$\cos{45^\circ}$ value

Exact value

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

Introduction

The value of cosine in a $45$ degrees right triangle is called the cosine of angle forty five degrees. According to sexagesimal system, the angle forty five degrees is written as $45^\circ$ mathematically and the cosine of $45$ degrees is written as $\cos{45^\circ}$. It is time to learn what the cos $45$ degrees in trigonometry is.

Fraction form

The cos $45$ degrees value is an irrational number, it can be written as a radical fraction, and its exact value is equal to $1$ divided by the square root of $2$.

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

Decimal form

The exact value of cos $45^\circ$ can also be calculated in decimal form and it is an infinite extended number. However, the cos of $45$ degrees value can be considered approximately as follows.

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

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

Other forms

Alternatively, the cos of $45$ degrees is written in two distinct mathematical forms in trigonometry.

Circular system

In circular system, the cosine of angle $45$ degrees is written as cos of pi divided by four radians. The exact value of cos $\pi$ divided by $4$ radians in fraction form is equal to $1$ divided by the square root of $2$ and it is approximate value in decimal form is $0.7071$.

$\cos{\Big(\dfrac{\pi}{4}\Big)}$ $\,=\,$ $\dfrac{1}{\sqrt{2}}$ $\,\approx\,$ $0.7071$

Centesimal system

Likewise, the cosine of forty five degrees is also written as cos of $50$ gradians in centesimal system, So, the exact value of cos of $50$ grades in fraction form is equal to one divided by $\sqrt{2}$. In decimal form, the cos of $50$ grads is equal to $0.7071$ approximately.

$\cos{(50^g)}$ $\,=\,$ $\dfrac{1}{\sqrt{2}}$ $\,\approx\,$ $0.7071$

Proof

Learn how to find the exact value of cos $45$ degrees as an irrational number $1$ divided by $2$ by constructing a right angled triangle with an angle $45^\circ$.