This question already has an answer here: Finite Sum $\sum\limits_{k=1}^{m-1}\frac{1}{\sin^2\frac{k\pi}{m}}$ 5 answers

If we have a circle of radius $r$ with an $n$-gon inscribed within this circle (i.e. with the same circumradius), we can find the difference of the areas using: $$A_n =\overbrace{\pi r^2}^\text{Area of circle}-\overbrace{\frac{1}{2} r^2 n \sin (\frac{2 \pi}{n})}^\text{Area of n-gon} =r^2(\pi-\frac{1}{2} n \sin (\frac{2 \pi}{n}))$$ I want to find the following sum (starting with […]

In formal, does there exist $k\in\mathbb{N}$ such that $\sin n\leq\sin k$ for all $n\in\mathbb{N}$?

Find $\theta$ with $\sec(3\theta/2)=-2$ on the interval $[0, 2\pi]$. I started off with $\cos(3 \theta/2)=-1/2$, thus $3\theta/2 = 2\pi/3$, but I don’t know what to do afterwards, the answer should be a huge list of $\theta$s, which I cannot seem to get.

This question already has an answer here: A “fast” way for computing $ \prod \limits_{i=1}^{45}(1+\tan i^\circ) $? 3 answers

Problem : If $ \cos x +2 \cos y+3 \cos z=0 , \sin x+2 \sin y+3 \sin z=0$ and $x+y+z=\pi$. Find $\sin 3x+8 \sin 3y+27 \sin 3z$ Solution: Adding $ \cos x +2 \cos y+3 \cos z=0$ and $\sin x+2 \sin y+3 \sin z=0$,we get $ (\cos x+\sin x) +2(\cos y+\sin y)+3(\cos z+\sin z) =0$ […]

Why does $\sum_{n=1}^{\infty}\frac{\cos\frac{1}{n}}{n}$(I’m told it can be compared to the harmonic sequence, but I don’t see the tangible comparison) diverge but $\sum_{n=1}^{\infty}\frac{\sin\frac{1}{n}}{n}$ converges?

Given $\sin^2\alpha+\sin^2\beta+\sin^2\gamma=2 $. I have to prove that $ \left| \begin{matrix} \cos\alpha & \cos\beta & \sin\gamma\\\sin\alpha & \cos\beta & \cos\gamma\\\cos\alpha & \sin\beta & \cos\gamma \end{matrix} \right| \leq 2\sqrt2 \sin\alpha \sin\beta \sin\gamma $. I decided to directly expand the determinant, the left becomes $ |2\cos \alpha\cos\beta\cos\gamma-\sin(\alpha+\beta+\gamma)| $. This is quite different from what I encountered before […]

What does $$\lim_{x\to\pi/6}\frac{1-\sqrt{3}\tan x}{\pi-6x}$$ evaluate to? This very likely needs substitution.

I’ve been learning about the unit circle, sine, cosine, and the like in my introduction to trigonometry course, but I’m drawing a blank here. If I have a circle centered at the origin, with radius r and point(x,y), how do I find the measure of the angle from (r,0) to (x,y)? For example, if the […]

Intereting Posts

Finding a space with given homology groups and fundamental group
Show a stochastic process is a martingale using Ito's lemma
How to use the Mean Value Theorem to prove the following statement:
Generating the special linear group of 2 by 2 matrices over the integers.
Number of ring homomorphisms from $\mathbb Z_{12}$ to $\mathbb Z_{28}$.
Are Lie algebras $u_n$ and $su_n$ simple?
The direct sum of two closed subspace is closed? (Hilbert space)
Proving that a polynomial is not solvable by radicals.
Affine scheme $X$ with $\dim(X)=0$ but infinitely many points
Selection problem: how to solve?
A proof that $C$ is separable
Two closest sums of pairs of reciprocals
Integral with Undefined Endpoint (Complex Variables)
Question about trigonometry/trigonometry question?
Does the square or the circle have the greater perimeter? A surprisingly hard problem for high schoolers