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October 8th, 2019, 08:09 AM   #1
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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.
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October 8th, 2019, 08:47 AM   #2
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Math Focus: Area of Circle
1. $\displaystyle \sin\left( \frac{z_1 +z_2}{2}\right) > \frac{1}{2}(\sin(z_1) + \sin(z_2)) = \sin\left( \frac{z_1 +z_2}{2}\right) \times \cos\left( \frac{z_1 -z_2}{2}\right) \Rightarrow 1 > \cos\left( \frac{z_1 -z_2}{2}\right)$
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Last edited by skipjack; October 8th, 2019 at 11:48 PM.
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October 8th, 2019, 08:56 AM   #3
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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...
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Last edited by skipjack; October 8th, 2019 at 11:51 PM.
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October 8th, 2019, 09:06 AM   #4
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Math Focus: Area of Circle
And 4. is wrong...
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October 8th, 2019, 09:18 AM   #5
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What about (3) ?
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October 8th, 2019, 10:26 AM   #6
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Quote:
Originally Posted by idontknow View Post
(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.
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Last edited by skipjack; October 8th, 2019 at 11:57 PM.
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October 9th, 2019, 12:24 AM   #7
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Quote:
Originally Posted by tahirimanov19 View Post
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.
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October 9th, 2019, 04:50 AM   #8
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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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