0
votes
2answers
27 views

Express $\sin 3\theta$ and $\cos 3\theta$ as functions of $\sin \theta$ and $\cos \theta$ using Euler's identity

Using Euler's identity ($e^{in\theta}=\cos n\theta+i \sin n\theta$), express $\sin 3\theta$ and $\cos 3\theta$ as functions of $\sin \theta$ and $\cos \theta$. Any ideas?
1
vote
1answer
30 views

Problem calculating the argument of a complex variable

In Signals & Systems 2nd Ed. written by A. V. Oppenheim, there is a result of Fourier transformation: $ \begin{align} H (j \omega) = \frac{1 + (j \omega / \omega_{0})^2 - 2 j \zeta (\omega / ...
3
votes
4answers
131 views

Picture/intuitive proof of $\cos(3 \theta) = 4 \cos^3(\theta)-3\cos(\theta)$?

Is there a nice geometric, intuitive or picture proof as to why the easily algebraically provable identity $\cos(3 \theta) = 4 \cos^3(\theta)-3\cos(\theta)$ is true? Note I'm not looking for a ...
1
vote
1answer
58 views

Find the 6th root of $-3+4i$ and plot on complex plane

So I have a rough idea on how to get the answer but I'm getting stuck on the angle or argument for the equation. The question is: Find the 6th root of $-3+4i$. I first find the $r$ value which ...
2
votes
1answer
48 views

Do sine and cosine of complex numbers have anything to do with right-triangles or circles?

I've recently been working on a web application that draws iterating function generated fractals. I've noticed that the sine and cosine functions can be used to draw exquisite plots using an ...
1
vote
2answers
50 views

Trigonometric equation with complex numbers

Let $x$, $y$, and $z$ be real numbers such that $\cos x+\cos y+\cos z=\sin x+\sin y+\sin z=0$. Prove that $\cos 2x+\cos 2y+\cos 2z=\sin 2x+\sin 2y+\sin 2z=0$. Starting with the given equation, I got ...
4
votes
4answers
56 views

Proof of trigonometric identity $\frac{\cos x+i\sin x+1}{\cos x+i\sin x-1}= -\frac{i}{\tan \frac{x}{2}}$

I was given a task of proving the following identity: $$\frac{\cos x+i\sin x+1}{\cos x+i\sin x-1}= -\frac{i}{\tan \frac{x}{2}}$$ I am not looking for a solution, just some kind of a hint to start ...
1
vote
2answers
61 views

Expansion of $\sin^5 \theta$ using the Complex Exponential

How do I expand $\sin^5\theta$ using the complex exponential, in order to obtain: $$\frac{1}{16}\sin 5\theta - \frac{5}{16}\sin 3\theta + \frac{5}{8}\sin\theta$$ Thank you.
1
vote
1answer
49 views

Summation of cos (2n-1) theta

By considering $\sum\limits_{n=1}^N z^{2n-1}$, where $z=e^{i\theta},$ show that $$ \sum\limits_{n=1}^N \cos{(2n-1)} \theta = \frac{\sin(2N\theta)}{2\sin\theta}, $$ where $\sin\theta\neq0$ I ...
0
votes
1answer
43 views

De Moivre and trignometry question

I showed this first result and after that for $x^4-10x^2+5=0$, I solved for $\tan 5\theta=0$, I understand all this , but then I get $\theta=\pi/5$. I know I have to multiply by $n$ to get 5 ...
1
vote
2answers
71 views

Given that $x$ is a rational number, is $\sin(x\pi)$ always expressible through radicals?

This is a theory I just thought of and I am wondering if there is truth to it. Here is the logic that I am working upon: Using Euler's formula, you can deduce that $$ (-1)^x = ...
2
votes
1answer
34 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 ...
1
vote
2answers
42 views

How to compute the sine of a complex number in floating-point arithmetic?

What is the most efficient way to numerically compute the sine of a complex number? Suppose I want to calculate the sine of a complex number a + bi on a computer. Suppose that a and b are both ...
1
vote
1answer
22 views

Complex argument and nyquist plot

I'm trying to sketch the nyquist plot of $$\frac{j\omega-1}{j\omega+1}$$ but can't seem to calculate the argument correctly. I think it should be $$\arctan(-\omega) - \arctan(\omega) = ...
0
votes
2answers
49 views

Using the formulas de Moivre to deduce trigonometric identities.

