# Complex numbers: Argument

## Homework Statement

Hi all.

When finding the argument of a complex number using tan(\theta) = y/x (where z = x + iy), sometimes I do not get the correct answer. I assume this is because tangent is only defined for -pi/2 to pi/2 (and from this it is periodic).

So is it a good idea always to find the argument of a complex number using cosine and sine? Or am I missing something here?

Thanks in advance.

Regards,
Niles.

## Answers and Replies

if you use cos and sine, you should still get the same theta if you use tan. Think of it as a triangle, where z is the hypotenuse, x is the base, and y is the height.

berkeman
Mentor
if you use cos and sine, you should still get the same theta if you use tan. Think of it as a triangle, where z is the hypotenuse, x is the base, and y is the height.

As in the figure at the top of this page:

http://en.wikipedia.org/wiki/Complex_plane

.

HallsofIvy
Science Advisor
Homework Helper

## Homework Statement

Hi all.

When finding the argument of a complex number using tan(\theta) = y/x (where z = x + iy), sometimes I do not get the correct answer. I assume this is because tangent is only defined for -pi/2 to pi/2 (and from this it is periodic).

So is it a good idea always to find the argument of a complex number using cosine and sine? Or am I missing something here?

Thanks in advance.

Regards,
Niles.
There is nothing wrong with using tangent (more correctly arctangent) as long as you keep track of the signs of x and y. For example if you find that y/x= 1, then arctan(1) could be $\pi/4$ or $5\pi/4$. If you know that x and y are positive then you know the argument is $\pi/4$, if negative, $5\pi/4$.

First, thanks to all for replying.

But I can avoid all of this "confusion" by using sine or cosine? I mean, what if only x is negative and y is positive. Then using arctangent I have to add pi/2 to get the correct result.

HallsofIvy
Science Advisor
Homework Helper
What do you mean by "using" sine and cosine? Yes, $tan(\theta)= y/x$ can be interpreted as $sin(\theta)= y/r$ and $cos(\theta)= x/r$. If you only find $arcsin(y/r)$ where y is positive, you still have two possible answers $\theta$ between 0 and $\pi/2$ and $\pi/2- \theta$. You can, of course, decide which you want by looking at $arccos(x/r)$ but it seems to me simpler to just find $\arctan(y/x)$ and look at the signs of x and y. That way you don't have to calculate $r= \sqrt{x^2+ y^2}$.

Perhaps getting to know the handy-dandy http://en.wikipedia.org/wiki/Unit_circle" [Broken] will help you avoid this confusion? That way you'll get a sense on what happens when either X or Y is positive or negative.

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So what I can conclude from your answers is that there is no way using arctan, arccos or arcsin to find the "correct" angle all the time, but it depends on the values of x and y?

no, you can find the angle using arctan, arccos, arcsin if you know x,y, and z.
are you familiar with the term: SOHCAHTOA?

just remember that the answer can range from 0 to 2pi, because that depends on the signs of x and y.

HallsofIvy
Science Advisor
Homework Helper
So what I can conclude from your answers is that there is no way using arctan, arccos or arcsin to find the "correct" angle all the time, but it depends on the values of x and y?
You cannot determine it by using any ONE of those without taking the signs of x and y separately into account.

no, you can find the angle using arctan, arccos, arcsin if you know x,y, and z.
are you familiar with the term: SOHCAHTOA?

just remember that the answer can range from 0 to 2pi, because that depends on the signs of x and y.
I think you have misunderstood the question.