$\cos{(45^\circ)}$ 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.