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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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 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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 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:{$\displaystyle x_1 , x_2 ,x_n$}. $\displaystyle \prod_{i=1}^{n} tan(x_i ) < \prod_{i=1}^{n} x_i < \underbrace{x\cdot x \cdot .... \cdot x }_{n} =\lfloor x^{n} \rfloor \leq n^n$. Tags problems, trigonometry Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Drasik Trigonometry 4 October 12th, 2014 11:36 PM chocolatesheep Trigonometry 2 May 20th, 2013 10:45 PM jorgerocha Trigonometry 3 September 21st, 2012 12:34 AM swm06 Trigonometry 4 February 15th, 2012 01:17 AM Debjani Algebra 3 April 20th, 2009 10:01 PM

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