# 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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### Complex roots of $z^3 + \bar{z} = 0$

I'm trying to find the complex roots of $z^3 + \bar{z} = 0$ using De Moivre. Some suggested multiplying both sides by z first, but that seems wrong to me as it would add a root ( and I wouldn't know ...
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### The only two rational values for cosine and their connection to the Kummer Rings

I am trying to learn about Kummer Rings, and in particular what makes $n=3,4,6$ so special. (That is the Gaussian and Eisenstein integers) The only $\theta\in [0,\frac{\pi}{2}]$ which are rational ...
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### How do I prove that $\arccos(x) + \arccos(-x)=\pi$ when $x \in [-1,1]$? [closed]

Prove that $\arccos x + \arccos(-x) = \pi$ when $x \in [-1,1]$. How do I prove this? Where should I begin and what should I consider?
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### Need help with calculating this sum: $\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$

I need help with calculating this sum: $$\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$$
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### How does a calculator calculate the sine, cosine ,tangent using just a number?

Sine Θ = oposite/hypotenuse Cosine Θ = adjacent/hypotenuse Tangent Θ = oposite/adjacent So in order to calculate the Sine or the cosine or the tangent I need to ...
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### How prove this $\tan{\frac{2\pi}{13}}+4\sin{\frac{6\pi}{13}}=\sqrt{13+2\sqrt{13}}$

Nice Question: show that: The follow nice trigonometry $$\tan{\dfrac{2\pi}{13}}+4\sin{\dfrac{6\pi}{13}}=\sqrt{13+2\sqrt{13}}$$ This problem I have ugly solution,maybe someone have nice mthods?...
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### Showing $\tan\frac{2\pi}{13}\tan\frac{5\pi}{13}\tan\frac{6\pi}{13}=\sqrt{65+18\sqrt{13}}$

How can one show : $$\tan\frac{2\pi}{13}\tan\frac{5\pi}{13}\tan\frac{6\pi}{13}=\sqrt{65+18\sqrt{13}}?$$
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### Simplify the sum $\sum_{n=2}^N\frac{1}{n^2}\sin^2(\pi x)\csc^2(\frac{\pi x}{n})$? - a sum shows all primes $\le N^2$

I was looking for a closed form but it seemed too difficult. Now I'm seeking help to simplify this sum. The 50 bounty points or more will be awarded for any meaningful simplification of this sum. I ...
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### Method of proof of $\sum\limits_{n=1}^{\infty}\tfrac{\coth n\pi}{n^7}=\tfrac{19}{56700}\pi^7$

The following formula was stated by Ramanujan: $$\sum\limits_{n=1}^{\infty}\frac{\coth n\pi}{n^7}=\frac{19\pi^7}{56700}$$ Does anybody know the method of proof of this formula? I know that typically ...
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Some years ago, I came across the following question: Find the value of $\tan5°\tan55°\tan65°\tan75°$. The value is 1 and I generalized the question as follows. $\tan a°\tan b°\tan c°\tan d° ... 2answers 499 views ### Reference for a tangent squared sum identity Can anyone help me find a formal reference for the following identity about the summation of squared tangent function: $$\sum_{k=1}^m\tan^2\frac{k\pi}{2m+1} = 2m^2+m,\quad m\in\mathbb{N}^+.$$ I ... 2answers 212 views ### A question by Ramanujan about a relational expression of a triangle I found the following question in a book without any proof: Question : Suppose that each length of three edges of a triangle$ABC$are$BC=a, CA=b, AB=c$respectively. If$$\frac1a=\frac1b+\frac1c, \... 5answers 4k views ### Why aren't the graphs of$\sin(\arcsin x)$and$\arcsin(\sin x)$the same? (source for above graph) (source for above graph) Both functions simplify to x, but why aren't the graphs the same? 2answers 6k views ### What is the optimum angle of projection when throwing a stone off a cliff? You are standing on a cliff at a height$h$above the sea. You are capable of throwing a stone with velocity$v$at any angle$a$between horizontal and vertical. What is the value of$a$when the ... 1answer 756 views ### Continued fraction for$\tan(nx)$I found this beautiful continued fraction expansion of$\tan(nx)$,$n$being a positive integer, online but I don't remember the source now:$\displaystyle \tan(nx) = \cfrac{n\tan x}{1 -\cfrac{(n^{2} ...
Triangle $ABC$ is an equilateral triangle and $P$ is an arbitrary point inside it. The distance from $P$ to $A$ is $4$ and the distance from $P$ to $B$ is $6$ and the distance from $P$ to $C$ is $5$. ...