# Sine

## Definition

The ratio of lengths of opposite side to hypotenuse at the respective angle of a right angled triangle is called sine.

Sine is a representative to disclose the relation between angle and the ratio of length of opposite side to length of hypotenuse in the right angled triangle. Hence, it is also called as trigonometric function.

### Expression

It is expressed as a ratio of lengths of opposite side to hypotenuse to express sine in mathematical form.

$$\frac{Length \, of \, Opposite \, Side}{Length \, of \, Hypotenuse}$$

But the ratio is calculated at the angle of the right angled triangle. So, the relation of ratio with angle of the right angled triangle is expressed as sine of angle. Mathematically, sine of angle is written by writing $\sin$ first and then angle of the right angled triangle.

According to this theory, the relation of ratio of lengths of opposite side to hypotenuse with angle is expressed in mathematical form as follows.

$$\sin \, (angle) = \frac{Length \, of \, Opposite \, side}{Length \, of \, Hypotenuse}$$

Sine is defined in trigonometry from ratio. So, it is also called as a trigonometric ratio.

For example, in $\Delta BAC$, the angle of the right angled triangle is $\theta$. So, sine of angle is written as $\sin \theta$. Lengths of opposite side and hypotenuse are $BC$ and $AC$ respectively.

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

#### Example

$\Delta DEF$ is another right angle triangle.

1. Length of the opposite side $(\overline{DF})$ is $DF = 3$ meters
2. Length of the hypotenuse $(\overline{DE})$ is $DE = 5$ meters
3. Angle of this right angled triangle is $36.87^°$

Express sine in mathematical form for this triangle.

$$\sin 36.87^\circ = \frac{DF}{DE}$$

$$\implies \sin 36.87^\circ = \frac{3}{5} = 0.6$$

It is read as the value of sine of $36.87^\circ$ is $0.6$.