The value of tangent when the angle of the right angled triangle equals to $30^\circ$, is called $\tan 30^\circ$

The value of $\tan 30^\circ$ is obtained mathematically by calculating the ratio of length of opposite side to length of adjacent side when the angle of the right angled triangle equals to $30^\circ$.

The value of $\tan 30^\circ$ can be obtained in two mathematical approaches but they both are developed on the basis of fundamental geometry.

1

The length of the hypotenuse is twice the length of the opposite side and the length of the adjacent side is $\frac{\sqrt{3}}{2}$ times to length of the hypotenuse, according to properties of the right angled triangle when its angle equals to $30^\circ$.

The properties can be expressed in mathematical form.

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

$$Length \, of \, Adjacent \, side = \frac{\sqrt{3}}{2} \times Length \, of \, Hypotenuse$$

The value of $\tan 30^\circ$ can be derived by considering these two properties. It is actually done by eliminating length of the hypotenuse from these two expressions to obtain relation between lengths of opposite and adjacent sides.

$$Length \, of \, Adjacent \, side = \frac{\sqrt{3}}{2} \times (2 \times Length \, of \, Opposite \, side)$$

$$\implies Length \, of \, Adjacent \, side = \Bigg(\frac{\sqrt{3}}{2} \times 2\Bigg) \times Length \, of \, Opposite \, side$$

$$\implies Length \, of \, Adjacent \, side = \sqrt{3} \times Length \, of \, Opposite \, side$$

$$\implies \frac{1}{\sqrt{3}} = \frac{Length \, of \, Opposite \, side}{Length \, of \, Adjacent \, side}$$

$$\implies \frac{Length \, of \, Opposite \, side}{Length \, of \, Adjacent \, side} = \frac{1}{\sqrt{3}}$$

Lengths of opposite and adjacent side are in ratio form and the angle of the triangle is $30^\circ$. So, the ratio is $\tan 30^\circ$.

$$\therefore \,\, \tan 30^\circ = \frac{1}{\sqrt{3}}$$

$$\implies \tan 30^\circ = \frac{1}{\sqrt{3}} = 0.577350269…$$

2

The value of $\tan 30^\circ$ can be obtained geometrically by constructing a right angled triangle with $30^\circ$ angle.

- Use ruler and draw a line in horizontal direction. The left endpoint of the horizontal line is called as point $H$.
- Take protractor and mark a point at $30^\circ$ angle after coinciding point $H$ with centre of the protractor and horizontal line with right side base line of protractor.
- Take ruler and draw a line from point $H$ through the $30^\circ$ angle marked point.
- Take compass and adjust it to have $7$ centimeters length between needle point and point of pencil lead by considering the ruler. Put needle point on point $H$ and draw an arc on $30^\circ$ angle line. The arc cuts the $30$ degrees angle line at point $I$.
- Take set square and draw a line perpendicular to horizontal line and it meets the horizontal line at point $J$ perpendicularly.

As the result of this geometric procedure, a right angled triangle, known as $\Delta IHJ$ is constructed. The $7$ centimetres line become hypotenuse to this triangle but the lengths of opposite and adjacent sides are unknown.

Take centimeters ruler and measure the length of $\overline{IJ}$ to get the length of the opposite side and also measure the length of $\overline{HJ}$ to get the length of the adjacent side. The length of the opposite side will be $3.5$ centimetres exactly and the length of the adjacent side will be $6.05$ centimetres.

Now calculate the $\tan 30^\circ$ by using the lengths of them.

$$\tan 30^\circ = \frac{Length \, of \, Opposite \, side}{Length \, of \, Adjacent \, side}$$

$$\implies \tan 30^\circ = \frac{IJ}{HJ}$$

$$\implies \tan 30^\circ = \frac{3.5}{6.05}$$

$\implies \tan 30^\circ = 0.578512396…$

Compare values of $\tan 30^\circ$ of both geometrical methods. The value of $\tan 30^\circ$ is $0.577350269…$ from fundamental geometric approach and the value of $\tan 30^\circ$ is $0.578512396…$ from direct geometric method.

The values of $\tan 30^\circ$ from both methods are approximately same but there is slight difference between them. The value of $\tan 30^\circ$, obtained from fundamental geometric approach is exact value because it is obtained on the basis of properties of the triangle but the value of $\tan 30^\circ$, obtained from direct geometric approach cannot be exact value due to problem in measuring length of the adjacent side exactly.

$$\therefore \,\, \tan 30^° = \frac{1}{\sqrt{3}}$$

It is expressed in three different angle measuring systems in mathematics.

It is expressed in mathematics in sexagesimal system as follows.

$$\tan 30^° = \frac{1}{\sqrt{3}}$$

It is expressed in mathematics in circular system as follows.

$$\tan \Bigg(\frac{\pi}{6}\Bigg) = \frac{1}{\sqrt{3}}$$

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

$$\tan {33\frac{1}{3}}^g = \frac{1}{\sqrt{3}}$$