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

The value of $\cos 30^\circ$ can be evaluated by calculating the ratio of length of adjacent side to length of hypotenuse in a right angled triangle when its angle is equal to $30^\circ$.

The value of $\cos 30^\circ$ can be calculated by considering properties of triangle when its angle is $30^\circ$ and it also can obtain by constructing a right angled triangle with $30$ degrees.

1

According to properties of the right angled triangle, the length of the hypotenuse is equal to twice the length of the opposite side in the right angled triangle angle whose angle equals to $30^\circ$.

On the basis of this principle, it is proved that the length of the adjacent side is $\frac{\sqrt{3}}{2}$ times to length of hypotenuse by applying Pythagorean Theorem to the right angled triangle whose angle is $30^\circ$.

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

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

The ratio of length of adjacent side to length of hypotenuse is cosine and the angle of the triangle is $30^\circ$. Therefore, the ratio is $\cos 30^\circ$.

$$\therefore \,\, \cos 30^\circ = \frac{\sqrt{3}}{2}$$

$$\implies \,\, \cos 30^\circ = \frac{\sqrt{3}}{2} = 0.866025403…$$

2

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

- Draw a horizontal line by using ruler and the left side point of the line is named as point $E$.
- Use protractor, coincide point $E$ with centre of the protractor and also coincide horizontal line with right side of the base line. Identify $30^\circ$ angle by considering bottom scale and mark it.
- Draw a straight line from point $E$ through $30$ degrees marked point by using ruler.
- Take compass, and set it to have a distance of $9$ centimeters between pencil lead point and needle point of the compass by using ruler. Later, draw an arc from point $E$ on $30^\circ$ angle line. The intersection point of line by the arc is called as point $F$.
- Use set square and draw a perpendicular line to horizontal line from point $F$. The perpendicular line intersects the horizontal line at point $G$.

In this way, the right angled triangle ($\Delta FEG$) is constructed geometrically. The $9$ centimetres line become hypotenuse of the right angled triangle but the length of the adjacent side is unknown. However, it can be obtained by measuring its length using a ruler. In this triangle, the length of the adjacent side will be exactly $7.8$ centimetres.

The angle of this right angled triangle is $30^\circ$. So, calculate value of $\cos30^\circ$ for this triangle.

$$\cos 30^\circ = \frac{Length \, of \, Adjacent \, side}{Length \, of \, Hypotenuse}$$

$$\implies \cos 30^\circ = \frac{EG}{EF}$$

$$\implies \cos 30^\circ = \frac{7.8}{9}$$

$ \implies \cos 30^\circ = 0.8666… $

Observe the results of both methods. The value of $\cos 30^\circ$ from fundamental geometric approach is $0.866025403…$ and the value of $\cos 30^\circ$ from direct geometric approach is $0.8666…$

The values of $\cos 30^\circ$ from both methods are almost equal but there is very slight difference between them. The value of $\cos 30^\circ$ obtained from fundamental geometric approach is actual value because it is derived on the basis of geometric properties of the triangle but direct geometrical approach can only give approximate value and cannot give exact value due to problem in considering the exact value between one division.

The value is expressed in three different forms in trigonometry.

In sexagesimal system, it can be written in mathematical form as follows.

$$\cos 30^° = \frac{\sqrt{3}}{2}$$

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

$$\cos \Bigg(\frac{\pi}{6}\Bigg) = \frac{\sqrt{3}}{2}$$

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

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