2
votes
1answer
33 views

Proving $\left(\frac{1+\sin x+i\cos x}{1+\sin x-i\cos x}\right)^n=\cos n\left(\frac{\pi }{2}-x\right)+i\sin n\left(\frac{\pi }{2-x}\right)$

How to solve the following question? If $n$ is an integer, show that \begin{eqnarray} \left(\frac{1+\sin x+i\cos x}{1+\sin x-i\cos x}\right)^n=\cos n\left(\frac{\pi }{2}-x\right)+i\sin ...
0
votes
0answers
22 views

Can a complex number have two arguments

Now, the reason why I wrote two $\theta$s is because my answer is the answer we get from $\theta$2 and the answer in the book is given the value of $\theta$1. So, I was just wondering whether both ...
1
vote
2answers
39 views

Trignometric problem (using De Movier's Theorem)

Ok so this question, I started out writing tan as sin and cos in the right side of the equation, simplified as much as possible and ended up with a very (sort of) fascinating equation which is ...
0
votes
3answers
30 views

Question regarding in periodic function

I have question I know that $\cos(x+2\pi)=\cos x$ and $\sin(x+2\pi)=\sin x$ but if we have $\cos(x+\pi)=?$ and $\sin(x+\pi)=?$ with explaination thanks
4
votes
3answers
210 views

Find all values for cos(i)

In my Differential Equations class recently we have learned about Euler's Formula and Fourier Series. I am given the problem ...
0
votes
1answer
21 views

how should I think about the complex-exponential form of sinusoid waves?

Say there's a sinusoid wave with amplitude $A$, frequency $\omega$, and phase shift $\psi$, then one way to write it is $A cos(\omega t - \psi)$. But it can also be written as $Re(Ae^{i(\omega t - ...
0
votes
3answers
53 views

Find the sum $1+\cos (x)+\cos (2x)+\cos (3x)+…+\cos (n-1)x$ [duplicate]

By considering the geometric series $1+z+z^{2}+...+z^{n-1}$ where $z=\cos(\theta)+i\sin(\theta)$, show that $1+\cos(\theta)+\cos(2\theta)+\cos(3\theta)+...+\cos(n-1)\theta$ = ...
1
vote
2answers
33 views

Sum of two trig function's identity

We all know that $\sin(x) + \sin(y) = 2\sin((x+y)/2)\cos((x-y)/2)$ But is there an identity for $\sin(x) + z\sin(y) = ?$ Or do I need to figure it out using Euler's formula $\sin(x) = (e^{ix} - ...
0
votes
2answers
29 views

Lengthy Product of trigonometric ratios

What is the value of the product $\sin(10) \sin(20) \sin(30) \sin(40) \sin(50) \sin(60) \sin(70) \sin80$, where all the angles are in degrees? Solve using complex numbers. I found this in a book of ...
0
votes
2answers
21 views

Find angle $\alpha$ from a complex vector

I'm trying to solve this problem from a Russian book: Find the angle which is needed to rotate the vector $3\sqrt{2} + i2\sqrt{2}$ to obtain the vector $-5+i$. EDIT: $\tan\dfrac{\pi}{6} \neq ...
3
votes
3answers
102 views

Problems with trigonometry getting the power of this complex expression

I'm here because I can't finish this problem, that comes from a Russian book: Calculate $z^{40}$ where $z = \dfrac{1+i\sqrt{3}}{1-i}$ Here $i=\sqrt{-1}$. All I know right now is I need to use ...
1
vote
1answer
57 views

Sum of the trigonometric series

I'm studying de Moivre's theorem's application on the summation of trigonometric series. Here's what I have so far: \begin{align*} \sum_{k=0}^n \cos(k\theta)&= \text{Re}\sum_{k=0}^n e^{ki\theta} ...
2
votes
2answers
25 views

Rectangular to polar form using exact values.

I'm in a first year math course at university, and we've been asked to convert a rectangular form complex number into polar form, using exact values only. I have the modulus, that's all good. But I ...
2
votes
1answer
168 views

Polar form of complex numbers.

Write the given number in polar form $re^{i\theta}$ i) $z = -8\pi (1 + \sqrt{3}i)$ So I thought that $\theta = \arctan(-\sqrt{3}/-1) = \frac{4\pi}{3}$ and it would be $z = 8\pi ...
0
votes
2answers
40 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
33 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 / ...
5
votes
5answers
191 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
61 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
50 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
59 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
65 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
68 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
66 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
49 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
76 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
36 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
47 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
26 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
1answer
38 views

Discontinuity of principal argument in nonpositive real axis

Let $\operatorname{Arg}(z)$ be principal argument function defined in branch $(-\pi, \pi]$. Prove that $\operatorname{Arg}(z)$ is discontinuous in every point in nonpositive real axis. "Solution": ...
0
votes
2answers
56 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
141 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
43 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
51 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
48 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
50 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
355 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
162 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
126 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
111 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
67 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
117 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
164 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
66 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
110 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
163 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
86 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
191 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
95 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 $x=i$ ...