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

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.

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.

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

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.

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

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

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

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.

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

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

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 $45^\circ$.

The math problems with solutions to learn how to solve a problem.

The math worksheets with answers for your practice with examples.

The math videos tutorials with visual graphics to learn every concept.

Get the latest math updates from the Math Doubts by subscribing us.

A free math education service for students to learn every math concept easily, for teachers to teach mathematics understandably and for mathematicians to share their maths researching projects.

Copyright © 2012 – 2023 Math Doubts, All Rights Reserved