Sine, Cosine & Tangent Ratios

SOH CAH TOA is the memory tool for the three basic trig ratios in a right triangle. Each chunk names a ratio and its two sides: SOH means $\sin A = \dfrac{\text{opposite}}{\text{hypotenuse}}$ , CAH means $\cos A = \dfrac{\text{adjacent}}{\text{hypotenuse}}$ , and TOA means $\tan A = \dfrac{\text{opposite}}{\text{adjacent}}$ .

These ratios work because of similar triangles: every right triangle with, say, a $35^\circ$ angle is similar to every other one, so the ratio of any two sides is locked in by the angle alone. That's what makes trig so powerful — one angle plus one side is enough to find everything else in the triangle. The only real skill to build first is labeling the three sides correctly.

Labeling the sides: opposite, adjacent, hypotenuse

Everything is measured from the angle you're working with. The hypotenuse is easy — it's the side across from the right angle, always the longest side, and it never changes. The opposite side is the one directly across from your angle, not touching it. The adjacent side is the leg that touches your angle (the hypotenuse also touches it, but the hypotenuse already has its own name).

The figure below shows the labels for the angle $\theta$ at the left vertex. If you switched to the other acute angle instead, opposite and adjacent would trade places — which is why you must label the sides fresh for each angle, every time.

Picking the right ratio

In any problem, exactly two sides matter: the one you know and the one you want (or the two you know, if you're computing a ratio). Name those two sides relative to the angle, then pick the ratio that uses exactly that pair — SOH for opposite and hypotenuse, CAH for adjacent and hypotenuse, TOA for opposite and adjacent.

So if a problem gives the hypotenuse and asks for the side across from the angle, that's opposite and hypotenuse: sine. If it gives both legs, that's opposite and adjacent: tangent. The third side never enters the equation.

The mistakes that cost points

Mixing up opposite and adjacent is the classic error, and it silently swaps sine and cosine. Before writing anything, put your finger on the angle and trace: across the triangle is opposite, along the leg touching the angle is adjacent.

Two more to watch: the hypotenuse is always the side across from the right angle — a leg never plays hypotenuse no matter which acute angle you use. And when you eventually reach for a calculator, make sure it's in degree mode if the angle is in degrees; radian mode produces answers that look plausible and are completely wrong.

Worked examples

Example 1: all three ratios in a 3-4-5 triangle

In a right triangle, the side opposite $\angle A$ is $3$ , the side adjacent to $\angle A$ is $4$ , and the hypotenuse is $5$ . Find $\sin A$ , $\cos A$ , and $\tan A$ .

Label the sides for

\angle A

\text{opp} = 3, \quad \text{adj} = 4, \quad \text{hyp} = 5

SOH: sine is opposite over hypotenuse

\sin A = \dfrac{3}{5}

CAH: cosine is adjacent over hypotenuse

\cos A = \dfrac{4}{5}

TOA: tangent is opposite over adjacent

\tan A = \dfrac{3}{4}

Answer: $\sin A = \dfrac{3}{5}$ , $\cos A = \dfrac{4}{5}$ , $\tan A = \dfrac{3}{4}$

Example 2: same triangle, other angle

Using the same $3$ - $4$ - $5$ triangle, find $\sin B$ , $\cos B$ , and $\tan B$ , where $\angle B$ is the other acute angle.

Relabel the sides for

\angle B

— opposite and adjacent trade places

\text{opp} = 4, \quad \text{adj} = 3, \quad \text{hyp} = 5

SOH

\sin B = \dfrac{4}{5}

CAH

\cos B = \dfrac{3}{5}

TOA

\tan B = \dfrac{4}{3}

Notice

\sin B = \cos A

— the sine of one acute angle equals the cosine of the other

Answer: $\sin B = \dfrac{4}{5}$ , $\cos B = \dfrac{3}{5}$ , $\tan B = \dfrac{4}{3}$

Example 3: a 5-12-13 triangle

In a right triangle, the side opposite $\theta$ is $5$ and the side adjacent to $\theta$ is $12$ . Find $\tan \theta$ and $\sin \theta$ .

TOA uses the two given legs directly

\tan \theta = \dfrac{5}{12}

Sine needs the hypotenuse, so find it with the Pythagorean theorem

5^2 + 12^2 = 25 + 144 = 169

Take the square root

\text{hyp} = \sqrt{169} = 13

SOH

\sin \theta = \dfrac{5}{13}

Answer: $\tan \theta = \dfrac{5}{12}$ , $\sin \theta = \dfrac{5}{13}$

Example 4: finding a missing side

A kite string is $12$ meters long and makes a $30^\circ$ angle with the ground. How high is the kite?

Identify the sides: the height is opposite the

30^\circ

angle, and the string is the hypotenuse — that pair is SOH

\sin 30^\circ = \dfrac{h}{12}

Use the exact value

\sin 30^\circ = \dfrac{1}{2}

\dfrac{1}{2} = \dfrac{h}{12}

Multiply both sides by

12

h = 6

Check with the 30-60-90 pattern: the side opposite

30^\circ

is half the hypotenuse, and half of

12

6

✓

Answer: $h = 6$ meters

Try one yourself

Common questions

How do I know whether to use sine, cosine, or tangent?

Look at which two sides are involved — the side you know and the side you want. Opposite and hypotenuse means sine, adjacent and hypotenuse means cosine, opposite and adjacent means tangent. The angle's job is just to define which side is opposite and which is adjacent.

Do opposite and adjacent change if I use the other acute angle?

Yes — they swap. The side that was opposite one acute angle is adjacent to the other, and vice versa. Only the hypotenuse keeps its name, because it's defined by the right angle, not by the angle you're using.

Does SOH CAH TOA work on triangles without a right angle?

No. These three ratios are defined using the hypotenuse and legs of a right triangle, so the triangle must have a right angle. For other triangles, geometry has separate tools — the Law of Sines and the Law of Cosines — that handle any triangle.

Want the video version?

Allday Everyday Math has video lessons, practice, and an AI tutor for every topic, Pre-Algebra through Algebra 2.

Try it for $1