SOH CAH TOA: How to Know When to Use Sin, Cos or Tan

Quick answer: SOH CAH TOA tells you which ratio to use in a right-angled triangle: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. Label the three sides from the angle in the question, then circle the two sides that appear (the one you know and the one you want). That pair names the ratio.

I got trig wrong for most of a term because I memorised the rhyme without ever labelling a triangle. I could chant SOH CAH TOA perfectly and still reach for cos when I needed tan, because I was labelling sides from the right angle instead of from the angle in the question. The fix was embarrassingly small: write O, A and H on the diagram before touching the calculator.

Label the sides first, and label them from your angle

Right-angled trig depends on three labels, and two of them move depending on which angle you are using.

• Hypotenuse (H) — the longest side, always opposite the right angle. It never moves.
• Opposite (O) — the side directly facing the angle you are working with. Not the right angle. Your angle.
• Adjacent (A) — the side left over: it touches your angle but is not the hypotenuse.

Take a triangle with the right angle at B and the marked angle at A. Then AC is the hypotenuse, BC the opposite, AB the adjacent. If the next part uses angle C instead, AC is still the hypotenuse but opposite and adjacent swap: opposite is now AB, adjacent BC. That swap is where most lost marks begin. Pencil the letters on every time. Four seconds, and the guessing disappears.

The three ratios, and a two-second way to choose one

The ratios are:

• sin = O / H
• cos = A / H
• tan = O / A

To choose, look at what you are given and what is asked. Two sides are always involved, one known and one unknown.

1. Opposite and hypotenuse involved, adjacent not needed → sin
2. Adjacent and hypotenuse involved, opposite not needed → cos
3. Opposite and adjacent involved, hypotenuse not needed → tan

The shortcut I actually use is the leftover side. Whichever side is not mentioned names the ratio: no adjacent means sin, no opposite means cos, no hypotenuse means tan. Same rule read backwards, but faster.

One boundary worth naming: this only works when there is a right angle. Without a 90 degree corner, SOH CAH TOA does not apply and you need the sine or cosine rule. Examiners across Cambridge IGCSE, GCSE and CBSE Class 10 like sneaking a non-right-angled triangle into a question that looks like standard trig, so check for the little square first.

Worked example: finding a missing side

Case one, the unknown on top. A right-angled triangle has an angle of 38 degrees and a hypotenuse of 12 cm. Find the side opposite the 38 degree angle.

1. Sides involved: opposite (want) and hypotenuse (have). Adjacent unused, so use sin.
2. Write the statement: sin 38 = x / 12
3. Multiply both sides by 12: x = 12 times sin 38
4. Calculator: sin 38 = 0.61566, so x = 7.3879
5. Answer: 7.39 cm to 3 significant figures.

Case two, the unknown on the bottom. The angle is 40 degrees, the opposite side is 9 cm, and you want the hypotenuse.

1. Opposite and hypotenuse again, so sin.
2. sin 40 = 9 / h
3. The unknown is a denominator, so swap it with the thing you are dividing by: h = 9 / sin 40
4. sin 40 = 0.64279, so h = 14.0015
5. Answer: 14.0 cm to 3 significant figures.

That step 3 is the most common place I see people drop a mark. When the unknown sits underneath, you divide by the trig value; when it sits on top, you multiply. If you are unsure about your rearranging, put the numbers into the math solver and compare its line-by-line working with yours, not just the final number.

Finding a missing angle with the inverse buttons

If you know two sides and want the angle, the method is identical up to the last step, then you use sin^-1, cos^-1 or tan^-1 (usually SHIFT then the ratio button).

Worked example: the opposite side is 7 cm, the adjacent is 4 cm, find the angle.

1. Opposite and adjacent, no hypotenuse, so tan.
2. tan x = 7 / 4 = 1.75
3. x = tan^-1(1.75)
4. x = 60.255 degrees
5. Answer: 60.3 degrees to 1 decimal place.

Sanity check that has saved me more than once: the opposite is bigger than the adjacent, so the angle must be over 45 degrees. If I had got 29.7 degrees, I would know I had divided the wrong way round. Do this on every angle answer; it catches most flipped fractions.

Elevation, depression and the word-problem versions

Most real exam marks here come wrapped in a story: a ladder against a wall, a kite string, someone looking up at a tower. The trig is unchanged, but two things trip people up.

An angle of elevation is measured up from the horizontal and an angle of depression down from the horizontal, not from the vertical. Draw the horizontal dashed line and mark the angle against it, because the depression angle at the top of a cliff equals the elevation angle at the bottom (alternate angles) and questions love that swap.

The second trap is the extra bit. If the observer's eyes are 1.6 m above the ground, your triangle gives the height above eye level, so the answer needs 1.6 added on. Underline what is actually being asked. If setting up word problems is the hard part for you, ask for the setup only in chat and do the arithmetic yourself.

The mistakes that quietly cost marks

• Calculator in the wrong mode. Radians or gradians gives nonsense. Check for a small D or DEG before question one, and test with sin 30 = 0.5.
• Rounding too early. Keep the full value in the calculator through the chain and round only at the end. Rounding sin 38 to 0.62 midway can push your answer outside the accepted range.
• Not writing the trig statement. Mark schemes typically give a method mark for a correct statement like sin 38 = x/12 and a separate accuracy mark for the answer. Do it in your head, slip up, and you lose both.
• Ignoring the rounding instruction. Sides usually go to 3 significant figures, angles to 1 decimal place, unless the paper says otherwise. WAEC and Cambridge papers state it and enforce it.
• Reaching for the calculator when exact values are wanted. Know these cold: sin 30 = 1/2, cos 60 = 1/2, sin 60 = sqrt(3)/2, cos 30 = sqrt(3)/2, tan 45 = 1, tan 30 = 1/sqrt(3), tan 60 = sqrt(3).

To check these have actually stuck, a short mixed set on quiz beats rereading notes, and the maths hub covers the surrounding topics if your gaps sit further back than trig.

Test yourself

1. A right-angled triangle has an angle of 52 degrees and an adjacent side of 8 cm. Which ratio finds the hypotenuse, and what is it to 3 significant figures?
2. The opposite side is 5 cm and the hypotenuse is 9 cm. Find the angle to 1 decimal place.
3. Angle of depression from a cliff top is 24 degrees and the boat is 150 m from the base. How tall is the cliff to 3 significant figures?

FAQ

How do I remember which ratio is which under pressure?

Write SOH CAH TOA in the margin the moment you open the paper, before reading a single question. Then use the leftover-side rule: the side the question never mentions names the ratio. Reading a rhyme you wrote down calmly beats digging one up mid-question.

Why does my calculator give a strange answer for sin 30?

Almost always the angle mode. Radians mode gives sin 30 = negative 0.988, which looks like a plausible number and slips past unnoticed. Switch to degrees and test with sin 30 = 0.5.

Can I use SOH CAH TOA on any triangle?

No. Only when the triangle contains a 90 degree angle. Without one you need the sine or cosine rule. If a diagram has no right angle marked and you cannot split it into two right-angled halves, switch method.

Do I still need Pythagoras if I know trig?

Yes, often in the same question. Pythagoras uses three sides and no angles; trig uses two sides and an angle. Given two sides and asked for the third with no angle involved, that is Pythagoras.

In short: label the sides from your angle, spot which two the question involves, pick the ratio the leftover side points to, and write the statement down before you calculate. Check the calculator is in degrees, keep full accuracy until the last line, and sanity-check angles against 45 degrees. That routine is most of the marks in this topic.