Найти значение выражения 2*cos(x)*sin(x)еслиx=-2 (2 умножить на косинус от (х) умножить на синус от (х)если х равно минус 2) [Есть ответ!]

2*cos(x)*sin(x)еслиx=-2 (упростите выражение)

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Решение

Вы ввели [src]
2*cos(x)*sin(x)
$$2 \sin{\left(x \right)} \cos{\left(x \right)}$$
Подстановка условия [src]
2*cos(x)*sin(x) при x = -2
подставляем
2*cos(x)*sin(x)
$$2 \sin{\left(x \right)} \cos{\left(x \right)}$$
sin(2*x)
$$\sin{\left(2 x \right)}$$
переменные
x = -2
$$x = -2$$
sin(2*(-2))
$$\sin{\left(2 (-2) \right)}$$
-sin(4)
$$- \sin{\left(4 \right)}$$
Степени [src]
   / I*x    -I*x\                 
   |e      e    | /   -I*x    I*x\
-I*|---- + -----|*\- e     + e   /
   \ 2       2  /                 
$$- i \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right) \left(e^{i x} - e^{- i x}\right)$$
Численный ответ [src]
2.0*cos(x)*sin(x)
Рациональный знаменатель [src]
2*cos(x)*sin(x)
$$2 \sin{\left (x \right )} \cos{\left (x \right )}$$
Объединение рациональных выражений [src]
2*cos(x)*sin(x)
$$2 \sin{\left (x \right )} \cos{\left (x \right )}$$
Общее упрощение [src]
sin(2*x)
$$\sin{\left(2 x \right)}$$
Собрать выражение [src]
2*cos(x)*sin(x)
$$2 \sin{\left (x \right )} \cos{\left (x \right )}$$
sin(2*x)
$$\sin{\left (2 x \right )}$$
Комбинаторика [src]
2*cos(x)*sin(x)
$$2 \sin{\left (x \right )} \cos{\left (x \right )}$$
Общий знаменатель [src]
2*cos(x)*sin(x)
$$2 \sin{\left (x \right )} \cos{\left (x \right )}$$
Тригонометрическая часть [src]
  //  0     for And(im(x) = 0, x mod pi = 0)\ //  1     for And(im(x) = 0, x mod 2*pi = 0)\
2*|<                                        |*|<                                          |
  \\sin(x)             otherwise            / \\cos(x)              otherwise             /
$$2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right)$$
  //     0       for And(im(x) = 0, x mod pi = 0)\ //     1        for And(im(x) = 0, x mod 2*pi = 0)\
  ||                                             | ||                                                |
  ||       /x\                                   | ||        2/x\                                    |
  ||  2*cot|-|                                   | ||-1 + cot |-|                                    |
2*|<       \2/                                   |*|<         \2/                                    |
  ||-----------             otherwise            | ||------------              otherwise             |
  ||       2/x\                                  | ||       2/x\                                     |
  ||1 + cot |-|                                  | ||1 + cot |-|                                     |
  \\        \2/                                  / \\        \2/                                     /
$$2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
        2         
------------------
          /pi    \
csc(x)*csc|-- - x|
          \2     /
$$\frac{2}{\csc{\left(x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}}$$
  //     0       for And(im(x) = 0, x mod pi = 0)\                                              
  ||                                             | //  1     for And(im(x) = 0, x mod 2*pi = 0)\
2*|<   /    pi\                                  |*|<                                          |
  ||cos|x - --|             otherwise            | \\cos(x)              otherwise             /
  \\   \    2 /                                  /                                              
$$2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\cos{\left(x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right)$$
/   0      for And(im(x) = 0, 2*x mod pi = 0)
<                                            
\sin(2*x)              otherwise             
$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}$$
                                              //     1       for And(im(x) = 0, x mod 2*pi = 0)\
  //  0     for And(im(x) = 0, x mod pi = 0)\ ||                                               |
  ||                                        | ||     1                                         |
2*|<  1                                     |*|<-----------              otherwise             |
  ||------             otherwise            | ||   /pi    \                                    |
  \\csc(x)                                  / ||csc|-- - x|                                    |
                                              \\   \2     /                                    /
$$2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{1}{\csc{\left(x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
  2*tan(x) 
-----------
       2   
1 + tan (x)
$$\frac{2 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}$$
  //     0       for And(im(x) = 0, x mod pi = 0)\ //     1       for And(im(x) = 0, x mod 2*pi = 0)\
  ||                                             | ||                                               |
  ||       /x\                                   | ||       2/x\                                    |
  ||  2*tan|-|                                   | ||1 - tan |-|                                    |
2*|<       \2/                                   |*|<        \2/                                    |
  ||-----------             otherwise            | ||-----------              otherwise             |
  ||       2/x\                                  | ||       2/x\                                    |
  ||1 + tan |-|                                  | ||1 + tan |-|                                    |
