# cot 45°

## Definition

The value of cotangent when the angle of right angled triangle equals to $45$ degrees is called $\cot 45^\circ$.

The value of $\cot 45^\circ$ is calculated by calculating the ratio of length of adjacent side to length of opposite side when the angle of the right angled triangle equals to $45^\circ$.

### Proof

The value of $\cot 45^\circ$ can be calculated in mathematics in two different mathematical systems.

1

#### Algebraic approach

Algebraically, the value of $\cot 45^\circ$ can be derived by considering the properties of the triangle.

On the basis of properties of the triangle, the length of the opposite side is equal to length of the adjacent side in the case of a right angled triangle when the angle of the right angled triangle is $45$ degrees.

$$\cot 45^\circ = \frac{Length \, of \, Adjacent \, side}{Length \, of \, Opposite \, side}$$

In right angled triangle ($\Delta QPR$), the length of the opposite and adjacent sides are equal to $l$.

$$\implies \cot 45^\circ = \frac{l}{l}$$

$\therefore \,\, \cot 45^\circ = 1$

2

#### Geometric approach

Geometrically, the value of cotangent of angle $45^\circ$ can be derived by constructing a right angled triangle on the basis of drawing a straight line of any length with $45^\circ$ angle.

1. Draw a horizontal line, which will be later used as a base line while using protractor. The left side point of the horizontal line is denoted by point $S$.
2. Take protractor. Identity a point at $45$ degrees angle and mark it by coinciding the point $S$ with centre of the protractor and horizontal line with right side base line of protractor.
3. Draw a straight line from point $S$ through $45^\circ$ angle marked point by using a straight scale.
4. Take compass and set the length to $7.5$ Centimeters, and then draw an arc on the $45$ degrees angle line. The arc cuts the $45$ degrees angle line at point $T$.
5. Finally, draw a line from point $T$ to horizontal line perpendicularly and it touches the horizontal line at point $U$.

A right angled triangle, denoted as $\Delta TSU$, is constructed geometrically and the angle of the triangle is $45^\circ$. So, calculate the cotangent of angle $45^\circ$.

$$\cot 45^\circ = \frac{Length \, of \, Adjacent \, side}{Length \, of \, Opposite \, side}$$

Measure the lengths of both opposite and adjacent sides by using scale. You will observe that the lengths of both opposite and adjacent sides are equal to $5.3$ centimetres.

$$\implies \cot 45^\circ = \frac{5.3}{5.3}$$

$\therefore \,\, \cot 45^\circ = 1$

##### Result

Algebraic and geometric methods both have proved that the value of $\cot 45^\circ$ is $1$.

###### Representation

It is expressed in three different forms by considering the three angle measuring systems.

It is frequently expressed in sexagesimal system as,

$\cot 45^° = 1$

It is also expressed in circular system as,

$$\cot \Bigg(\frac{\pi}{4}\Bigg) = 1$$

It is also expressed in centesimal system as,

$\cot 50^g = 1$