Yesterday I made a test of complex variables, and this contained a question (in which I could not solve) that asked to use the de Moivre formulas to deduce the following trigonometric identities: ...
0
votes
3answers
140 views

Compute the Integral

Compute the integral. $$\int_{-\infty}^\infty \frac{x^4}{1+x^8} \, dx$$ The answer at the back of the book is $$\frac{\pi}{4\sin(\frac{3\pi}{8})}$$
2
votes
1answer
40 views

Is the complex cosine function surjective?

Let $\cos z=\frac{e^{iz} - e^{-iz}}{2}$ be the complex cosine function. Then is $\cos:\mathbb{C}\rightarrow \mathbb{C}$ surjective? If so, how do i prove this?
3
votes
2answers
45 views

Product of $1-\operatorname{cis}(2k\pi/n)$

I'm in a question about polygonals and got stuck at a part. I have to prove that $$\prod_{k=1}^{n-1} \left(1 - \operatorname{cis}(\frac{2k\pi}{n})\right) = n$$ I've tried to multiply it to make ...
0
votes
2answers
46 views

Complex and Trigonometric Identities

How can I get this result: $$\frac{1+cis\theta}{1-cis\theta}=-\frac{1}{i\tan(\theta/2)}$$ I've tried to expand $1-cis\theta$ as $(1+cis(\theta/2))(1-cis(\theta/2))$, but it doesn't help.
2
votes
1answer
47 views

Neat way to prove $\sin(\alpha+\beta)$ using complex exponential

I am supposed to prove that $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\sin\beta\cos\alpha$ using complex exponentials: $$ \begin{align} \sin\theta&=-\frac{1}{2}i(e^{i\theta}-e^{-i\theta})\\ ...
3
votes
2answers
261 views

Proof of $\sin^2(x) + \cos^2(x)=1$ using series

I have to prove the following identity $\sin^2 (x) + \cos^2(x)=1$. I can easily prove this, but this exercise is given in the section introducing the series expansions for $\sin(x)$ and $\cos(x)$ and ...
1
vote
4answers
132 views

Show $1+\cosθ+\cos(2θ)+\cdots+\cos(nθ)=\frac{1}{2}+\frac{\sin[(n+1/2)θ]}{2\sin(θ/2)}$ [duplicate]

Show $$1+\cosθ+\cos(2θ)+\cdots+\cos(nθ)=\frac12+\frac{\sin\left(\left(n+\frac12\right)θ\right)}{2\sin\left(\frac\theta2\right)}$$ I want to use De Moivre's formula and ...
1
vote
3answers
122 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 ...
2
votes
2answers
102 views

Integral of complex questions?

$$\int_0^{\pi/4} \frac {\sin x + \cos x}{\sin^4x+\cos^2x}dx$$ $$\int e^x\cot x(\csc x-1)dx$$ These two integrals are impossible to find. If anyone knows how to integrate them please help me. I am ...
0
votes
1answer
95 views

How do I solve $|\sinh(x+iy)|^2 = (\sin(y))^2+(\sinh(x))^2$

How do I solve this ? $|\sinh (x+iy)|^2 = ( \sin (y))^2+ ( \sinh (x))^2$ I'm not sure how to solve the left hand side.
1
vote
2answers
63 views

The angle made by the complex number $\frac{1}{(\sqrt{3}+i)^{100}}$ with the positive real axis is ..

Problem : The angle made by the complex number $\frac{1}{(\sqrt{3}+i)^{100}}$ with the positive real axis is ( options) (a) $240^{\circ}$ (b) $140^{\circ}$ (c) $120^{\circ}$ (d) ...
1
vote
1answer
90 views

Complex numbers - exponential numbers - (double angles?)

