In a right triangle, the side ratios around an acute angle depend only on the angle, not on the overall size of the triangle. This gives the three basic trigonometric ratios:

$$\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}},$$

$$\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}},$$

$$\tan\theta=\frac{\text{opposite}}{\text{adjacent}}.$$

The words opposite and adjacent are always relative to the chosen angle $\theta$.

Finding a side

Suppose a right triangle has hypotenuse $10$ and an acute angle of $30^\circ$. If $x$ is the side opposite the angle, then

$$\sin30^\circ=\frac{x}{10}.$$

Since $\sin30^\circ=1/2$,

$$x=5.$$

The same reasoning works with cosine or tangent when different sides are known.

Ratios determine the angle

For similar right triangles, the same acute angle always gives the same three side ratios. Conversely, a side ratio determines the acute angle. For example, a triangle with

$$\frac{\text{opposite}}{\text{adjacent}}=\frac34$$

has an angle whose tangent is $3/4$.

Later, inverse trigonometric functions provide notation and a direct numerical method for recovering that angle.

Right-triangle trigonometry turns geometric relationships between sides and angles into numerical ratios.