2
votes
2answers
56 views

Different Definitions Of The Sine Function

I was hoping someone could give me a flow chart or high-level map connecting all of the definitions of the sine function, with some of the reasons why we care next to each. I've tried this but I'm not ...
0
votes
1answer
15 views

Limit when $y>>a$ of a derived solution

I am able to do part d), however I am very stuck on part e). If $y >> a$ then surely we get $\phi(x,y)$ $= \frac{1}{\pi} \Big[ tan^{-1} \Big( \frac{x+a}{y}\Big)-tan^{-1} ...
4
votes
3answers
160 views

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

In an example I had to prove that $\sin^2(x)+\cos^2(x)=1$ which is fairly easy using the unit circle. My teacher then asked me to show the same thing using the following power ...
1
vote
1answer
128 views

Applications of higher powers of trigonometric functions

I am after a reference (book, papers etc) about the practical applications of trigonometric functions raised to higher powers. An example is one that I have been using in my own studies: $\cos^4 ...
8
votes
5answers
263 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 ...
10
votes
3answers
206 views

Could we show $1-(x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\dots)^2=(1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}- \dots)^2$ if we didn't know about Taylor Expansion?

Suppose that humanity haven't discovered Taylor Series Expansion of trigonometric functions or of any function that would help us on this. Which means we are not allowed to replace the given infinite ...
5
votes
1answer
120 views

Power series for $\cos(n\theta)$ in terms of $\sin^{2i}(\theta/2)$?

Does anybody know an expression for the weights in $$ \cos(n\theta) = \sum_{i=0}^n c_i \sin^{2i}(\theta/2) $$ I checked the standard sources (Abramowitz & Stegun, Gradshteyn & Rhyzik) and ...
2
votes
1answer
138 views

Maclaurin expansion of $\sqrt{\cos 2x}$ and $\tan^2 x$ up to degree 4

Find the Maclaurin expansion $\sqrt{\cos(2x)}$ and $\tan^2x$ up to degree $4$. I tried differentiation but it gives me something really horrible.
13
votes
2answers
337 views

How to do a very long division: continued fraction for tan

I want to compute $$\tan(r) = \cfrac{r}{1 - \cfrac{r^2}{3 - \cfrac{r^2}{5 - \cfrac{r^2}{7 - {}\ddots}}}}$$ by dividing the power series for sin and cos as it is said can be done in ...
0
votes
1answer
153 views

Evaluate $ \ \int_{- \infty}^{\infty} \frac{x}{( x^2 + 4x + 13 )^2}\,dx . \ $ using contour integration and the calculus of residues

How do I evaluate $ \ \int_{- \infty}^{\infty} \frac{x}{( x^2 + 4x + 13 )^2}\,dx . \ $ using contour integration and the calculus of residues? I have many more problems of this kind need to be done. ...
1
vote
2answers
151 views

Use Residue Theorem to evaluate $ \ \oint_{C_3 (0)} \frac{z+7}{z^4 + z^3 - 2 z^2}\,dz \ $?

How do I use Residue Theorem to evaluate $ \ \oint_{C_3 (0)} \frac{z+7}{z^4 + z^3 - 2 z^2}\,dz \ $ where $C_3(0)$ is the circle of radius 3 centered at the origin, oriented in the counter- clockwise ...
1
vote
1answer
330 views

Cauchy product and the exponential function

Simplify the following series using the Cauchy product $$\sum\limits_{k=1}^\infty\frac{1}{k!}\cdot\sum\limits_{j=1}^\infty\frac{1}{j!}$$ ...
5
votes
2answers
183 views

How to transform the factored form of $\sin(x)$?

We know $\sin(x)=0$ has solutions $0,\pm\pi,\pm2\pi,\pm3\pi,\dots$. So $\sin(x)$, if interpreted as a polynomial, could be written as: $a_0x^0+a_1x^1+a_2x^2+\cdots$ and we know this polynomial too: ...
2
votes
2answers
304 views

Identify this power series / solve this trig equation

I was asked to find a solution to $$\frac{\sin^2(nx)}{n^2\sin^2(x)}=2^{-1/2}$$ where $n$ is a fixed integer greater than 1. Numerically, there's a solution just above 1/n so I decided to find this ...