Trigonometry: SOHCAHTOA

In a right-angled triangle, three ratios connect each acute angle to the side lengths. These are sine, cosine, and tangent, remembered with the mnemonic SOH CAH TOA:

$\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}}\cos \theta =\frac{\text{adjacent}}{\text{hypotenuse}}\tan \theta =\frac{\text{opposite}}{\text{adjacent}}$

Mnemonic part	Ratio	Formula
SOH	Sine	Opposite ÷ Hypotenuse
CAH	Cosine	Adjacent ÷ Hypotenuse
TOA	Tangent	Opposite ÷ Adjacent

These ratios are the same for any right-angled triangle containing that angle — the side lengths scale, but the ratios do not change. A 35° angle always produces the same sine value, no matter how large the triangle is.

SOHCAHTOA is not given on the formula sheet. You must know all three ratios.

The three sides are named relative to the angle θ you are working with — not the right angle.

Hypotenuse — always the longest side; always opposite the right angle. Does not change when you switch angles.

Opposite — the side directly facing angle θ.

Adjacent — the remaining side, next to θ (not the hypotenuse).

If you move to a different angle in the same triangle, opposite and adjacent swap — the hypotenuse stays the same.

Worked example — In a right-angled triangle with sides 5, 12, and 13, label all three sides for the angle θ at the bottom-left (the angle facing the side of length 5):

Side	Length	Reason
Hypotenuse	13	Longest side; opposite the right angle
Opposite	5	Directly facing θ
Adjacent	12	Remaining side; next to θ

If instead θ is the angle facing the side of length 12: opposite = 12, adjacent = 5, hypotenuse = 13 (unchanged).

Always label all three sides before choosing a ratio. Skipping this step is the single most common reason students pick the wrong ratio.

Choose the ratio that links the known angle, the known side, and the unknown side. Substitute, then rearrange.

Worked example 1 — Find the side opposite a 35° angle in a right-angled triangle with hypotenuse 12 cm.

Sides involved: opposite (unknown) and hypotenuse (12 cm) → use sin.

$\sin 35°=\frac{\text{opp}}{12}⟹\text{opp}=12\times \sin 35°=12\times 0.5736=6.88\text{ cm (3 s.f.)}$

Worked example 2 — The side adjacent to a 48° angle is 9 cm. Find the hypotenuse.

Sides involved: adjacent (9 cm) and hypotenuse (unknown) → use cos.

$\cos 48°=\frac{9}{\text{hyp}}⟹\text{hyp}=\frac{9}{\cos 48°}=\frac{9}{0.6691}=13.5\text{ cm (3 s.f.)}$

When the unknown side is on the bottom of the fraction, divide the known value by the trig ratio. Multiplying instead is one of the most common errors on this topic.

To find an angle from a known ratio, use the inverse trig functions: ${\sin }^{-1}$, ${\cos }^{-1}$, ${\tan }^{-1}$ (also written arcsin, arccos, arctan on some calculators).

If $\sin \theta =x$, then $\theta ={\sin }^{-1}(x)$.

Worked example 1 — Find angle θ where opposite = 7 cm and adjacent = 10 cm.

Sides involved: opposite and adjacent → use tan.

$\tan \theta =\frac{7}{10}=0.7⟹\theta ={\tan }^{-1}(0.7)=35.0°\text{ (1 d.p.)}$

Worked example 2 — Find angle θ where the adjacent side is 9 cm and hypotenuse is 15 cm.

$\cos \theta =\frac{9}{15}=0.6⟹\theta ={\cos }^{-1}(0.6)=53.1°\text{ (1 d.p.)}$

Before pressing any trig key, confirm your calculator shows D (degrees), not R (radians) or G (gradians). In radian mode, sin(35) = 0.428 not 0.574 — answers will be wrong by a large margin.

For specific angles, exact trig values must be known without a calculator. These appear in Paper 1 (non-calculator) and in questions requiring exact answers.

θ	sin θ	cos θ	tan θ
0°	$0$	$1$	$0$
30°	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$
45°	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$
60°	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
90°	$1$	$0$	—

G21 requires tan values for 0°, 30°, 45°, and 60° only. Tan 90° is undefined (adjacent side has zero length) but is not required by the specification.

Key patterns: sin and cos for 30° and 60° are swapped ($\sin 30°=\cos 60°=\frac{1}{2}$); $\tan 45°=1$ because opposite = adjacent in a 45–45–90 triangle.

Worked example — Without a calculator, find the exact hypotenuse in a right-angled triangle where the opposite side to 60° is 8 cm.

$\sin 60°=\frac{8}{\text{hyp}}⟹\frac{\sqrt{3}}{2}=\frac{8}{\text{hyp}}⟹\text{hyp}=\frac{16}{\sqrt{3}}=\frac{16\sqrt{3}}{3}\text{ cm}$

The two sides you are working with (one known, one unknown) determine the ratio. Use the table below to read off the ratio directly.

Sides involved	Ratio	Standard form
Opposite + Hypotenuse	sin	$\sin \theta =\frac{\text{opp}}{\text{hyp}}$
Adjacent + Hypotenuse	cos	$\cos \theta =\frac{\text{adj}}{\text{hyp}}$
Opposite + Adjacent	tan	$\tan \theta =\frac{\text{opp}}{\text{adj}}$

Process to use every time:

1. Mark the right angle, then mark angle θ.
2. Label hypotenuse (longest), opposite (facing θ), adjacent (next to θ).
3. Identify the two sides you know or need.
4. Read the ratio from the table.
5. Write the equation, substitute, solve.

Worked example — A ladder 6 m long leans against a wall with its base 2 m from the wall. Find the angle the ladder makes with the ground.

The 6 m ladder is the hypotenuse. The 2 m base is adjacent to the ground angle θ. Sides: adjacent and hypotenuse → use cos.

$\cos \theta =\frac{2}{6}=\frac{1}{3}⟹\theta ={\cos }^{-1}\left(\frac{1}{3}\right)=70.5°\text{ (1 d.p.)}$

1. Calculator set to radians instead of degrees

Trig values in radian mode are completely different from degree mode. Check for D on the display before every trig calculation. This error produces answers that are wrong by a large, hard-to-spot margin.

2. Mislabelling opposite and adjacent

The labels depend on which angle θ you are using. Students often label the sides once and forget to re-label when a question asks about a different angle. Hypotenuse is always safe — it never changes.

3. Forgetting the inverse function when finding an angle

$\sin \theta =0.6$ gives a ratio, not an angle. You must press ${\sin }^{-1}(0.6)$ to get $\theta =36.9°$. Writing $\theta =\sin (0.6)$ applies the wrong operation.

4. Dividing in the wrong direction

If $\cos 40°=\frac{8}{\text{hyp}}$, then $\text{hyp}=\frac{8}{\cos 40°}$, not $8\times \cos 40°$. A quick sanity check: the hypotenuse is always the longest side, so the result must be greater than 8.

5. Not giving exact answers on non-calculator questions

When a question involves 30°, 45°, or 60° and says "give an exact answer", leave the result in surd form — for example $\frac{16\sqrt{3}}{3}$, not a decimal approximation.