Trigonometric Ratios: Sine, Cosine and Tangent

The ratio belongs to the angle

Draw a right triangle with an acute angle of $30°$. Draw another, ten times bigger, also with a $30°$ angle.

Both have a right angle and both have a $30°$ angle. That is two matching pairs, so by AA they are similar — and similar triangles have all corresponding side ratios equal.

$$\frac{\text{opposite}}{\text{hypotenuse}} \;\text{ is the same in both}$$

Nothing about the size survives. The ratio is a property of the $30°$ angle itself, not of the triangle you happened to draw.

That is what makes trigonometry possible. Each acute angle carries a fixed set of numbers, they can be tabulated once, and every right triangle in the world can then be solved from them.

Naming the three sides

For a chosen acute angle, the three sides get names:

Side	Which one
hypotenuse	opposite the right angle — always the longest
opposite	across the triangle from the chosen angle
adjacent	the remaining side, touching the chosen angle

The hypotenuse never changes. Opposite and adjacent swap when you switch to the other acute angle, and that swap is the source of most sign-free errors in this topic.

The three ratios

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

The traditional way to hold them is SOH CAH TOA:

	Stands for
SOH	Sine = Opposite / Hypotenuse
CAH	Cosine = Adjacent / Hypotenuse
TOA	Tangent = Opposite / Adjacent

For the triangle above, taking the angle at the bottom right, the opposite side is $3$, the adjacent is $4$ and the hypotenuse is $5$:

$$\sin \theta = \frac{3}{5} = 0.6 \qquad \cos \theta = \frac{4}{5} = 0.8 \qquad \tan \theta = \frac{3}{4} = 0.75$$

Why sine and cosine are never above 1

The hypotenuse is the longest side, so both the opposite and the adjacent side are shorter than it.

$$0 < \sin \theta < 1 \qquad 0 < \cos \theta < 1 \qquad \text{for an acute } \theta$$

Tangent has no such ceiling, because it compares two legs and either one can be the longer.

This is a fast sanity check. A sine of $1.4$ means a side was put in the wrong place.

Why sin θ equals cos of its complement

The two acute angles of a right triangle add to $90°$, since the third angle takes $90°$ of the $180°$ total.

Call them $\theta$ and $90° - \theta$. Now look at one leg. It is opposite one of those angles and adjacent to the other — the same segment, wearing two labels depending on which angle you stand at.

$$\sin \theta = \cos(90° - \theta)$$ $$\sin 30° = \cos 60° = 0.5$$

This is why the two functions are called co-functions. The prefix records exactly this relationship: the cosine of an angle is the sine of its complement.

Finding a side

With one angle and one side, every other side follows.

A ramp rises at $15°$ and is $20$ ft long along the slope. How high does it rise?

The $20$ ft is the hypotenuse. The rise is opposite the $15°$ angle. Opposite and hypotenuse means sine:

$$\sin 15° = \frac{h}{20} \quad\Rightarrow\quad h = 20 \sin 15° \approx 5.2 \text{ ft}$$

The method is always the same three steps:

1. Label the sides relative to the known angle.
2. Pick the ratio that uses the side you know and the side you want.
3. Solve for the unknown.

Step 2 is the decision. Choose the ratio by which two sides are involved, not by habit — sine is not the default.

Known	Wanted	Ratio
hypotenuse	opposite	sine
hypotenuse	adjacent	cosine
adjacent	opposite	tangent

Worked examples

Common mistakes

Practice problems