  \\        \2/                                  / \\        \2/                                    /
$$2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
sin(2*x)
$$\sin{\left(2 x \right)}$$
  //     0       for And(im(x) = 0, x mod pi = 0)\                                              
  ||                                             | //  1     for And(im(x) = 0, x mod 2*pi = 0)\
  ||     1                                       | ||                                          |
2*|<-----------             otherwise            |*|<  1                                       |
  ||   /    pi\                                  | ||------              otherwise             |
  ||sec|x - --|                                  | \\sec(x)                                    /
  \\   \    2 /                                  /                                              
$$2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(x \right)}} & \text{otherwise} \end{cases}\right)$$
        2         
------------------
          /    pi\
sec(x)*sec|x - --|
          \    2 /
$$\frac{2}{\sec{\left(x \right)} \sec{\left(x - \frac{\pi}{2} \right)}}$$
                                              //     1       for And(im(x) = 0, x mod 2*pi = 0)\
  //  0     for And(im(x) = 0, x mod pi = 0)\ ||                                               |
2*|<                                        |*|<   /    pi\                                    |
  \\sin(x)             otherwise            / ||sin|x + --|              otherwise             |
                                              \\   \    2 /                                    /
$$2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\sin{\left(x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$
      2      
-------------
csc(x)*sec(x)
$$\frac{2}{\csc{\left(x \right)} \sec{\left(x \right)}}$$
   /      pi\
cos|2*x - --|
   \      2 /
$$\cos{\left(2 x - \frac{\pi}{2} \right)}$$
            /    pi\
2*sin(x)*sin|x + --|
            \    2 /
$$2 \sin{\left(x \right)} \sin{\left(x + \frac{\pi}{2} \right)}$$
  //                    0                      for And(im(x) = 0, x mod pi = 0)\ //                     1                       for And(im(x) = 0, x mod 2*pi = 0)\
  ||                                                                           | ||                                                                               |
2*|
$$2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
  //                      0                         for And(im(x) = 0, x mod pi = 0)\ //                        1                          for And(im(x) = 0, x mod 2*pi = 0)\
  ||                                                                                | ||                                                                                     |
  ||/     0       for And(im(x) = 0, x mod pi = 0)                                  | ||/     1        for And(im(x) = 0, x mod 2*pi = 0)                                    |
  |||                                                                               | |||                                                                                    |
  |||       /x\                                                                     | |||        2/x\                                                                        |
2*|<|  2*cot|-|                                                                     |*|<|-1 + cot |-|                                                                        |
  ||<       \2/                                                otherwise            | ||<         \2/                                                  otherwise             |
  |||-----------             otherwise                                              | |||------------              otherwise                                                 |
  |||       2/x\                                                                    | |||       2/x\                                                                         |
  |||1 + cot |-|                                                                    | |||1 + cot |-|                                                                         |
  \\\        \2/                                                                    / \\\        \2/                                                                         /
$$2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
      1      
-------------
   /      pi\
sec|2*x - --|
   \      2 /
$$\frac{1}{\sec{\left(2 x - \frac{\pi}{2} \right)}}$$
  /       2/x\\    /x\
4*|1 - tan |-||*tan|-|
  \        \2//    \2/
----------------------
                 2    
    /       2/x\\     
    |1 + tan |-||     
    \        \2//     
$$\frac{4 \cdot \left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \tan{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}$$
   1    
--------
csc(2*x)
$$\frac{1}{\csc{\left(2 x \right)}}$$
            /    pi\
2*cos(x)*cos|x - --|
            \    2 /
$$2 \cos{\left(x \right)} \cos{\left(x - \frac{\pi}{2} \right)}$$
/     0       for And(im(x) = 0, 2*x mod pi = 0)
|                                               
|  2*cot(x)                                     
<-----------              otherwise             
|       2                                       
|1 + cot (x)                                    
\                                               
$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}$$
Раскрыть выражение [src]
2*cos(x)*sin(x)
$$2 \sin{\left (x \right )} \cos{\left (x \right )}$$