Table of Contents

Welcome to **arcctg -1**, our post aboutthe arccotangent of -1.

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

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

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

Read on to learn all about the arcctg of -1.

## Arcctg of -1

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

arccotangent -1 = 3pi/4 rad = 135 °

arccotangent of -1 = 3pi/4 radians = 135 degrees

The arcctg of -1 is 3pi/4 radians, and the value in degrees is 135°. To change the result from the unit radian to the unit degree multiply the angle by 180° / $\pi$ and obtain 135°.

Our results above contain fractions of pi for the results in radian, and are exact values otherwise. If you compute arcctg(-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 ctg -1 in radians, then you must press the swap units button.

### Calculate arcctg x

The identities of arccotangent -1 are as follows: arcctg(-1) =

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

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

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

## Derivative of arcctg -1

The derivative of arcctg -1 is particularly useful to calculate the inverse cotangent -1 as an integral.

The formula for x is (arcctg x)’ = – $\frac{1}{1 + x^{2}}$, x ≠ -1,1, so for x = -1 the derivative equals -0.5.

Using the arcctg -1 derivative, we can calculate the angle as a definite integral:

arcctg -1 = $\frac{\pi}{2}$ – $\int_{-1}^{\infty}\frac{1}{{1+z^{2}}}dz$.

The relationship of arcctg of -1 and the trigonometric functions sin, cos and tg is:

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

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

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

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

## What is arcctg -1?

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 = -1. The angle whose cotangent value equals -1 is α.

In the interval ]0, pi[ or ]0°, 180°[, there is only one α whose arccotangent value equals -1. For that interval we define the function which determines the value of α as

From the definition of arcctg(-1) follows that the *inverse* function y^{-1} = ctg(y) = -1. 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(-1). 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 -1 degrees* and *what is the inverse cotangent -1* for example; reading our content they are no-brainers.

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– Article written by Mark