All Questions
6
questions
1
vote
1
answer
36
views
Summation of infinite cos series and determining theta
Question: In the figure,
$A_0A_1,A_2A_3,A_4A_3...$
are all perpendicular to $L_1$
$A_1A_2,A_3A_4,A_5A_6...$
are all perpendicular to $L_2$
If $A_0A_1=1$
And $A_0A_1+A_1A_2+A_2A_3+A_3A_4......\...
8
votes
2
answers
166
views
Given $\varphi$ is golden ratio, how do I prove this $\sum \limits_{j=1}^{\infty}\frac{(1-\varphi)^j}{j^2}\cos{\frac{3j\pi}{5}}=\frac{\pi^2}{100}$?
Given $ \varphi$ is golden ratio, how do I prove this:
$ \displaystyle \tag*{}\sum \limits_{j=1}^{\infty}\frac{(1-\varphi)^j}{j^2}\cos{\frac{3j\pi}{5}}=\frac{\pi^2}{100}$
My approach:
We can reduce ...
user1024407
3
votes
1
answer
203
views
How do I simplify this sum of arccosines?
After trying to solve a geometry problem, represented by the following image
I've arrived at this expression:
$\alpha=\arccos\left(\frac{d+r \cos \left(\varphi+\frac{\vartheta}2\right)}{\sqrt{d^2+r^2+...
- 507
2
votes
2
answers
1k
views
Proving $\sum_{k=0}^{n-1} \text{cos}\left(\frac{2\pi k}{n} \right) = \sum_{k=0}^{n-1} \text{sin}\left(\frac{2\pi k}{n} \right) = 0$ [duplicate]
I would like to prove the following, $$\sum_{k=0}^{n-1} \text{cos}\left(\frac{2\pi k}{n} \right) = \sum_{k=0}^{n-1} \text{sin}\left(\frac{2\pi k}{n} \right) = 0$$ This is equivalent to showing that if ...
user445909
1
vote
1
answer
148
views
Geometry formulas, how to show identities.
Given $d$ is integer:
How do I show:
$$\frac{1}{(e^{\frac{2i\pi p}{d}}-1)}=\frac{-i}{2\tan(\frac{\pi p}{d})}-\frac{1}{2}$$
How do I rewrite and show, for $k$ is an integer:
$$ \sum_{p=1}^{d-1}\frac{\...
- 593
47
votes
1
answer
2k
views
Why does this ratio of sums of square roots equal $1+\sqrt2+\sqrt{4+2\sqrt2}=\cot\frac\pi{16}$ for any natural number $n$?
Why is the following function $f(n)$ constant for any natural number $n$?
$$f(n)=\frac{\sum_{k=1}^{n^2+2n}\sqrt{\sqrt{2n+2}+{\sqrt{n+1+\sqrt k}}}}{\sum_{k=1}^{n^2+2n}\sqrt{\sqrt{2n+2}-{\sqrt{n+1+\...
- 144k