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Welcome to **arcctg -√(3)/3**, our post aboutthe arccotangent of -√(3)/3.

For the inverse trigonometric function of cotangent -√(3)/3 we usually employ the abbreviation *arcctg* and write it as arcctg -√(3)/3 or arcctg(-√(3)/3).

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

In this post you can find the angle arccotangent of -√(3)/3, along with identities.

Read on to learn all about the arcctg of -√(3)/3, and note that the term -√(3)/3 is approximately -0.57735026 as a decimal number.

## Arcctg of -√(3)/3

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

arccotangent -√(3)/3 = 2pi/3 rad = 120 °

arccotangent of -√(3)/3 = 2pi/3 radians = 120 degrees

The arcctg of -√(3)/3 is 2pi/3 radians, and the value in degrees is 120°. To change the result from the unit radian to the unit degree multiply the angle by 180° / $\pi$ and obtain 120°.

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

### Calculate arcctg x

The identities of arccotangent -√(3)/3 are as follows: arcctg(-√(3)/3) =

- $\frac{\pi}{2}$ – arctg(-√(3)/3) ⇔ 90°- arctg(-√(3)/3)
- $\pi$ – arcctg(√(3)/3) ⇔ 180°- arctg(√(3)/3)
- arctg(1/-√(3)/3) + $\pi $

The infinite series of arcctg -√(3)/3 is: $\frac{\pi}{2}$ – $\sum_{n=0}^{\infty} \frac{(-1)^{n}(-\frac{\sqrt{3}}{3})^{2n+1}}{(2n+1)}$.

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

## Derivative of arcctg -√(3)/3

The derivative of arcctg -√(3)/3 is particularly useful to calculate the inverse cotangent -√(3)/3 as an integral.

The formula for x is (arcctg x)’ = – $\frac{1}{1 + x^{2}}$, x ≠ -1,1, so for x = -√(3)/3 the derivative equals -0.75.

Using the arcctg -√(3)/3 derivative, we can calculate the angle as a definite integral:

arcctg -√(3)/3 = $\frac{\pi}{2}$ – $\int_{-\frac{\sqrt{3}}{3}}^{\infty}\frac{1}{{1+z^{2}}}dz$.

The relationship of arcctg of -√(3)/3 and the trigonometric functions sin, cos and tg is:

- sin(arccotangent(-√(3)/3)) = $\frac{1}{\sqrt{1 + (-\frac{\sqrt{3}}{3})^{2}}}$
- cos(arccotangent(-√(3)/3)) = $\frac{-\frac{\sqrt{3}}{3}}{\sqrt{1 + (-\frac{\sqrt{3}}{3})^{2}}}$
- tg(arccotangent(-√(3)/3)) = 1/-√(3)/3

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

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

In the next part of this article we discuss the trigonometric significance of arccotangent -√(3)/3, and there we also explain the difference between the inverse and the reciprocal of ctg -√(3)/3.

## What is arcctg -√(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, and 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, sin α = y / r = y, cos α = x / r = x, and ctg α = x / y = -√(3)/3. The angle whose cotangent value equals -√(3)/3 is α.

In the interval ]0, pi[ or ]0°, 180°[, there is only one α whose arccotangent value equals -√(3)/3. For that interval we define the function which determines the value of α as

From the definition of arcctg(-√(3)/3) follows that the *inverse* function y^{-1} = ctg(y) = -√(3)/3. Observe that the *reciprocal* function of ctg(y),(ctg(y))^{-1} is 1/ctg(y).

Avoid misconceptions and remember (ctg(y))^{-1} = 1/ctg(y) ≠ ctg^{-1}(y) = arcctg(-√(3)/3). And make sure to understand that the trigonometric function y=arcctg(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 = arcctg(x) if and only if ctg y = x and 0 < y < pi. The domain of x is $\mathbb{R}$.

## Conclusion

The frequently asked questions in the context include *what is arcctg -√(3)/3 degrees* and *what is the inverse cotangent -√(3)/3* for example; reading our content they are no-brainers.

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