Table of Contents

Welcome to **arctg 0**, our post aboutthe arcotangent of 0.

For the inverse trigonometric function of cotangent 0 we usually employ the abbreviation *arctg* and write it as arctg 0 or arctg(0).

If you have been looking for *what is arctg 0*, either in degrees or radians, or if you have been wondering about the inverse of ctg 0, then you are right here, too.

In this post you can find the angle arcotangent of 0, along with identities.

Read on to learn all about the arctg of 0.

## Arcctg of 0

If you want to know *what is arctg 0* in terms of trigonometry, check out the explanations in the last paragraph; ahead in this section is the value of arcotangent(0):

arcotangent 0 = pi/2 rad = 90 °

arcotangent of 0 = pi/2 radians = 90 degrees

The arctg of 0 is pi/2 radians, and the value in degrees is 90°. To change the result from the unit radian to the unit degree multiply the angle by 180° / $\pi$ and obtain 90°.

Our results above contain fractions of pi for the results in radian, and are exact values otherwise. If you compute arctg(0), 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 0 as decimal in the field labelled “x”. However, if you want to be given the angle of ctg 0 in radians, then you must press the swap units button.

### Calculate arcctg x

The identities of arcotangent 0 are as follows: arctg(0) =

- $\frac{\pi}{2}$ – arctg(0) ⇔ 90°- arctg(0)
- $\pi$ – arctg(-0) ⇔ 180°- arctg(-0)

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

Next, we discuss the derivative of arctg 0 for x = 0. In the following paragraph you can additionally learn what the search calculations form in the sidebar is used for.

## Derivative of arctg 0

The derivative of arctg 0 is particularly useful to calculate the inverse cotangent 0 as an integral.

The formula for x is (arctg x)’ = – $\frac{1}{1 + x^{2}}$, x ≠ -1,1, so for x = 0 the derivative equals -1.

Using the arctg 0 derivative, we can calculate the angle as a definite integral:

arctg 0 = $\frac{\pi}{2}$ – $\int_{0}^{\infty}\frac{1}{{1+z^{2}}}dz$.

The relationship of arctg of 0 and the trigonometric functions sin and cos is:

- sin(arcotangent(0)) = $\frac{1}{\sqrt{1 + (0)^{2}}}$
- cos(arcotangent(0)) = $\frac{0}{\sqrt{1 + (0)^{2}}}$

Note that you can locate many terms including the arcotangent(0) value using the search form. On mobile devices you can find it by scrolling down. Enter, for instance, arctg0 angle.

Using the aforementioned form in the same way, you can also look up terms including derivative of inverse cotangent 0, inverse cotangent 0, and derivative of arctg 0, just to name a few.

In the next part of this article we discuss the trigonometric significance of arcotangent 0, and there we also explain the difference between the inverse and the reciprocal of ctg 0.

## What is arctg 0?

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 = 0. The angle whose cotangent value equals 0 is α.

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

From the definition of arcctg(0) follows that the *inverse* function y^{-1} = ctg(y) = 0. 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(0). And make sure to understand that the trigonometric function y=arctg(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 = arctg(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 arctg 0 degrees* and *what is the inverse cotangent 0* for example; reading our content they are no-brainers.

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