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 October 8th, 2019, 08:09 AM #1 Senior Member   Joined: Dec 2015 From: somewhere Posts: 734 Thanks: 98 Trigonometry problems (1) If $\displaystyle z_1 \neq z_2$ and $\displaystyle z_1 , z_2 \in (0,\pi )$. Prove that $\displaystyle \: \sin\left(\frac{z_1 +z_2 }{2}\right) > \frac{\sin(z_1 ) +\sin(z_2 )}{2}$. (2) Find all pairs $\displaystyle (x,y)$ such that $\displaystyle x^2 +2x\cdot \cos(xy)+1=0$. (3) Find the number of solutions of the equation $\displaystyle \sin(x)=\frac{x}{n} \;$, where $\displaystyle n\in \mathbb{N} \: , n\neq 1$. (4) Prove that $\displaystyle \: \tan(a_0 )\cdot \tan(a_1 ) \cdot ... \cdot \tan(a_n )\geq n^{1+n}$. Last edited by skipjack; October 8th, 2019 at 11:48 PM.
 October 8th, 2019, 08:47 AM #2 Senior Member   Joined: Mar 2015 From: Universe 2.71828i3.14159 Posts: 132 Thanks: 49 Math Focus: Area of Circle 1. $\displaystyle \sin\left( \frac{z_1 +z_2}{2}\right) > \frac{1}{2}(\sin(z_1) + \sin(z_2)) = \sin\left( \frac{z_1 +z_2}{2}\right) \times \cos\left( \frac{z_1 -z_2}{2}\right) \Rightarrow 1 > \cos\left( \frac{z_1 -z_2}{2}\right)$ Thanks from idontknow Last edited by skipjack; October 8th, 2019 at 11:48 PM.
 October 8th, 2019, 08:56 AM #3 Senior Member   Joined: Mar 2015 From: Universe 2.71828i3.14159 Posts: 132 Thanks: 49 Math Focus: Area of Circle 2. It is easy. For any non-zero real number x, find the value of y. And it is $$y= \pm \frac{1}{x} \times \arccos\left( \frac{-1-x^2}{2x}\right)$$ It is not a precise answer... Thanks from idontknow Last edited by skipjack; October 8th, 2019 at 11:51 PM.
 October 8th, 2019, 09:06 AM #4 Senior Member   Joined: Mar 2015 From: Universe 2.71828i3.14159 Posts: 132 Thanks: 49 Math Focus: Area of Circle And 4. is wrong...
 October 8th, 2019, 09:18 AM #5 Senior Member   Joined: Dec 2015 From: somewhere Posts: 734 Thanks: 98 What about (3) ?
October 8th, 2019, 10:26 AM   #6
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Quote:
 Originally Posted by idontknow (3) Find the number of solutions of the equation $\displaystyle \sin(x)=\frac{x}{n} \;$, where $\displaystyle n\in \mathbb{N} \: , n\neq 1$.
$\displaystyle N = 4 \left( \left\lceil \frac{n-1}{2\pi} \right\rceil - \varepsilon_n \right) - 1$
where $\varepsilon_n \in \mathbb{Z}$ is an approximation error that is usually zero for small $n$ but is nonzero increasingly more frequently as $n$ increases.

Last edited by skipjack; October 8th, 2019 at 11:57 PM.

October 9th, 2019, 12:24 AM   #7
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Quote:
 Originally Posted by tahirimanov19 2. It is easy. . . . It is not a precise answer...
For $y$ to be real, $x$ must be 1 or -1. Now it's easy.

 October 9th, 2019, 04:50 AM #8 Senior Member   Joined: Dec 2015 From: somewhere Posts: 734 Thanks: 98 By inspecting problem with a similiar example I got (4) : (4) $\displaystyle tan(x_i ) < x_i \: \Rightarrow \prod_{i=1}^{n} tan(x_i ) < \prod_{i=1}^{n} x_i$ . Let $\displaystyle x$ denote the largest number in set S:{$\displaystyle x_1 , x_2 ,x_n$}. $\displaystyle \prod_{i=1}^{n} tan(x_i ) < \prod_{i=1}^{n} x_i < \underbrace{x\cdot x \cdot .... \cdot x }_{n} =\lfloor x^{n} \rfloor \leq n^n$.

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