Math Doubts

$\csc{45^\circ}$ value

Exact value

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


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.

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

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

cosec 45 degrees value

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.

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

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


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 $45^\circ$.

Math Doubts

A best free mathematics education website that helps students, teachers and researchers.

Maths Topics

Learn each topic of the mathematics easily with understandable proofs and visual animation graphics.

Maths Problems

A math help place with list of solved problems with answers and worksheets on every concept for your practice.

Learn solutions

Subscribe us

You can get the latest updates from us by following to our official page of Math Doubts in one of your favourite social media sites.

Copyright © 2012 - 2022 Math Doubts, All Rights Reserved