$\csc{45^\circ}$ value

Exact value

$\csc{45^\circ} \,=\, \sqrt{2}$

Introduction

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

The angle forty five degrees is written $45^\circ$ in the sexagesimal system and the cosecant of forty five degrees is written $\csc{45^\circ}$. Similarly, it is also written as $\operatorname{cosec}{45^\circ}$ in mathematics.

Radical form

The exact value of cosecant for a standard angle $45$ degrees is a number in radical form and it is exactly equal to square root of two.

$\csc{(45^\circ)}$ $\,=\,$ $\sqrt{2}$

Decimal form

The cosec $45$ degrees value is an irrational number, and its exact value can be written as a number with infinitely extended digits in decimal form. However, the exact value of cosecant $45$ degrees can be written as $1.4142$ approximately.

$\csc{(45^\circ)}$ $\,=\,$ $1.4142135624\ldots$

$\implies$ $\csc{(45^\circ)}$ $\,\approx\,$ $1.4142$

Other forms

The cosecant of $45$ degrees is alternatively written in trigonometry in two distinct mathematical forms.

Circular system

In circular system, the cosecant of angle $45$ degrees is written as cosecant of pi divided by four. So, the exact value of cosecant $\pi$ divided by $4$ is equal to $\sqrt{2}$ and approximately equal to $1.4142$ in decimal form.

$(1).\,\,$ $\csc{\Big(\dfrac{\pi}{4}\Big)}$ $\,=\,$ $\sqrt{2}$ $\,\approx\,$ $1.4142$

$(2).\,\,$ $\operatorname{cosec}{\Big(\dfrac{\pi}{4}\Big)}$ $\,=\,$ $\sqrt{2}$ $\,\approx\,$ $1.4142$

Centesimal system

Likewise, the trigonometric ratio cosecant of $45$ degrees is written as cosecant of fifty grades in the centesimal system. Therefore, the csc $50$ gradians is equal to square root of $2$ and its exact value is approximately $1.4142$ in decimal form.

$(1).\,\,$ $\csc{(50^g)}$ $\,=\,$ $\sqrt{2}$ $\,\approx\,$ $1.4142$

$(2).\,\,$ $\operatorname{cosec}{(50^g)}$ $\,=\,$ $\sqrt{2}$ $\,\approx\,$ $1.4142$

Proof

Learn how to find the cosecant of $45$ degrees value as the square root of $2$ by constructing a right angled angle with angle of $45^\circ$.