Let and . If , find $u$ in polar form.
I've managed to subtitute z and w into the equation for u to find the correct |u| value, but I can't find arg(u).
Suppose that $x\cdot y\ne 0 $ then
$Arg(x + yi) = \left\{ {\begin{array}{{lr}} {\arctan \left( {\frac{y}{x}} \right),}&{x > 0} \\ {\arctan \left( {\frac{y}{x}} \right) + \pi ,}&{x < 0\;\& \;y > 0} \\ {\arctan \left( {\frac{y}{x}} \right) - \pi ,}&{x < 0\;\& \;y < 0} \end{array}} \right. $
To be single valued, -Π/2 < tan^{-1}x < Π/2, all x.
Given x+iy, tan^{-1}(y/x) gives the same value for x&y in 1st and 3rd quadrants as for x&y in 2nd and 4th quadrants. So you have to know where x&y actually are on complex plane to pick the right angle.
(explanation of rule in previous post.)
Sorry for the late reply, I've had a busy week since posting this thread.
Yet again, the notation you've used is unfamiliar to me. We're yet to cover arctan or the large brackets. Is there another way to explain it?
If z=a+bi, let z’=a-bi
Multiply 1/z by z’/z’ and 1/w by w’/w’ and collect real and imaginary parts into u=a+bi.
Then plot u on the complex plane. lul is the distance from origin to u, and theta is the angle from positive x axis to radius.
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I have told you this before, here it is again. For the life of me, I cannot understand what sort of course you are doing in which you are asked questions having been given not reals tools to use.
For any complex number to give the polar form it is necessary to find the argument.
That is associated with the tangent function.
If then the argument is .
If then the quantity is the tangent of the argument given the correct sign.
To find the correct sine, you need to see where the number is located on the graph.
YOU need to know the tangents of each of the following:
etc.
Once again you lack of depth is in evidence.
Tell me if $w=-5+3i$ then how " converting (x,y) to polar coordinates (r,theta) without thinking about it" can possibility give a accurate $\arg(z)~?$
You see I would then require the student to give exact representations of the the six sixth roots of $z$.
What are they?
I understand all of this fine (although it's written differently to what I'm familiar with). My problem was that couldn't grasp your specific use of arctan and the large brackets, but I have since cleared up my unfamiliarity with such notation. As for the course I'm doing -- it's early days. I'm not sure what the real tools are, but we've covered (from the textbook): The complex number system, C; Argand diagrams; Operations using complex numbers; Division of complex numbers and the complex conjugate; Polar form (modulus and argument) of complex numbers; Multiplication and division in polar form. This has involved completing exercises from the textbook, as well as a range of extended response questions (such as this one). My teacher is not going through the textbook in the order of the chapters, so that is the most likely reason why such things as arctan (something covered in chapter 2; complex numbers is chapter 3 -- the first chapter we're doing) are unfamiliar to me. This is also the first time this teacher has taught this course.
I thank you all for your willingness to help me through these problems. When contact time with my teacher is limited, the feedback and assistance MHF provides is extremely valuable.