# $\cos{(45^\circ)}$ value

$\cos{(45^\circ)} \,=\, \dfrac{1}{\sqrt{2}}$

The value of cosine in a forty five degrees right triangle is called the cosine of angle forty five degrees.

## Introduction

In a forty five degrees right angled triangle, the cosine of angle forty five degrees is a value that expresses the ratio of the length of adjacent side to the length of hypotenuse.

In the sexagesimal system, the cosine of angle forty five degrees is written as $\cos{(45^\circ)}$ mathematically and its exact value in fraction form is one by square root of two. Therefore, it is written in the following form in trigonometry mathematics.

$\cos{(45^\circ)} \,=\, \dfrac{1}{\sqrt{2}}$

Actually, the cosine of angle $45$ degrees is an irrational number and its exact value can be written in decimal form as follows.

$\implies$ $\cos{(45^\circ)} \,=\, 0.7071067812\ldots$

The exact value of cosine of angle forty five degrees in decimal form is approximately taken in some cases and its approximate value is given below.

$\implies$ $\cos{(45^\circ)} \,\approx\, 0.7071$

In fact, the triangle whose angle is forty five degrees is a special triangle as per its properties. Hence, the cosine of angle forty five degrees is generally called the trigonometric ratio for standard angle.

### Circular system

The cos of angle forty five degrees is expressed as $\cos{\Big(\dfrac{\pi}{4}\Big)}$ in circular system. It is read as the cosine of angle pi by four.

$\cos{\Big(\dfrac{\pi}{4}\Big)}$ $\,=\,$ $\dfrac{1}{\sqrt{2}}$ $\,=\,$ $0.7071067812\ldots$

### Centesimal system

Similarly, the cos of angle $45$ degrees is also written as $\cos{\big(50^g\big)}$ in centesimal system. It is read as the cosine of angle fifty gradians or grades.

$\cos{\big(50^g\big)}$ $\,=\,$ $\dfrac{1}{\sqrt{2}}$ $\,=\,$ $0.7071067812\ldots$

#### Proof

Learn how to prove the cosine of angle pi by four is equal to the quotient of one by square root of two in a geometric method.

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