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 = π rad = 180 °

arcsecant of -1 = π radians = 180 degrees

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

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, arcsec-1 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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