How to Use SOHCAHTOA

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1. 01

   Label the three sides from the angle you are using

   hyp = opposite the right angle; opp = opposite the marked angle; adj = the remaining side

   Only the hypotenuse is fixed. Opposite and adjacent swap places depending on which angle is marked, which is why labelling has to happen after the angle is chosen rather than before. Two triangles with the same shape and different marked angles have different opposite sides.

2. 02

   Tick the two sides the question involves

   known: hyp = 12 cm wanted: opp = x angle = 35°

   Exactly two of the three sides matter in any single question — the one you are given and the one you want. Marking them on the diagram makes the choice of ratio a reading rather than a recollection, and it stops the third side from pulling you towards the wrong formula.

3. 03

   Pick the ratio that uses exactly those two sides

   SOH: sin = opp/hyp CAH: cos = adj/hyp TOA: tan = opp/adj

   The two ticked sides appear together in exactly one of the three ratios, so there is nothing to decide once they are marked. Here opposite and hypotenuse are ticked, which is sine. This is the whole function of the mnemonic — it turns a choice into a lookup.

4. 04

   Write the equation and rearrange before calculating

   sin 35° = x / 12 → x = 12 × sin 35°

   Rearranging on paper before touching the calculator is what keeps the unknown out of a denominator. When the unknown is on the bottom — for instance sin 40° = 8 / x — the rearrangement is x = 8 / sin 40°, and doing that in your head is where the division gets inverted.

5. 05

   Evaluate with the calculator in degree mode

   x = 12 × 0.5736 = 6.88 cm (3 s.f.)

   A calculator left in radians returns an answer that is wrong but not obviously so, and it is the single most common cause of lost marks in this topic. Check for the DEG indicator once at the start of the paper. Rounding only at the final line keeps the answer accurate.

6. 06

   For a missing angle, use the inverse function

   tan θ = 7/10 = 0.7 → θ = tan⁻¹(0.7) = 35.0° (1 d.p.)

   Finding an angle reverses the process: form the ratio from the two known sides, then apply the inverse. The inverse functions sin⁻¹, cos⁻¹ and tan⁻¹ are what convert a ratio back into an angle, and forgetting them leaves an answer between 0 and 1 where degrees were wanted.

01

The labelling is the difficult part

Students rarely go wrong on the ratios themselves. They go wrong on which side is opposite and which is adjacent, because both depend on the angle being used and neither is a property of the triangle alone.

The reliable procedure is mechanical. Find the right angle and mark the hypotenuse opposite it. Then find the angle named in the question, look straight across from it, and mark that side opposite. Whatever is left is adjacent. Doing this in the same order every time removes the judgement from it.

• Hypotenuse — opposite the right angle, longest side, never changes
• Opposite — directly across from the angle in the question
• Adjacent — the remaining side, touching both the angle and the right angle

02

Finding an angle rather than a side

When two sides are known and the angle is wanted, form the ratio first and then apply the inverse function. With an opposite side of 7 cm and an adjacent side of 10 cm, tan θ = 7/10 = 0.7, so θ = tan⁻¹(0.7) = 35.0° to one decimal place.

The same logic works with the other ratios. In a 5-12-13 triangle, the angle whose adjacent side is 5 has cos θ = 5/13 ≈ 0.3846, giving θ = 67.4°. As a check, the same angle has tan θ = 12/5 = 2.4, and tan⁻¹(2.4) = 67.4° as well.

03

Checks that catch a wrong answer

Trigonometry gives answers that look plausible when they are wrong, so the checks matter more here than in most topics. Three of them take seconds and between them catch the majority of errors.

The first is structural: the hypotenuse must come out longest. The second is about the ratios: for an acute angle, sine and cosine are always between 0 and 1, so an answer of 1.6 for a sine means something has been inverted. The third is about the angle: the three angles of a triangle sum to 180°, so with the right angle taken out, the two remaining angles must add to 90°.

• The hypotenuse must be the longest side
• sin θ and cos θ lie between 0 and 1 for any acute angle
• The two non-right angles must add to 90°
• The larger side always faces the larger angle

04

How we teach this

We teach the labelling as a fixed sequence — hypotenuse, then opposite, then adjacent — and we ask for it on the diagram every time, including on questions a student could do without it. The habit is what holds when a triangle turns up rotated inside a larger figure on a Higher paper.

Degree mode is checked at the start of every session, out loud, because a student who has met the radian error once tends not to meet it again. It is a small ritual that protects several marks a paper.