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Question 1

Question 2

1.

$\sin{x^2} = 0 \ ;$

$\left\{\begin{array}{l} x^2 = \pi n \ , \ n \in Z \ ; \\ x^2 \geq 0 \ . \end{array}\right$

$x^2 = \pi n \ , \ n \in ( \{ 0 \} \cup N ) \ ;$

О т в е т :    $x = \pm \sqrt{ \pi } \cdot \sqrt{ n } \ \ , \ n \in ( \{ 0 \} \cup N ) \ ;$

2.

$\cos{ ( \cos{x} ) } = 1 \ ;$

$\left\{\begin{array}{l} \cos{x} = 2 \pi n \ , \ n \in Z \ ; \\ | \cos{x} | \leq 1 \ . \end{array}\right$

$\cos{x} = 0 \ ;$

О т в е т :    $x = \frac{ \pi }{2} + \pi n \ , \ n \in Z \ ;$

3.

$\frac{ \cos{x} }{ \sin{x} - 1 } = 0 \ ;$

$\left\{\begin{array}{l} \cos{x} = 0 \ ; \\ \sin{x} \neq 1 \ . \end{array}\right$

$\left\{\begin{array}{l} x = \frac{ \pi }{2} + \pi n \ , \ n \in Z \ ; \\ x \neq \frac{ \pi }{2} + 2 \pi n \ , \ n \in Z \ . \end{array}\right$

О т в е т :    $x = -\frac{ \pi }{2} + 2 \pi n \ , \ n \in Z \ ;$

4.

$\sin{x} + \sin{|x|} = 0 \ ;$

$\left[\begin{array}{l} \left\{\begin{array}{l} x < 0 \ , \\ \sin{x} - \sin{x} = 0 \ ; \end{array}\right \\\\ \left\{\begin{array}{l} x \geq 0 \ , \\ \sin{x} + \sin{x} = 0 \ ; \end{array}\right \end{array}\right$

$\left[\begin{array}{l} \left\{\begin{array}{l} x < 0 \ , \\ \sin{x} = \sin{x} \ ; \end{array}\right \\\\ \left\{\begin{array}{l} x \geq 0 \ , \\ 2\sin{x} = 0 \ ; \end{array}\right \end{array}\right$

$\left[\begin{array}{l} x < 0 \ , \\\\ \left\{\begin{array}{l} x \geq 0 \ , \\ x = \pi n \ , \ n \in Z \ ; \end{array}\right \end{array}\right$

О т в е т :    $\left[\begin{array}{l} x \leq 0 \ , \\\\ x = \pi n \ , \ n \in N \ ; \end{array}\right$

5.

$| \cos{x} | = \cos{x} + 2 \sin{x} \ ;$

$\left[\begin{array}{l} \left\{\begin{array}{l} \cos{x} < 0 \ , \\ -\cos{x} = \cos{x} + 2 \sin{x} \ ; \end{array}\right \\\\ \left\{\begin{array}{l} \cos{x} \geq 0 \ , \\ \cos{x} = \cos{x} + 2 \sin{x} \ ; \end{array}\right \end{array}\right$

$\left[\begin{array}{l} \left\{\begin{array}{l} \cos{x} < 0 \ , \\ 2 \cos{x} + 2 \sin{x} = 0 \ ; \end{array}\right \\\\ \left\{\begin{array}{l} \cos{x} \geq 0 \ , \\ 2 \sin{x} = 0 \ ; \end{array}\right \end{array}\right$

$\left[\begin{array}{l} \left\{\begin{array}{l} \cos{x} < 0 \ , \\ \cos{x} + \cos{ ( \frac{ \pi }{2} - x ) } = 0 \ ; \end{array}\right \\\\ \left\{\begin{array}{l} \cos{x} \geq 0 \ , \\ \sin{x} = 0 \ ; \end{array}\right \end{array}\right$

$\left[\begin{array}{l} \left\{\begin{array}{l} \cos{x} < 0 \ , \\ 2 \cos{ \frac{ x + [ \pi/2 - x ] }{2} } \cos{ \frac{ x - [ \pi/2 - x ] }{2} } = 0 \ ; \end{array}\right \\\\ \left\{\begin{array}{l} \cos{x} \geq 0 \ , \\ x = \pi n \ , \ n \in Z \ ; \end{array}\right \end{array}\right$

<img src="https://tex.z-dn.net/?f=+%5Cleft%5B%5Cbegin%7Barray%7D%7Bl%7D+%5Cleft%5C%7B%5Cbegin%7Barray%7D%7Bl%7D+%5Ccos%7Bx%7D+%3C+0+%5C+%2C+%5C%5C%5C%5C+%5Cleft%7C%5Cbegin%7Barray%7D%7Bl%7D+%5Ccos%7B+%5Cfrac%7B+%5Cpi%2F2+%7D%7B2%7D+%7D+%5Ccos%7B+%5Cfrac%7B+2x+-+%5Cpi%2F2+%7D%7B2%7D+%7D+%3D+0+%5C+%2C+%5C%5C%5C%5C+%5Ccos%7B+%5Cfrac%7B+%5Cpi+%7D%7B4%7D+%7D+%5Ccos%7B

gartenzie_zn · Answer 1 · 2018-04-18T14:43:42+0000

1.

