0
votes
2answers
37 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
130 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
35 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
37 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
42 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
43 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
149 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
105 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
114 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
93 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
59 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
53 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
66 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
117 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
55 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
87 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
150 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 ...
0
votes
4answers
62 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
76 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
137 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
82 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
96 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
139 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
32 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
194 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
51 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
174 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
56 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
41 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
139 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
123 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
274 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
60 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
403 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 ...
0
votes
1answer
117 views

Using De Moivre's theorem

Hello I want to solve this $x^5=32$ using De Moivre's theorem $$z^n=r(\cos(n\theta) + i \sin(n \theta))$$ I want help to find the solution and more specifically to find the missing $\theta$. ...
1
vote
0answers
155 views

Proving Question (Complex Numbers, De Moivre's)

Prove that $(1+\cos \theta + i \sin \theta)^n + (1+\cos \theta - i \sin \theta)^n=2^{n+1}\cos^n\frac{\theta}{2}\cos\frac{n\theta}{2}$ I want to avoid using the $e^{i\theta}$ form since I haven't ...
0
votes
1answer
33 views

Quadrant problem

$(-1+i)^{\frac{1}{3}}$ here, $\tan\theta=-1$ so, $\theta=\tan^{-1}(-1)=\tan^{-1}(\tan(-\frac{\pi}{4}))=\tan^{-1}(\tan(\pi-\frac{\pi}{4}))=\pi-\frac{\pi}{4}$ My question is why can't i write ...
1
vote
1answer
45 views

For what $k$ is $P(k):=\prod_{j=1}^{13}\cos\frac{\pi kj}{13}$ negative?

Let $k>0$ be an integer. For what $k$ is $P(k):=\prod_{j=1}^{13}\cos\frac{\pi kj}{13}$ negative? Since $13$ is prime, and for $\gcd(m,13)=1$, $P(2m)=P(2)=2^{-12}$ (can be shown by considering the ...
4
votes
3answers
109 views

Calculate $\tan(1+i)$

Calculate $\tan(1+i)$. I use the expression $\tan x = -i\dfrac{e^{ix}-e^{-ix}}{e^{ix}+e^{-ix}}$. So it only remains to calculate $e^{i(1+i)}$ (and then $e^{-i(1+i)}$ follows by taking the ...
2
votes
1answer
235 views

Tangent half-angles and linear fractional transformations

Suppose $z=x+iy$, $x$ and $y$ are real, and $|z|=x^2+y^2=1$ so that $z=e^{i\alpha}$ for some real $\alpha$. Then for some real $\gamma$, $$ \begin{align} e^{i\gamma} = f(e^{i\alpha}) = f(z) & ...
0
votes
3answers
118 views

Can you get the exact real value of $ \left((-1)^{\frac{1}{180}}\right)^{89}-\left((-1)^{\frac{1}{180}}\right)^{91}$?

By using euler formula,one can obtain: $$ 2\sin\left(\frac{\pi}{180}\right)=\left((-1)^{\frac{1}{180}}\right)^{89}-\left((-1)^{\frac{1}{180}}\right)^{91}. $$ In order to get the exact real value of ...
1
vote
1answer
95 views

Trigonometric manipulation of complex number, how does this step occur?

I was reading the section about DeMoivre, and my book showed how to derive his formulas. The next part is supposed to be about finding roots of complex and real numbers. Roughly, it says: "Let $z$ be ...
4
votes
1answer
98 views

Prove a Trigonometric Series

Question: $$\cot^{2}\frac{\pi }{2m+1}+\cot^{2}\frac{2\pi }{2m+1}+\cdots+\cot^{2}\frac{m\pi }{2m+1}=\frac{m(2m-1)}{3}$$ $m$ is a positive integer. Attempt: I started by showing that ...
1
vote
5answers
134 views

How does $Ae^{4ix}+Be^{-4ix}=A\cos(4x)+B\sin(4x)$?

$e^{ix}=\cos(x)+i\sin(x)$ $Ae^{4ix}=A(\cos(4x)+i\sin(4x))$ $Be^{-4ix}=B(-\cos(4x)-i\sin(4x))$ What am I doing wrong? I am trying to find the complimentary function of $\frac{d^2y}{dx^2} ...
2
votes
2answers
124 views

trigonometric representation of a complex number.

Let $z=e^{it}+1$ where $0\leq t\leq \pi$, Find the trigonometric representation of $z^2+z+1$. (The trigonometric representation should be in the form of : $r(\cos \theta +i \sin \theta)$, where ...
1
vote
2answers
61 views

Trigonometric problem

I'm trying to get the roots for a complex number $x^2+1$ $x^2+1=0\rightarrow x^2=-1 \rightarrow x = \sqrt{-1} \rightarrow i$ So, $w^2 = 0 + 1i$ $p = \sqrt{0^2+1^2} = 1$ $\theta = \tan^{-1} \left( ...
8
votes
5answers
261 views

When are we (not) allowed to replace $x$ by $ix$?

It seems to be quite a common manipulation to replace $x$ by $ix$. Every time I see it's being done in a textbook, I blindly trust the author without really understanding when are we allowed to do so ...
1
vote
0answers
57 views

tangent function of rational angle

Can any ne help me to prove this problem? $x$ is called rational angle if $x=a\pi$ for $a\in \mathbb{Q}$. Let $0<x<\pi/4$ be a rational angle, prove that $\tan x$ is irrational. Let ...