A term that represents a ratio of lengths of opposite side to hypotenuse at an angle of a right triangle is called the sine.

Sine is a name and it is used to represent the ratio of lengths of opposite side to hypotenuse at a particular angle in a right triangle. It is written in ratio form and also written as sine with angle alternatively.

The value of sine at an angle is calculated by the ratio of lengths of opposite side to hypotenuse.

$\dfrac{Length \, of \, Opposite \, side}{Length \, of \, Hypotenuse}$

So, sine is called as a trigonometric ratio generally.

Alternatively, the value of sine at an angle is expressed mathematically by writing sine in its short form $\sin$ and then corresponding angle of the right triangle.

For example, if angle of right triangle is denoted by $x$, then sine of angle $x$ is written as $\sin{x}$ in trigonometry. $\sin{x}$ is a function form. So, it is usually called as sin function in mathematics.

Thus, sin functions like $\sin{A}$, $\sin{\alpha}$, $\sin{\beta}$, and etc. are defined in mathematics.

$\Delta BAC$ is a right angled triangle and its angle is theta ($\theta$).

sine of angle is written as $\sin{\theta}$ in this case.

$\sin{\theta} \,=\, \dfrac{Length \, of \, Opposite \, side}{Length \, of \, Hypotenuse}$

It is used as a formula to calculate the value of sine at any angle of the right triangle.

In this example, $BC$ is length of opposite side (perpendicular) and $AC$ is length of hypotenuse.

$\,\,\, \therefore \,\,\,\,\,\, \sin{\theta} \,=\, \dfrac{BC}{AC}$

The list of exact values of sine functions in fraction and decimal forms in a table with proofs.

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