$\sin{x^2} = 0 \ ;$

$\left\{\begin{array}{l} x^2 = \pi n \ , \ n \in Z \ ; \\ x^2 \geq 0 \ . \end{array}\right$

$x^2 = \pi n \ , \ n \in ( \{ 0 \} \cup N ) \ ;$

О т в е т :    $x = \pm \sqrt{ \pi } \cdot \sqrt{ n } \ \ , \ n \in ( \{ 0 \} \cup N ) \ ;$

2.

$\cos{ ( \cos{x} ) } = 1 \ ;$

$\left\{\begin{array}{l} \cos{x} = 2 \pi n \ , \ n \in Z \ ; \\ | \cos{x} | \leq 1 \ . \end{array}\right$

$\cos{x} = 0 \ ;$

О т в е т :    $x = \frac{ \pi }{2} + \pi n \ , \ n \in Z \ ;$

3.

$\frac{ \cos{x} }{ \sin{x} - 1 } = 0 \ ;$

$\left\{\begin{array}{l} \cos{x} = 0 \ ; \\ \sin{x} \neq 1 \ . \end{array}\right$

$\left\{\begin{array}{l} x = \frac{ \pi }{2} + \pi n \ , \ n \in Z \ ; \\ x \neq \frac{ \pi }{2} + 2 \pi n \ , \ n \in Z \ . \end{array}\right$

О т в е т :    $x = -\frac{ \pi }{2} + 2 \pi n \ , \ n \in Z \ ;$

4.

$\sin{x} + \sin{|x|} = 0 \ ;$

$\left[\begin{array}{l} \left\{\begin{array}{l} x < 0 \ , \\ \sin{x} - \sin{x} = 0 \ ; \end{array}\right \\\\ \left\{\begin{array}{l} x \geq 0 \ , \\ \sin{x} + \sin{x} = 0 \ ; \end{array}\right \end{array}\right$

$\left[\begin{array}{l} \left\{\begin{array}{l} x < 0 \ , \\ \sin{x} = \sin{x} \ ; \end{array}\right \\\\ \left\{\begin{array}{l} x \geq 0 \ , \\ 2\sin{x} = 0 \ ; \end{array}\right \end{array}\right$

$\left[\begin{array}{l} x < 0 \ , \\\\ \left\{\begin{array}{l} x \geq 0 \ , \\ x = \pi n \ , \ n \in Z \ ; \end{array}\right \end{array}\right$

О т в е т :    $\left[\begin{array}{l} x \leq 0 \ , \\\\ x = \pi n \ , \ n \in N \ ; \end{array}\right$

5.

$| \cos{x} | = \cos{x} + 2 \sin{x} \ ;$

$\left[\begin{array}{l} \left\{\begin{array}{l} \cos{x} < 0 \ , \\ -\cos{x} = \cos{x} + 2 \sin{x} \ ; \end{array}\right \\\\ \left\{\begin{array}{l} \cos{x} \geq 0 \ , \\ \cos{x} = \cos{x} + 2 \sin{x} \ ; \end{array}\right \end{array}\right$

$\left[\begin{array}{l} \left\{\begin{array}{l} \cos{x} < 0 \ , \\ 2 \cos{x} + 2 \sin{x} = 0 \ ; \end{array}\right \\\\ \left\{\begin{array}{l} \cos{x} \geq 0 \ , \\ 2 \sin{x} = 0 \ ; \end{array}\right \end{array}\right$

$\left[\begin{array}{l} \left\{\begin{array}{l} \cos{x} < 0 \ , \\ \cos{x} + \cos{ ( \frac{ \pi }{2} - x ) } = 0 \ ; \end{array}\right \\\\ \left\{\begin{array}{l} \cos{x} \geq 0 \ , \\ \sin{x} = 0 \ ; \end{array}\right \end{array}\right$

$\left[\begin{array}{l} \left\{\begin{array}{l} \cos{x} < 0 \ , \\ 2 \cos{ \frac{ x + [ \pi/2 - x ] }{2} } \cos{ \frac{ x - [ \pi/2 - x ] }{2} } = 0 \ ; \end{array}\right \\\\ \left\{\begin{array}{l} \cos{x} \geq 0 \ , \\ x = \pi n \ , \ n \in Z \ ; \end{array}\right \end{array}\right$

<img src="https://tex.z-dn.net/?f=+%5Cleft%5B%5Cbegin%7Barray%7D%7Bl%7D+%5Cleft%5C%7B%5Cbegin%7Barray%7D%7Bl%7D+%5Ccos%7Bx%7D+%3C+0+%5C+%2C+%5C%5C%5C%5C+%5Cleft%7C%5Cbegin%7Barray%7D%7Bl%7D+%5Ccos%7B+%5Cfrac%7B+%5Cpi%2F2+%7D%7B2%7D+%7D+%5Ccos%7B+%5Cfrac%7B+2x+-+%5Cpi%2F2+%7D%7B2%7D+%7D+%3D+0+%5C+%2C+%5C%5C%5C%5C+%5Ccos%7B+%5Cfrac%7B+%5Cpi+%7D%7B4%7D+%7D+%5Ccos%7B