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2answers
35 views

Complex numbers in polar form

If we have two complex numbers, in polar form, as the numerator and denominator of a fraction, and we are asked to write them as a single complex number, is there an easier way to deal with them ...
1
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2answers
95 views

Polar form of a complex number

Question: Write the polar form of $$\frac{(1+i)^{13}}{(1-i)^7}$$ Well its obviously impractical to expand it and try and solve it. Multiplying the denominator by $(1+i)^7$ will simplify the ...
2
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1answer
35 views

Representation of cardiod in the complex plane

I noticed that the complex function $$f(z) = \frac{2}{(z+i)^2}$$ seems to map the real line onto the cardioid given by the polar equation: $$r = 1- \cos(\theta).$$ I was wondering if there is a simple ...
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votes
1answer
31 views

How to calculate Polar coordinates for Complex Polynomials of Higher Degree?

When such I have a complex number such as $3 - 4i$, I can calculate the $r$ with $r=\sqrt{X^2+Y^2} = \sqrt{3^2+4^2}$. But how do I solve this when I have a complex number such as $(2+6i)^6$
2
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1answer
33 views

Sinusoids closed under addition, Euler's Formula

Real sinusoids with the same frequency are closed under addition. If $$f(\omega) = A_1 \cos(\omega + \phi_1) + A_2 \cos(\omega + \phi_2)$$ Then there is some $A_3$ and $\phi_3$ so that: $$f(\omega) ...
0
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2answers
57 views

Complex number polar form equation

I've been struggling with a complex numbers algebra question for a few days now, and the tutor says I still haven't got it right. Express $z_4 =−\sqrt{3} + i$ in polar form. Hence solve the ...
2
votes
1answer
42 views

Complex number to polar representation

I'm trying to change the complex number, $-3i$ to polar representation. What I did: $a=0$ $b=-3$ $r=\sqrt{a^2+b^2} = \sqrt{0+3^2} = 3$ $\theta = \frac{b}{a} = \frac{-3}{0}$ But after that I'm ...
1
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1answer
37 views

How to find polar values of complex number as quick as possible?

I need to calculate these kind of values in exams in best speedy way. Convert $1.46 + 3.17j$ to polar form ($r∠θ$) Is there is any solution to find of the values as quick as possible? By the way, ...
0
votes
1answer
63 views

Problem understanding solution of complex nth-root of unity

a while ago we had the solution for a complex number task about the nth-root of unity in the complex. But now I am having some difficulties to fully understand it: The task was to find all complex ...
1
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3answers
123 views

How to calculate $\theta$ when we know $\tan \theta$.

Hej I'm having difficulties calculating the angle given the tangent. Example: In a homework assignement I'm to express a complex variable $z = \sqrt{3} -i$ in polar form. I know how to solve this ...
0
votes
1answer
46 views

Complex number in polar coordinates

I have to get $\Im$, $\Re$, the absolut value as well as the argument $\phi$ of the complex number $$z = \left(-\frac{1}{\sqrt2}+\sqrt\frac{3}{2}i\right)^8$$ I do this by transforming $z' = ...
1
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1answer
55 views

Determining the polar form for all n-th roots of unity.

By definition $ z \in\mathbb{C}$ is a n-th root of unity iff $z^n = 1$. My assignment is to (iv) List all n-th roots of unity in their polar form. You may use that there are only $n$ Elements with ...
1
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2answers
72 views

How to solve a complex equation $w^4 = \sqrt{3} -i$

$z = \sqrt{3} -i$ How do I solve a complex equation $w^4 = \sqrt{3} -i$ I know that I first have to rewrite z to polar format which I have done as $z = 2(cos (-π/6) + sin (-π/6))$ but I do not know ...
1
vote
0answers
107 views

Correct way to write the polar form of a complex number

What is the most correct way to write the polar form of a complex number? For example, given the complex number: $\dfrac{\sqrt{3}}{2} + \dfrac{1}{2}i$ I would write the polar form as follows: ...
0
votes
0answers
80 views

