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Welcome to **arccsc -2sqrt(3)/3**, our post aboutthe arccosecant of -2sqrt(3)/3.

For the inverse trigonometric function of arccsc -2sqrt(3)/3 we usually employ the abbreviation *arccsc* and write it as arccsc -2sqrt(3)/3 or arccsc(-2sqrt(3)/3).

If you have been looking for *what is arccsc -2sqrt(3)/3*, either in degrees or radians, or if you have been wondering about the inverse of csc -2sqrt(3)/3, then you are right here, too.

In this post you can find the angle arccosecant of -2sqrt(3)/3, along with identities.

Read on to learn all about the arccsc of -2sqrt(3)/3, and note that the term -2sqrt(3)/3 is approximately -1.15470053 as a decimal number.

## Arccsc of -2sqrt(3)/3

If you want to know *what is arccsc -2sqrt(3)/3* in terms of trigonometry, check out the explanations in the last paragraph; ahead in this section is the value of arccosecant(-2sqrt(3)/3):

arccosecant -2sqrt(3)/3 = -pi/3 rad = -60 °

arccosecant of -2sqrt(3)/3 = -pi/3 radians = -60 degrees

The arccsc of -2sqrt(3)/3 is -pi/3 radians, and the value in degrees is -60°. To change the result from the unit radian to the unit degree multiply the angle by 180° / $\pi$ and obtain -60°.

Our results above contain fractions of pi for the results in radian, and are exact values otherwise. If you compute arccsc(-2sqrt(3)/3), 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 -2sqrt(3)/3 as decimal in the field labelled “x”. However, if you want to be given the angle of csc -2sqrt(3)/3 in radians, then you must press the swap units button.

### Calculate arccsc x

The identities of arccosecant -2sqrt(3)/3 are as follows: arccsc(-2sqrt(3)/3) =

- $\frac{\pi}{2}$ – arcsec(-2sqrt(3)/3) ⇔ 90°- arcsec(-2sqrt(3)/3)
- -arccsc(2sqrt(3)/3)
- arcsin(1/-2sqrt(3)/3)

The infinite series of arccsc -2sqrt(3)/3 is: $\sum_{n=0}^{\infty}\frac{\binom{2n}{n}(-2\frac{\sqrt{3}}{3})^{-(2n+1)}}{4^{n}(2n+1)}$.

Next, we discuss the derivative of arccsc -2sqrt(3)/3 for -2sqrt(3)/3 = -2sqrt(3)/3. In the following paragraph you can additionally learn what the search calculations form in the sidebar is used for.

## Derivative of arccsc -2sqrt(3)/3

The derivative of arccsc -2sqrt(3)/3 is particularly useful to calculate the inverse arccsc -2sqrt(3)/3 as an integral.

The formula for x is (arccsc x)’ = -$\frac{1}{|x|\sqrt{x^{2}-1}}$, |x| > 1, so for x = -2sqrt(3)/3 the derivative equals -1.5.

Using the arccsc -2sqrt(3)/3 derivative, we can calculate the angle as a definite integral:

arccsc -2sqrt(3)/3 = $\int_{-2\frac{\sqrt{3}}{3}}^{\infty}\frac{1}{z\sqrt{z^{2}-1}}dz$.

The relationship of arccsc of -2sqrt(3)/3 and the trigonometric functions sin, cos and tan is:

- sin(arccosecant(-2sqrt(3)/3)) = $\frac{1}{(-2\frac{\sqrt{3}}{3})}$
- cos(arccosecant(-2sqrt(3)/3)) = $\frac{\sqrt{(-2\frac{\sqrt{3}}{3})^{2}-1}}{-2\frac{\sqrt{3}}{3}}$
- tan(arccosecant(-2sqrt(3)/3)) = $\frac{1}{\sqrt{(-2\frac{\sqrt{3}}{3})^{2}-1}}$

Note that you can locate many terms including the arccosecant(-2sqrt(3)/3) value using the search form. On mobile devices you can find it by scrolling down. Enter, for instance, arccsc-2sqrt(3)/3 angle.

Using the aforementioned form in the same way, you can also look up terms including derivative of inverse arccsc -2sqrt(3)/3, inverse arccsc -2sqrt(3)/3, and derivative of arccsc -2sqrt(3)/3, just to name a few.

In the next part of this article we discuss the trigonometric significance of arccosecant -2sqrt(3)/3, and there we also explain the difference between the inverse and the reciprocal of csc -2sqrt(3)/3.

## What is arccsc -2sqrt(3)/3?

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 = -2sqrt(3)/3. The angle whose arccsc value equals -2sqrt(3)/3 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 -2sqrt(3)/3. For that interval we define the function which determines the value of α as

From the definition of arccsc(-2sqrt(3)/3) follows that the *inverse* function y^{-1} = csc(y) = -2sqrt(3)/3. 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(-2sqrt(3)/3). 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 -2sqrt(3)/3 degrees* and *what is the inverse arccsc -2sqrt(3)/3* for example; reading our content they are no-brainers.

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