then solve for the unknown.

Worked example — finding the opposite side:

In a right-angled triangle, angle θ = 35° and the hypotenuse = 12 cm. Find the opposite side.

We have θ, H, and want O → use SOH: sin 35° = O ÷ 12

Rearrange: O = 12 × sin 35°

Calculate: O = 12 × 0.5736 = 6.88 cm (to 3 s.f.)

How do you find a missing angle?

When you know two sides and need the angle, use the inverse trigonometric function (sin⁻¹, cos⁻¹, or tan⁻¹ on your calculator, often labelled "arcsin", "arccos", "arctan" or accessed via the SHIFT / 2nd key).

Worked example — finding a missing angle:

A right-angled triangle has opposite = 7 cm and adjacent = 9 cm. Find angle θ.

We have O and A → use TOA: tan θ = 7 ÷ 9 = 0.7778

Apply inverse tan: θ = tan⁻¹(0.7778)

Calculate: θ = 37.9° (to 1 d.p.)

How do you choose between SOH, CAH, and TOA?

Ask yourself: which two sides are involved?

Sides involved | Ratio to use
Opposite and Hypotenuse | SOH (sin)
Adjacent and Hypotenuse | CAH (cos)
Opposite and Adjacent | TOA (tan)

If you mark the known side and the unknown side on a quick sketch, the choice becomes obvious.

How do you rearrange the formula to find each side?

From sin θ = O ÷ H you can derive two rearrangements:

To find O: O = H × sin θ

To find H: H = O ÷ sin θ

The same pattern applies to CAH and TOA. A useful memory trick is the formula triangle: write O on top, H and sin θ on the bottom — cover the quantity you want to see the formula for it.

What calculator settings must you check?

Make sure your calculator is set to degree mode (not radian mode) for GCSE questions, unless the question specifies radians. Look for a "D" indicator on screen. Pressing MODE or SETUP lets you switch. Using radian mode by accident gives completely wrong answers for angle calculations.

Frequently asked questions

What does SOHCAHTOA stand for?

SOHCAHTOA stands for Sine = Opposite over Hypotenuse, Cosine = Adjacent over Hypotenuse, Tangent = Opposite over Adjacent. It is a memory aid for the three basic trigonometric ratios in right-angled triangles.

When do you use sin, cos, or tan?

Use sin when the question involves the opposite side and the hypotenuse; use cos when it involves the adjacent side and the hypotenuse; use tan when it involves the opposite and adjacent sides. Label the sides H, O, A first — the correct ratio follows automatically.

How do you find an angle using SOHCAHTOA?

Write the appropriate ratio equation (e.g. tan θ = O ÷ A), calculate the decimal value of the right-hand side, then apply the inverse function on your calculator (tan⁻¹, sin⁻¹, or cos⁻¹ using the SHIFT key) to find the angle in degrees.

Does SOHCAHTOA work for all triangles?

No — SOHCAHTOA only works for right-angled triangles. For triangles without a right angle, you need the sine rule or cosine rule, which are GCSE Higher topics. Always check for the right-angle symbol (a small square in the corner) before applying SOHCAHTOA.

For Socratic trigonometry practice at KS3 and GCSE, see aitutors.me.