$\sec{45^\circ}$ value

Exact value

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

Introduction

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

According to the sexagesimal system, the angle forty five degrees is mathematically written as $45^\circ$ and the sec of $45$ degrees is written as $\sec{45^\circ}$ in mathematical form.

Radical form

The exact value of trigonometric function secant for a standard angle $45$ degrees is an irrational number and it is equal to the square root of two.

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

Decimal form

In decimal form, the sec $45$ degrees value is obtained as an irrational number with infinitely extended digits and its approximate value is equal to $1.4142$.

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

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

Other forms

Alternatively, the secant of $45$ degrees is written in trigonometry in two different forms.

Circular system

According to the circular system, the secant of angle $45$ degrees is written as secant of pi divided by four. The exact value of sec $\pi$ divided by $4$ is equal to $\sqrt{2}$ and its approximate value in decimal form is $1.4142$.

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

Centesimal system

In the same way, the sec of standard angle $45$ degrees is written as sec of fifty gradians as per centesimal system. So, the exact value of sec $50$ grades is equal to the square root of $2$ and its value in decimal form is $1.4142$ approximately.

$\sin{(50^g)}$ $\,=\,$ $\sqrt{2}$ $\,\approx\,$ $1.4142$

Proof

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

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