Table of Contents

Welcome to **arcsec 1**, our post aboutthe arcsecant of 1.

For the inverse trigonometric function of secant 1 we usually employ the abbreviation *arcsec* and write it as arcsec 1 or arcsec(1).

If you have been looking for *what is arcsec 1*, either in degrees or radians, or if you have been wondering about the inverse of sec 1, then you are right here, too.

In this post you can find the angle arcsecant of 1, along with identities.

Read on to learn all about the arcsec of 1.

## Arcsec of 1

If you want to know *what is arcsec 1* in terms of trigonometry, check out the explanations in the last paragraph; ahead in this section is the value of arcsecant(1):

arcsecant 1 = 0 rad = 0 °

arcsecant of 1 = 0 radians = 0 degrees

The arcsec of 1 is 0 radians, and the value in degrees is 0°. To change the result from the unit radian to the unit degree multiply the angle by 180° / $\pi$ and obtain 0°.

Our results above contain fractions of π for the results in radian, and are exact values otherwise. If you compute arcsec(1), and any other angle, using the calculator below, then the value will be rounded to ten decimal places.

To obtain the angle in degrees insert 1 as decimal in the field labelled “x”. However, if you want to be given the angle of sec 1 in radians, then you must press the swap units button.

### Calculate arcsec x

The identities of arcsecant 1 are as follows: arcsec(1) =

- $\frac{\pi}{2}$ – arccsc(1) ⇔ 90°- arccsc(1)
- $\pi$ – arcsec(-1) ⇔ 180° – arcsec(-1)
- arccos($\frac{1}{1}$)

The infinite series of arcsec 1 is: $\frac{\pi}{2} – \sum_{n=0}^{\infty}\frac{\binom{2n}{n}(1)^{-(2n+1)}}{4^{n}(2n+1)}$.

Next, we discuss the derivative of arcsec 1 for 1 = 1. In the following paragraph you can additionally learn what the search calculations form in the sidebar is used for.

## Derivative of arcsec 1

The derivative of arcsec 1 is not defined because it would mean a division by zero. The same applies to its integral.

The relationship of arcsec of 1 and the trigonometric functions sin, cos and tan is:

- sin(arcsecant(1)) = $\frac{\sqrt{(1)^{2}-1}}{1}$
- cos(arcsecant(1)) = $\frac{1}{1}$
- tan(arcsecant(1)) = $\sqrt{(1)^{2}-1}$

Note that you can locate many terms including the arcsecant(1) value using the search form. On mobile devices you can find it by scrolling down. Enter, for instance, arcsec1 angle.

Using the aforementioned form in the same way, you can also look up terms including derivative of inverse secant 1, inverse secant 1, and derivative of arcsec 1, just to name a few.

In the next part of this article we discuss the trigonometric significance of arcsecant 1, and there we also explain the difference between the inverse and the reciprocal of sec 1.

## What is arcsec 1?

In a triangle which has one angle of 90 degrees, the cosine of the angle α is the ratio of the length of the adjacent side a to the length of the hypotenuse h: cos α = a/h.

In a circle with the radius r, the horizontal axis x, and the vertical axis y, α is the angle formed by the two sides x and r; r moving counterclockwise defines the positive angle.

As follows from the unit-circle definition on our homepage, assumed r = 1, in the intersection of the point (x,y) and the circle, cos α = x / r = x, and sec α = 1 / x = 1. The angle whose secant value equals 1 is α.

In the interval [0, π/2[ ∪ ]π/2, π] or [0°, 90°[ ∪ ]90°, 180°], there is only one α whose secant value equals 1. For that interval we define the function which determines the value of α as

From the definition of arcsec(1) follows that the *inverse* function y^{-1} = sec(y) = 1. Observe that the *reciprocal* function of sec(y),(sec(y))^{-1} is 1/sec(y) = cos(y).

Avoid misconceptions and remember (sec(y))^{-1} = 1/sec(y) ≠ sec^{-1}(y) = arcsec(1). And make sure to understand that the trigonometric function y=arcsec(x) is defined on a restricted domain, where it evaluates to a single value only, called the principal value:

In order to be injective, also known as one-to-one function, y = arcsec(x) if and only if sec y = x and 0 ≤ y < π/2 or sec y = x and π/2 < y ≤ π. The domain of x is x ≤ −1 or 1 ≤ x.

## Conclusion

The frequently asked questions in the context include *what is arcsec 1 degrees* and *what is the inverse secant 1* for example; reading our content they are no-brainers.

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