Table of Contents

Welcome to **arccsc -sqrt2**, our post aboutthe arccosecant of -sqrt2.

For the inverse trigonometric function of arccsc -sqrt2 we usually employ the abbreviation *arccsc* and write it as arccsc -sqrt2 or arccsc(-sqrt2).

If you have been looking for *what is arccsc -sqrt2*, either in degrees or radians, or if you have been wondering about the inverse of csc -sqrt2, then you are right here, too.

In this post you can find the angle arccosecant of -sqrt2, along with identities.

Read on to learn all about the arccsc of -sqrt2, and note that the term -sqrt2 is approximately -1.41421356 as a decimal number.

## Arccsc of -sqrt2

If you want to know *what is arccsc -sqrt2* in terms of trigonometry, check out the explanations in the last paragraph; ahead in this section is the value of arccosecant(-sqrt2):

arccosecant -sqrt2 = -pi/4 rad = -45 °

arccosecant of -sqrt2 = -pi/4 radians = -45 degrees

The arccsc of -sqrt2 is -pi/4 radians, and the value in degrees is -45°. To change the result from the unit radian to the unit degree multiply the angle by 180° / $\pi$ and obtain -45°.

Our results above contain fractions of pi for the results in radian, and are exact values otherwise. If you compute arccsc(-sqrt2), 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 -sqrt2 as decimal in the field labelled “x”. However, if you want to be given the angle of csc -sqrt2 in radians, then you must press the swap units button.

### Calculate arccsc x

The identities of arccosecant -sqrt2 are as follows: arccsc(-sqrt2) =

- $\frac{\pi}{2}$ – arcsec(-sqrt2) ⇔ 90°- arcsec(-sqrt2)
- -arccsc(sqrt2)
- arcsin(1/-sqrt2)

The infinite series of arccsc -sqrt2 is: $\sum_{n=0}^{\infty}\frac{\binom{2n}{n}(-\sqrt{2})^{-(2n+1)}}{4^{n}(2n+1)}$.

Next, we discuss the derivative of arccsc -sqrt2 for -sqrt2 = -sqrt2. In the following paragraph you can additionally learn what the search calculations form in the sidebar is used for.

## Derivative of arccsc -sqrt2

The derivative of arccsc -sqrt2 is particularly useful to calculate the inverse arccsc -sqrt2 as an integral.

The formula for x is (arccsc x)’ = -$\frac{1}{|x|\sqrt{x^{2}-1}}$, |x| > 1, so for x = -sqrt2 the derivative equals -0.7071067812.

Using the arccsc -sqrt2 derivative, we can calculate the angle as a definite integral:

arccsc -sqrt2 = $\int_{-\sqrt{2}}^{\infty}\frac{1}{z\sqrt{z^{2}-1}}dz$.

The relationship of arccsc of -sqrt2 and the trigonometric functions sin, cos and tan is:

- sin(arccosecant(-sqrt2)) = $\frac{1}{(-\sqrt{2})}$
- cos(arccosecant(-sqrt2)) = $\frac{\sqrt{(-\sqrt{2})^{2}-1}}{-\sqrt{2}}$
- tan(arccosecant(-sqrt2)) = $\frac{1}{\sqrt{(-\sqrt{2})^{2}-1}}$

Note that you can locate many terms including the arccosecant(-sqrt2) value using the search form. On mobile devices you can find it by scrolling down. Enter, for instance, arccsc-sqrt2 angle.

Using the aforementioned form in the same way, you can also look up terms including derivative of inverse arccsc -sqrt2, inverse arccsc -sqrt2, and derivative of arccsc -sqrt2, just to name a few.

In the next part of this article we discuss the trigonometric significance of arccosecant -sqrt2, and there we also explain the difference between the inverse and the reciprocal of csc -sqrt2.

## What is arccsc -sqrt2?

In a triangle which has one angle of 90 degrees, the sine of the angle α is the ratio of the length of the opposite side o to the length of the hypotenuse h: sin α = o/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, sin α = y / r = y, and csc α = 1 / y = -sqrt2. The angle whose arccsc value equals -sqrt2 is α.

In the interval [-pi/2, 0[ ∪ ]0, pi/2] or [-90°, 0[ ∪ ]0, 90°], there is only one α whose sine value equals -sqrt2. For that interval we define the function which determines the value of α as

From the definition of arccsc(-sqrt2) follows that the *inverse* function y^{-1} = csc(y) = -sqrt2. Observe that the *reciprocal* function of csc(y),(csc(y))^{-1} is 1/csc(y) = sin(y).

Avoid misconceptions and remember (csc(y))^{-1} = 1/csc(y) ≠ csc^{-1}(y) = arccsc(-sqrt2). And make sure to understand that the trigonometric function y=arccsc(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 = arccsc(x) if and only if csc y = x and -pi/2 ≤ y < 0 or csc y = x and 0 < y ≤ pi/2. The domain of x is x ≤ −1 or 1 ≤ x.

## Conclusion

The frequently asked questions in the context include *what is arccsc -sqrt2 degrees* and *what is the inverse arccsc -sqrt2* for example; reading our content they are no-brainers.

But, if there is something else about the topic you would like to know, fill in the form on the bottom of this post, or send us an email with a subject line such as arccosecant -sqrt2 in radians.

If our calculator and the information on csc -sqrt2 inverse have been helpful, please hit the sharing buttons to spread the word about our content, and don’t forget to bookmark us.

Or, even better, install absolutely free PWA app (see menu or sidebar)!

Thanks for visiting our post.

– Article written by Mark, last updated on February 4th, 2017