The value of sine when the angle of the right angled triangle equals to $30^\circ$, is called $\sin 30^\circ$.

The value of $\sin 30^\circ$ can possibly find in mathematics by calculating the ratio of length of opposite side to length of hypotenuse of right angled triangle when the angle of right angled triangle is $30^\circ$.

The value of $\sin 30^\circ$ can be derived in two different approaches but the system of development is common and it is geometric system.

1

In the case of right angled triangle whose angle is $30^\circ$, the length of hypotenuse is twice the length of the opposite side of the triangle. The value of sine of angle $30$ degrees can be derived on the basis of this principle.

$Length \, of \, Hypotenuse$ $=$ $2 \times Length \, of \, Opposite \, side$

$$\implies \frac{1}{2} = \frac{Length \, of \, Opposite \, side}{Length \, of \, Hypotenuse}$$

$$\implies \frac{Length \, of \, Opposite \, side}{Length \, of \, Hypotenuse} = \frac{1}{2}$$

The ratio of length of the opposite side to length of hypotenuse is called sine of an angle but the angle of the right angled triangle here is $30^\circ$. So, the ratio between two sides is sine of angle $30^\circ$.

$$\therefore \,\, \sin 30^\circ = \frac{1}{2}$$

The value of sine of angle $30^\circ$ is exactly real because the fundamental approach is derived by the system of geometric properties of the right angled triangle when the angle of the triangle is $30^\circ$.

2

The value of $\sin 30^\circ$ can also be derived practically by the direct geometric approach.

- Use ruler and draw a straight line horizontally. Call the left side point of this line as point $B$.
- Take protractor, coincide point $B$ with centre of the protractor and also coincide horizontal line with right side base line of the protractor. Then mark a point at $30$ degrees angle by considering the bottom scale of the protractor.
- Take ruler and draw a straight line from point $B$ through $30$ degrees angle point.
- Take compass and set it to have $8$ centimetres length between pencil’s lead and needle point by considering the ruler. Then, draw an arc from point $B$ on $30$ degrees angle line and call the intersecting point of arc and $30^\circ$ angle line as point $C$.
- Take set square, and draw a straight line from $C$ to horizontal line but the line should intersect it perpendicularly. Call the intersecting point as point $D$.

In this way, a right angled triangle ($\Delta CBD$) is constructed with $30^\circ$ angle and $8$ centimeters long line.

The $8$ centimetres long line is actually hypotenuse of this triangle but the length of opposite side is unknown. So, take ruler and measure the length of opposite side. You observe that the length of the opposite side is $4$ centimetres exactly.

$$\sin 30^\circ = \frac{Length \, of \, Opposite \, side}{Length \, of \, Hypotenuse}$$

$$\implies \sin 30^\circ = \frac{CD}{BC}$$

$$\implies \sin 30^\circ = \frac{4}{8}$$

$$\implies \sin 30^\circ = \frac{1}{2} = 0.5$$

The value of sine of angle $30$ degrees is derived in two methods and they both have given same value. So, it can be used anywhere without any doubt.

It is usually expressed in three different forms in mathematics.

In sexagesimal system, it is usually written in mathematical form as follows.

$$\sin 30^° = \frac{1}{2}$$

In circular system, it is also usually written in mathematics as follows.

$$\sin \Bigg(\frac{\pi}{6}\Bigg) = \frac{1}{2}$$

In centesimal system, it is also written in mathematical form as follows.

$$\sin {33\frac{1}{3}}^g = \frac{1}{2}$$