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

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

The value of $\cos 45^\circ$ can be calculated in two different mathematical methods.

1

Length of opposite side is equal to length of adjacent side if angle of the right angled triangle is $45^\circ$ according to properties of the triangle. So, the length of the hypotenuse should be $\sqrt{2}$ times to length of the adjacent side and it can be obtained by applying Pythagoras Theorem to this triangle.

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

$\Delta RPQ$ is a right angled triangle, which contains an angle $45^\circ$. So, lengths of both opposite and adjacent side are equal and length of each side is $l$. Length of hypotenuse is $r$.

$\therefore \, r = \sqrt{2} \times l$

$$\implies \frac{l}{r} = \frac{1}{\sqrt{2}}$$

$l$ is length of both opposite side and adjacent side and $r$ is length of hypotenuse. Here, we talking about cosine. So, replace $l$ as length of the adjacent side.

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

According to definition of cosine, the ratio of length of adjacent side to length of hypotenuse is cosine of an angle but angle of the right angled triangle here is $45^\circ$. The ratio between them can be represented as $\cos 45^\circ$.

$$\therefore \, \cos 45^\circ = \frac{1}{\sqrt{2}}$$

$$\therefore \, \cos 45^\circ = \frac{1}{\sqrt{2}} = 0.707106781…$$

2

The value of $\cos 45^\circ$ can also be obtained in geometrical method by drawing a straight line of any length with $45^\circ$ angle.

- Draw a horizontal line to use it as base line while using protractor. Call left side point of horizontal line as point $I$.
- Take protractor, coincide point $I$ with centre of the protractor and horizontal line with right side base line of the same protractor. Mark point at $45$ degrees angle by considering bottom angle scale of protractor.
- Draw a straight line from point $I$ through $45$ degree angle point.
- Take compass and set the length to $5.5$ centimeters. Then draw an arc on $45$ degrees angle line from point $I$. The arc cuts the $45$ degrees angle line and call it as point $J$.
- Use set square and draw a perpendicular line to horizontal line from point $J$ and it intersects the horizontal line perpendicularly at point $K$.

In this way, a right angled triangle, represented as $\Delta JIK$, is formed geometrically. In this right angled triangle, the line segment $\overline{IJ}$ is hypotenuse of the triangle and its length is $5.5$ centimetres but the lengths of both opposite side and adjacent side are unknown.

So, measure the length of opposite and adjacent sides by using centimetres scale and you observe that the lengths of both sides are same and the length of each side will be $3.9$ centimetres approximately. The angle of this triangle is $45^\circ$. Therefore, calculate the value of cosine of angle $45^\circ$ by the definition of cosine.

$$\cos 45^\circ = \frac{Length \, of \, Adjacent \, side}{Length \, of \, Hypotenuse}$$

$$\implies \cos 45^\circ = \frac{3.9}{5.5}$$

$\implies \cos 45^\circ = 0.709090…$

In algebraic method, the value of $\cos 45^\circ$ is $0.707106781…$ (or $\frac{1}{\sqrt{2}}$) but the value of $\cos 45^\circ$ is $0.709090…$ The two values are approximately same but there is some difference between them in value.

However, the value of $\cos 45^\circ$, which obtained from algebraic approach is absolute value because the value is actually calculated by considering the properties of the triangle. Due to either parallax error or problem in determining the decimal value while using scale to measure the length, geometrical approach may not give accurate value for $\cos 45^\circ$.

$$\therefore \, \cos 45^\circ = \frac{1}{\sqrt{2}}$$

The value is expressed in three different forms in mathematics.

It is expressed in sexagesimal system as,

$$\cos 45^° = \frac{1}{\sqrt{2}}$$

It is usually written in circular system as,

$$\cos \Bigg(\frac{\pi}{4}\Bigg) = \frac{1}{\sqrt{2}}$$

It is expressed in centesimal system as,

$$\cos 50^g = \frac{1}{\sqrt{2}}$$