I am half stumped on this rather confusing problem: Let x, y, and z be real numbers such that $\cos x + \cos y + \cos z = \sin x + \sin y + \sin z = 0$. Prove that $\cos 2x + \cos 2y + \cos 2z = \sin ...
3
votes
4answers
136 views

Complex numbers - Exponential numbers - Proof

Let $z$ be a complex number, and let $n$ be a positive integer such that $z^n = (z + 1)^n = 1$. Prove that $n$ is divisible by 6. For this problem I am stumped...how should I begin? Also there's a ...
1
vote
1answer
65 views

Calculus $T_1=\prod_{k=1}^{n-1} \cos\frac{k\pi}{2n}$

Calculus: $$T_1=\prod_{k=1}^{n-1} \cos\frac{k\pi}{2n}$$ and $$T_2=\prod_{k=1}^{n-1}\sin\frac{k\pi}{2n}$$ My tried: I use Euler's formal: $$z_k=e^{i\frac{k\pi}{2n}}=\cos\frac{k\pi}{2n}+i\sin ...
0
votes
2answers
109 views

How do I solve this integral?

As stated the title, I get to a point which I can't do anything, and I'm sure I've made a mistake some where, here is my full working out: $$ \int e^{ix}\cos(x)dx \\ u = e^{ix} \text{ | } u'= ie^{ie} ...
6
votes
2answers
160 views

Why isn't $\int\sin(ix)~dx$ equal to $i\cos(ix)+C$ ?

I was playing around with imaginary numbers, and I tried to solve $$\int\sin(ix)~dx$$ and ended up getting $$i\cos(ix)+C$$ But apparently the answer is $$i\cosh(x)+C$$ I was just wondering, is this ...
1
vote
4answers
78 views

factor $z^7-1$ into linear and quadratic factors and prove that $ \cos(\pi/7) \cdot\cos(2\pi/7) \cdot\cos(3\pi/7)=1/8$

Factor $z^7-1$ into linear and quadratic factors and prove that $$ \cos(\pi/7) \cdot\cos(2\pi/7) \cdot\cos(3\pi/7)=1/8$$ I have been able to prove it using the value of $\cos(\pi/7)$. Given here ...
2
votes
4answers
78 views

How to find the value of $\arctan(\frac{1}{1-x}) + \arctan(1-x)$?

I'm reading a book on complex analysis. In one step while evaluating a path integral, the author makes the following substitution: $$\arctan \left(\dfrac{1}{1-α} \right) + \arctan(1-α) = ...
3
votes
2answers
174 views

Prove that: $\sin{\frac{\pi}{n}} \sin{\frac{2\pi}{n}} …\sin{\frac{(n-1)\pi}{n}} =\frac{n}{2^{n-1}}$

Using that: $$ x^{n - 1} + x^{n - 2} + \cdots + x + 1 = \left(x - w\right)\left(x - w^{2}\right)\ldots\left(x - w^{n - 1}\right) $$ Prove that: $$ \sin\left(\pi \over n\right)\sin\left(2\pi \over ...
1
vote
2answers
92 views

Prove $p_0-p_2+p_4-\cdots=2^{n/2}\cos{\dfrac{n\pi}{4}}$ and $p_1-p_3+p_5-\dots=2^{n/2}\sin{\dfrac{n\pi}{4}}$

Consider: $$(1+x)^n= p_0 + p_1 x + p_2 x^2+\cdots$$ From where $$p_0=1,\quad p_1=\dfrac{n}{1},\quad p_2=\dfrac{n(n-1)}{2!},\ldots$$ Are the coefficients of the Newton´s Binomial expansion, using $x=i$ ...
2
votes
2answers
102 views

Understanding Euler's Identity

I would like to understand one specific moment in Euler's Identity, namely $$e^{j\theta}=\cos(\theta)+j\sin(\theta)$$ where $j=\sqrt{-1}$. We also know that $$e^{j2(\pi)}=\cos(2\pi)+j\sin(2\pi)$$ ...
2
votes
3answers
204 views

Limit n tends to infinity

How can i solve this: $$ \lim_{n\to\infty} \cos(1)\cos(0.5)\cos(0.25)\ldots \cos(1/2^n) $$ I tried using comlex numbers and logarithms but did'nt work out.Can anyone help please.
0
votes
1answer
40 views

