Articles of trigonometry

How prove that $\frac{1}{\sin^2\frac{\pi}{2n}}+\frac{1}{\sin^2\frac{2\pi}{2n}}+\cdots+\frac{1}{\sin^2\frac{(n-1)\pi}{2n}} =\frac{2}{3}(n-1)(n+1)$

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

Convergence and closed form of this infinite series?

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 […]

Does $\sin n$ have a maximum value for natural number $n$?

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

Trigonometric equation $\sec(3\theta/2) = -2$ – brain dead

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.

$X=(1 + \tan 1^{\circ})(1 + \tan 2^{\circ})(1 + \tan 3^{\circ})\ldots(1 + \tan {45}^{\circ})$. what is the value of X?

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

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 $\sin 3x+8 \sin 3y+27 \sin 3z$

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 $\sum_{n=1}^{\infty}\frac{\cos\frac{1}{n}}{n}$ diverge but $\sum_{n=1}^{\infty}\frac{\sin\frac{1}{n}}{n}$ converges?

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?

An unusual symmetric inequality of trigonometric functions

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 $\lim\limits_{x\to\pi/6}\frac{1-\sqrt{3}\tan x}{\pi-6x}$ evaluate to?

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

How do you find the angle of circle segment formed with points (x,y) and (radius,0)?

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 […]