0
votes
0answers
104 views

Integrating quotients with polynomials in numerator and denominator that are raised to fractional powers

I'm working through a paper on momentum in electrodynamics that requires the integration below and would greatly appreciate any help. I'm pretty sure it evaluates to $2/d$ but I can't quite figure ...
1
vote
2answers
40 views

Inequality with trigonometric functions

Find all values for $a$ such that the following inequality holds: $$\sin^6x + \cos^6x + a\sin x \cos x \ge 0$$ To be fair, I didn't manage to get anything helpful wiht my calculations. I tried to ...
0
votes
0answers
28 views

Trigonometric Power Formulas (or something more modest)

How does one begin to show (natural $n$): $$\cos^{2n}(x) =\frac{1}{2^{2n}} \binom{2n}{n}+ \frac{1}{2^{2n-1}} \sum_{k=0}^{n-1} \binom{2n}{k} \cos[2(n-k)x]$$ $$\cos^{2n+1}(x) =\frac{1}{4^{n}} ...
1
vote
1answer
48 views

moving trig function from the exponent?

Is there a form of $e^{\sin(x)}$ that does not have trig functions in the exponent? I've seen the "Euler Identity" form and looked into series expansions. (thanks for the answers -- sorry I didn't ...
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.
2
votes
3answers
543 views

Visual proof of the addition formula for $\sin^2(a+b)$?

Is there a visual proof of the addition formula for $\sin^2(a+b)$ ? The visual proof of the addition formula for $\sin(a+b)$ is here : Also it is easy to generalize (in any way: algebra , picture ...
15
votes
1answer
504 views

How to prove $\sum\limits_{n=1}^{\infty}\frac{\sin n}n=\frac{\pi-1}{2}$

One of my classmates challenged me to solve $\displaystyle\sum\limits_{n=1}^{\infty}\frac{\sin n}n=\;?$ With a simple c program I found that $\displaystyle\sum\limits_{n=1}^{1048576}\frac{\sin ...
0
votes
1answer
174 views

De Moivre's theorem question.

State De Moivre's theorem and use it to find integers $ A,B,C$ such that $$\sin^5\theta=A\sin5 \theta + B\sin3\theta + C\sin\theta.$$ I know De Moivre's theorem, how to prove it, and converting to ...
6
votes
2answers
639 views

A “Matrix Trigonometry”

$e^X$ for matrix $X$ is defined as an always-converging taylor series (provided that $X$ is a $n \times n $ complex matrix): $$e^X:=\sum_{k=0}^{\infty}\frac{X^k}{k!} $$ A thought occurred to me that ...
0
votes
2answers
624 views

Notation of inverse trigonometric functions and exponentiation [duplicate]

Possible Duplicate: $\arcsin$ written as $\sin^{-1}(x)$ I have worked a bit on trigonometry today, and something strikes me as inconsistent. In the book, the notation for the inverse sine ...
14
votes
5answers
2k views

$\sin(A)$, where $A$ is a matrix

If $A$ is an $n\times n$ matrix with elements $a_{ij}$ $i=$i'th row, $j=$j'th column. Then $e^A$ is also a matrix as can be seen by expanding it in a power series.Is $e^A$ always convergent and ...
3
votes
1answer
645 views

Integral of exponential function with trigonometric identities

I need help in solving the following definite integral. I could not find any example like this $$\int_{0}^{2\pi}\int_{0}^{d}\exp\!\Big(\frac{-r^2 +2\alpha\; r\cos\theta}{4\;\sigma^2}\Big)r\; dr\; ...
5
votes
6answers
385 views

Can you explain $(1 + iX/n)^{n}$ without using e, sin, or cos?

In my ongoing strugle to understand $e^{\pi i}$ I managed to narrow down my conceptual difficulty. I'm having intuitive trouble understanding why $(1 + iX/n)^{n}$ is conceptually the same as a ...