Multiply complex numbers to show trigonometric addition formulas

Use the rules for multiplication of two complex numbers written in the form $r(\cos\theta +i\sin\theta)$ to show that $\sin(\theta_1 +\theta_2)=\sin\theta_1\cos\theta_2 +\sin\theta_2\cos\theta_1$ ...
1
vote
3answers
417 views

Prove for $\cos (x+iy)$

I know it is such a foolish thing to do to ask this dumb question in this site. Please prove that $\cos (x + iy) = \cos x \cosh y - i\sin x \sinh y$ and $\cos (x - iy) = \cos x \cosh y + i\sin x ...
0
votes
0answers
59 views

Solving physics problems using real and imaginary numbers

I was working on a particular physics problem and like we usually in physics do - replaced $\cos(x)$ with $e^{ix}$ and worked the result. When I tried to solve it without complex numbers I stuck. In ...
0
votes
1answer
383 views

Finding angles using arctan

Given $z = 12 - 5j$ find the angle. The formula is angle $= \arctan(y/x)$. Because $y$ is negative, we need to subtract the answer from $2\pi$. (Or more accurately, because the sketch I made tells me ...
2
votes
2answers
87 views

Find the argument of $ \frac{-1 + \sqrt3 i}{2+2i} $

I rewrite equation $ \frac{-1 + \sqrt3 i}{2+2i} $ as $$ \frac{ \sqrt3 - 1}{4} + \frac{ \sqrt3 + 1}{4} i $$ using the conjugacy technique. And set forward to find the argument of this complex ...
1
vote
1answer
51 views

Complex exponential sumation: $\sum_{-M}^M e^{-2\pi ikf}$

Does someone know how to prove that $\displaystyle {\sum_{k = -M}^M e^{-2\pi ikf} = \frac{sin((2M + 1)\pi f)}{sin(2\pi f)}}$ where $f$ is a constant such that $-\frac{1}{2} < f < \frac{1}{2}$ ...
1
vote
3answers
221 views

Complex number - how to find the angle between the imaginary axis and real axis?

Assume I have complex number $z = a + ib$. $z$ can be represented by a polar representation as $r(\cos \theta+i\sin \theta)$, when $r$ is the ...
1
vote
2answers
44 views

Make the vector $[1,1]$ turn of an angle - $\pi/4$ , with complex numbers

We have $[1,1]$ and $\theta = -\pi/4$ here is my attempt: $(\cos(-\pi/4) + i \sin(-\pi/4)) * (x+iy)$ = $(\sqrt{2}/2 - i \sqrt{2}/2) (1+i)$ = $\sqrt{2}/2 - i^2\sqrt{2}/2 $ = $[\sqrt{2}/2 + ...
3
votes
1answer
166 views

Finding modulus and argument of z³ - 4√3 + 4i = 0

I think I am messing up somewhere as the principle argument should be a nice number from the standard triangles such as $\fracπ4$, $\fracπ3$ or $\fracπ6$ or something close. (That's what we have ...
0
votes
3answers
356 views

Find all complex solutions of $\sin(z)=1$ [closed]

Find all complex solutions of $\sin(z)=1$. How would I go about this?
0
votes
1answer
45 views

Trigonometry and complex numbers

Suppose $z_0=e^{i\theta_0}$ a complexe number as $\theta_0\in ]-\pi,\pi[ \setminus\{0\}$. For $n\in \mathbb{N}$, we pose $z_{n+1}=\dfrac{|z_n|+z_n}{2}$ and $z_n=r_ne^{i\theta_n}$ with ...
0
votes
1answer
63 views

Trigonometry with complex numbers [closed]

Express $(\cos(x))^5$ in terms of cosines of multiples of $x$. I've racked my brains for ages on this one! No notes to help me out, and I've failed to find any help online.
3
votes
3answers
484 views

Is it true that $ |\sin^2z+\cos^2z|=1, \forall z \in\Bbb C$?

We know that equation $ \sin^2z+ \cos^2z=1$ which holds $ \forall z \in\Bbb R$, actually holds $ \forall z \in\Bbb C$. Is it true that $ |\sin^2z+\cos^2z|=1, \forall z \in\Bbb C$? Thanks in ...