Ambiguity in converting from cartesian to polar coordinates

Just started looking into complex numbers in "The Art of Electronics" book. When converting from cartesian $(x,y)$ coordinates to polar (r,$\theta$), the conversions are carried out according to the ...
0
votes
3answers
2k views

Square root of complex number in polar or rectangular form

I am trying to find how to simplify: $$\sqrt{\frac{A+jb}{C+jd}}$$ My calculator errors out, giving a math error, and I don't know how else to solve this.
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votes
2answers
94 views

What is the inverse $z^{-1}(z)$ of $z(\varphi)=e^{i\varphi}$ with $\varphi\in\Bbb N_0$?

Suppose I am given a complex number $z=x+iy\in\Bbb C$, with $\left|z\right|=1$, and I am told that $z=e^{i\varphi}$ for some integer $\varphi\in\Bbb N_0$ (the value of which is not given to me). How ...
0
votes
3answers
148 views

Cartesian and Polar Coordinate

I should give the Cartesian Coordinates $(x,y)\in \mathbb{R\times R}$ and Polar Coordinates $(r,\varphi)\in R^+\times [0,2\pi)$ of the following Complex Numbers: a) $z_{1}=-i$ b) $z_{2}=\sqrt{3}+i$ ...
1
vote
1answer
1k views

Converting to polar form

Write each of the given numbers in the polar form $re^{i\theta}$. a.) $\frac{1-i}{3}$ b.) $-8\pi (1+\sqrt 3 i)$ For a, I got: r = $\frac{\sqrt 2}{3}$ and $e^{i7\pi /2}$ since ...
1
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2answers
61 views

Polar coordinates - issue with direction denoted by angle

Convert $1-\sqrt{3}i$ to polar coordinates $(r,\varphi)$. I started by computing $r=|1-\sqrt{3}i|=\sqrt{1^2+\sqrt{3}^2}=\sqrt{4}=2$. When I tried to compute the angle I did something like ...
0
votes
0answers
175 views

How to find the formula of a spiral using least-squares(regression)?

Assume data from a plane which are roughly showing a spiral. I want employ the rationale of regression to find the parameters for the best fit by some spiral. That means, I have to estimate the ...
1
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3answers
405 views

Expressing $e^z$ where $z=a+bi$ in polar form.

I am reading a passage of text that states: "We can use the fact that $e^{a+bi}=e^a(\cos b+i\sin b)$ has polar form $\left<e^a,b \right>$ to verify that complex exponentials have various ...
1
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1answer
209 views

Complex Numbers and polar form

I am given the following information: $$x[n]= s^n,\qquad n=0,\pm 1,\pm 2,\ldots$$ where $s=\sigma + j\omega = re^{i\theta}$ is a complex number in general. I was wondering how the following is ...
0
votes
1answer
60 views

Find complex $z$ such that $z$ has the largest possible real part, and satisfies: $z^7 = -18-18i$

Find complex $z$ such that $z$ has the largest possible real part, and satisfies the equation: $z^7 = -18 -18i$ So, the 7th roots of $z = 18\sqrt{2}e^{i\frac{\frac{\pi}{4} + 2\pi k}{7}}$ ...
0
votes
3answers
230 views

Write $\cos(9x)$ in terms of powers of $\cos(x)$ [duplicate]

Possible Duplicate: How to expand $\cos nx$ with $\cos x$? Write $\cos(9x)$ in terms of powers of $\cos(x)$ I realize I could solve this by using De Moivre's and binomial expansion: ...
3
votes
2answers
87 views

Using polar form to prove $|z| = 1 \implies \text{Re}\left(\frac{1-z}{1+z}\right) = 0$

This was an answer provided to a question I asked previously. I followed the other approaches to the question; however, I couldn't seem to follow this one: ...
1
vote
3answers
666 views

A square root of i with negative imaginary part

In an ODE class, one assignment question says find the “rectangular” expression z = a + bi (with a and b real) and the “polar” expression |z|, Arg(z) where z is "a square root of i with negative ...