The value of sine when the angle of the right angled triangle equals to $0^\circ$ is called $\sin 0^\circ$.

The exact value of $\sin 0^\circ$ is obtained mathematically by calculating the ratio of lengths of opposite side to hypotenuse when the angle of the right angled triangle is $0^\circ$.

The value of $\sin 0^\circ$ is derived in mathematics by using two fundamental geometrical approaches.

1

According to properties of the right angled triangle, when the angle of the right angled triangle is $0^\circ$, the length of the opposite side is zero but the length of the hypotenuse is not zero.

$Length \, of \, Opposite \, side = 0$

Calculate the ratio of length of opposite side to length of hypotenuse to obtain value of sine at $0^\circ$.

$$\frac{Length \, of \, Opposite \, side}{Length \, of \, Hypotenuse} = \frac{0}{d}$$

The ratio of length of opposite side to length of hypotenuse at angle $0^\circ$ is known as $\sin 0^\circ$.

$$\sin 0^\circ = \frac{0}{d}$$

$\implies \sin 0^\circ = 0$

2

Geometrically, the value of $\sin 0^\circ$ can be calculated by constructing a right angled triangle using geometric tools.

- Take a ruler and draw a straight line of any length in horizontal direction. The left side point of it is called as point $G$.
- Take protractor and coincide its centre with point $G$ and also coincide the right side base line with horizontal line. Identify $0^\circ$ angle by using bottom scale but the horizontal line direction is exact $0^\circ$ angle. So, there is no use of protractor in this case.
- Take ruler and compass, then set the distance between points of pencil’s lead and needle to $10$ centimetres. Later, draw an arc from point $G$ on $0$ degrees angle line and it cuts the line at point $H$.
- A perpendicular line should be drawn from point $H$ to horizontal line but it’s not possible to do it in this case because the $0^\circ$ angle line and horizontal line are at same position on the plane. Therefore, assume a perpendicular straight line is drawn from point $H$ to horizontal line and it meets the horizontal line at point $I$.

Right angled triangle ($\Delta HGI$) is constructed according to this geometrical procedure.

In right angled triangle $HGI$, the points $H$ and $I$ are at same location. So, the distance between them is zero. Therefore, the length of the opposite side ($HI$) is zero. The length of the hypotenuse ($GH$) is $10$ centimeters. The angle of the right angled triangle is $0^\circ$.

Now, calculate the value of $\sin 0^\circ$ by using this information.

$$\sin 0^\circ = \frac{HI}{GH}$$

$$\implies \sin 0^\circ = \frac{0}{10}$$

$\implies \sin 0^\circ = 0$

The fundamental geometrical approach and geometrical approach by using geometrical tools have given same result.

$\therefore \,\, \sin 0^\circ = 0$

It is expressed in mathematics in three possible forms as per different angle measuring systems.

In sexagesimal system, it is written as.

$\sin 0^° = 0$

In circular system, it is also written as.

$\sin 0 = 0$

In centesimal system, it is also expressed as.

$\sin 0^g = 0$