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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. $\displaystyle \sin\left( \frac{z_1 +z_2}{2}\right) > \frac{1}{2}(\sin(z_1) + \sin(z_2)) = \sin\left( \frac{z_1 +z_2}{2}\right) \times \cos\left( \frac{z_1 z_2}{2}\right) \Rightarrow 1 > \cos\left( \frac{z_1 z_2}{2}\right)$
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 nonzero real number x, find the value of y. And it is $$y= \pm \frac{1}{x} \times \arccos\left( \frac{1x^2}{2x}\right)$$ It is not a precise answer... 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  
Senior Member Joined: Jun 2019 From: USA Posts: 310 Thanks: 162  Quote:
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 
Global Moderator Joined: Dec 2006 Posts: 21,035 Thanks: 2271  
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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