Ambiguity in the Arc-Tangent Function

[bw0young0math]

This problem is about complex analysis.
Why arctan(5/-5) is different from arctan-1(-5/5)?
tan-1(5/-5)=3ㅠ/4 +2kㅠ
tan-1(-5/5)=-1ㅠ/4+2kㅠ (k :arbtrary integer.)

Why can't I calculate just arctan(-1)?I thought that it is related withe tangent peoriod ㅠ not 2ㅠ.
However I can't know that exactly.
Please help me...

 

[Ackbach]

Re: Ambiguity in the Arc Tangent Function

You are correct in that the problem arises from the tangent function having period $\pi$ and not $2\pi$. Because of that, your calculator may not give you the correct quadrant when it computes the inverse (arctan). You have to think to yourself, "Hmm. My $x$ is positive, so I have to be in either the first or the fourth quadrant. $y$ is negative; therefore, I must be in the fourth quadrant." Actually, if you ask your calculator to compute
$$\arctan(-5/5)\quad \text{and} \quad \arctan(5/(-5)),$$
it will give you the same result both times. One of them is in error, however. Some programming languages get around this by implementing the atan2 function, which takes two arguments and returns the correct quadrant.

 

[bw0young0math]

Re: Ambiguity in the Arc Tangent Function

$$\arctan(-5/5)\quad \text{and} \quad \arctan(5/(-5)),$$
it will give you the same result both times. One of them is in error, however. Some programming languages get around this by implementing the atan2 function, which takes two arguments and returns the correct quadrant.

At first, thanks for your help.
I understood your meaning that they have same values consequently. Right?
However, arctan(5/(-5)) can't express -ㅠ/4. Thus I can't understand why they are the same.Also,you mentioned about x, y, and quadrant.
My friend and I thought the reason arctan(5/(-5)) and arctan((-5)/5) are different as discussing x,y,and quadrant also.
However we didn't know why we put one of x and y denominator or numerator. Could you explain this?

 

[Ackbach]

Re: Ambiguity in the Arc Tangent Function

At first, thanks for your help.
I understood your meaning that they have same values consequently. Right?
However, arctan(5/(-5)) can't express -ㅠ/4. Thus I can't understand why they are the same.

It's essentially an order of operations problem. As it happens, $-5/5=5/(-5)=-1$, so the argument to the arctan function is the same in both cases.

Also,you mentioned about x, y, and quadrant.
My friend and I thought the reason arctan(5/(-5)) and arctan((-5)/5) are different as discussing x,y,and quadrant also.
However we didn't know why we put one of x and y denominator or numerator. Could you explain this?

This arises when you are attempting to find the magnitude and direction of a 2D vector from its components. Equivalently, you are converting from rectangular coordinates to polar, either with real number or with the complex numbers.

The forward direction is
\begin{align*}
x&=r \cos( \theta)\\
y&=r \sin( \theta).
\end{align*}
If you divide the second of these by the first, you get
$$ \frac{y}{x}= \tan( \theta).$$
Taking the arctangent of both sides yields the usual equation
$$ \theta= \arctan \left( \frac{y}{x} \right).$$
So the reason $y$ is in the numerator, is because this is the correct transformation from rectangular coordinates to polar.

 

[CellCoree]

The ambiguity in the arc-tangent function arises from the fact that it is a multi-valued function. This means that for a single input value, there can be multiple possible output values. In the case of arctan(5/-5) and arctan-1(-5/5), the difference in output values is due to the different branches of the arc-tangent function that are being used.

In the first case, arctan(5/-5), we are using the principal branch of the arc-tangent function, which is defined to have values between -π/2 and π/2. This is why the result is 3π/4.

In the second case, arctan-1(-5/5), we are using the other branch of the arc-tangent function, which is defined to have values between -π and π. This is why the result is -π/4.

As for calculating arctan(-1), it is important to specify which branch of the arc-tangent function you are using. If you are using the principal branch, the result would be -π/4. However, if you are using the other branch, the result would be 3π/4. It is important to clarify which branch is being used to avoid ambiguity.