# Tagged Questions

Trigonometric functions (both geometric and circular), relationships between lengths and angles in triangles, and other topics relating to measuring triangles.

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### Evaluating $\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}dx$

Evaluate $$\int_0^\pi \frac{x}{(\sin x)^{\sin (\cos x)}}dx$$ I tried using by parts and complex numbers along with series expansion but I was unable to find the answer. Please Help!
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### Trigonometry or inequality problem

Today, I saw this question: If $x,y,z \in [0,\frac\pi 2]$, $x+y+z=\frac{3\pi}{4}$ and $\sec^2(x)\sec^2(y)\sec^2(z)=8$, calculate $E=\tan x\tan y+\tan y\tan z+\tan z\tan x$ My first thought was ...
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### resizing rectangle within triangle

Imagine I have a parking lot that changes in width and length and in number of levels, and all of the levels need to be visible to a cctv camera at a fixed position, and I would want the camera to see ...
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### Maximum and Minimum value of an inverse function

Find the maximum and minimum value of $\arcsin \left(x\right)^3+\arccos \left(x\right)^3$. given that $-1\le x\le 1$ I have solved the problem but i am just curious to know if there are any ...
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### Explicit solutions to a digamma function equation

My main question: Can we obtain the exact solutions from the following equation? $$\sum_{k=1}^{n}\cfrac{1}{k-x-1}=0$$ Notation: This problem was reached from the digamma function $\psi$ as ...
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### Why is the unit circle definition of trig functions not rigorous enough?

It has recently come to my attention that the usual unit circle derivation of the elementary trigonometric functions isn't considered rigorous enough. Apparently, this has to do with problems ...
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### More on primes $p=u^2+27v^2$ and roots of unity

Given, $$p=u^2+27v^2=3m+1\tag1$$ and the cubic, $$x^3+x^2-mx+N=0\tag2$$ with its constant expressed in terms of $(1)$ as, $$N = \frac{1}{27}(1-3p\pm2pu)\tag3$$ and the sign $\pm u$ chosen ...
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### Dealing with absolute values after trigonometric substitution in $\int \frac{\sqrt{1+x^2}}{x} \text{ d}x$.

I was doing this integral and wondered if the signum function would be a viable method for approaching such an integral. I can't seem to find any other way to help integrate the $|\sec \theta|$ term ...
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### how understand if a segment is inside a lissajous curve

i am a programmer and not a math guru, but i like geometry. so if i'm not accurate in math terminology or i have folly question please sorry me. i'm drawing with a programming language the lissajous ...
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### How to get the distance from a top of an arm to a surface?

I am studying computer science and I am working on a project for my next semester. I am stuck with a problem and I am not that good in math. Imagine I have an "arm" shape line where it has 5 nodes <...
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### Sine identity involving (3/p) for prime p greater than 3.

I am working through Ireland and Rosen's "Classical Introduction to Modern Number Theory" and am very stuck on this problem (#34 in Chp 5, 2nd edition): Note that $(a/b)$ is the Legendre symbol (or ...
A friend of mine gave me the following question after struggling with it for quite some time: Consider a function defined as below: f^k (\theta) =\sum_{r=1}^n \left( \frac{\tan \left( \frac {\...