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sin(2*x)*cos(2*x)еслиx=3/2 (упростите выражение)

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Примеры

Решение

Вы ввели [src]
sin(2*x)*cos(2*x)
sin⁡(2x)cos⁡(2x)\sin{\left(2 x \right)} \cos{\left(2 x \right)}
Подстановка условия [src]
sin(2*x)*cos(2*x) при x = 3/2
подставляем
sin(2*x)*cos(2*x)
sin⁡(2x)cos⁡(2x)\sin{\left(2 x \right)} \cos{\left(2 x \right)}
sin(4*x)
--------
   2
sin⁡(4x)2\frac{\sin{\left(4 x \right)}}{2}
переменные
x = 3/2
x=32x = \frac{3}{2}
sin(4*(3/2))
------------
     2
sin⁡(4(3/2))2\frac{\sin{\left(4 (3/2) \right)}}{2}
sin(6)
------
  2
sin⁡(6)2\frac{\sin{\left(6 \right)}}{2}
Степени [src]
   / -2*I*x    2*I*x\                      
   |e         e     | /   -2*I*x    2*I*x\ 
-I*|------- + ------|*\- e       + e     / 
   \   2        2   /                      
-------------------------------------------
                     2
−i(e2ix2+e−2ix2)(e2ix−e−2ix)2- \frac{i \left(\frac{e^{2 i x}}{2} + \frac{e^{- 2 i x}}{2}\right) \left(e^{2 i x} - e^{- 2 i x}\right)}{2}
Численный ответ [src]
cos(2*x)*sin(2*x)
Общее упрощение [src]
sin(4*x)
--------
   2
sin⁡(4x)2\frac{\sin{\left(4 x \right)}}{2}
Собрать выражение [src]
sin(4*x)
--------
   2
12sin⁡(4x)\frac{1}{2} \sin{\left (4 x \right )}
Тригонометрическая часть [src]
        1        
-----------------
csc(2*x)*sec(2*x)
1csc⁡(2x)sec⁡(2x)\frac{1}{\csc{\left(2 x \right)} \sec{\left(2 x \right)}}
//   0      for And(im(x) = 0, 2*x mod pi = 0)\ //   1      for And(im(x) = 0, x mod pi = 0)\
|<                                            |*|<                                          |
\\sin(2*x)              otherwise             / \\cos(2*x)             otherwise            /
({0for im⁡(x)=0∧2x mod π=0sin⁡(2x)otherwise)({1for im⁡(x)=0∧x mod π=0cos⁡(2x)otherwise)\left(\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}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases}\right)
/      0        for And(im(x) = 0, 4*x mod pi = 0)
|                                                 
|  2*cot(2*x)                                     
<-------------              otherwise             
|       2                                         
|1 + cot (2*x)                                    
\                                                 
--------------------------------------------------
                        2
{0for im⁡(x)=0∧4x mod π=02cot⁡(2x)cot⁡2(2x)+1otherwise2\frac{\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 4 x \bmod \pi = 0 \\\frac{2 \cot{\left(2 x \right)}}{\cot^{2}{\left(2 x \right)} + 1} & \text{otherwise} \end{cases}}{2}
   /      pi\
cos|4*x - --|
   \      2 /
-------------
      2
cos⁡(4x−π2)2\frac{\cos{\left(4 x - \frac{\pi}{2} \right)}}{2}
            /pi      \
sin(2*x)*sin|-- + 2*x|
            \2       /
sin⁡(2x)sin⁡(2x+π2)\sin{\left(2 x \right)} \sin{\left(2 x + \frac{\pi}{2} \right)}
            /      pi\
cos(2*x)*cos|2*x - --|
            \      2 /
cos⁡(2x)cos⁡(2x−π2)\cos{\left(2 x \right)} \cos{\left(2 x - \frac{\pi}{2} \right)}
//     0       for And(im(x) = 0, 2*x mod pi = 0)\ //     1        for And(im(x) = 0, x mod pi = 0)\
||                                               | ||                                              |
||  2*cot(x)                                     | ||        2                                     |
|<-----------              otherwise             |*|<-1 + cot (x)                                  |
||       2                                       | ||------------             otherwise            |
||1 + cot (x)                                    | ||       2                                      |
\\                                               / \\1 + cot (x)                                   /
({0for im⁡(x)=0∧2x mod π=02cot⁡(x)cot⁡2(x)+1otherwise)({1for im⁡(x)=0∧x mod π=0cot⁡2(x)−1cot⁡2(x)+1otherwise)\left(\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}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right)
                                                //      1        for And(im(x) = 0, x mod pi = 0)\
//   0      for And(im(x) = 0, 2*x mod pi = 0)\ ||                                               |
|<                                            |*|<   /pi      \                                  |
\\sin(2*x)              otherwise             / ||sin|-- + 2*x|             otherwise            |
                                                \\   \2       /                                  /
({0for im⁡(x)=0∧2x mod π=0sin⁡(2x)otherwise)({1for im⁡(x)=0∧x mod π=0sin⁡(2x+π2)otherwise)\left(\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}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\sin{\left(2 x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)
//      0        for And(im(x) = 0, 2*x mod pi = 0)\                                              
||                                                 | //   1      for And(im(x) = 0, x mod pi = 0)\
||      1                                          | ||                                          |
|<-------------              otherwise             |*|<   1                                      |
||   /      pi\                                    | ||--------             otherwise            |
||sec|2*x - --|                                    | \\sec(2*x)                                  /
\\   \      2 /                                    /
({0for im⁡(x)=0∧2x mod π=01sec⁡(2x−π2)otherwise)({1for im⁡(x)=0∧x mod π=01sec⁡(2x)otherwise)\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\frac{1}{\sec{\left(2 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 \pi = 0 \\\frac{1}{\sec{\left(2 x \right)}} & \text{otherwise} \end{cases}\right)
//                       0                          for And(im(x) = 0, 2*x mod pi = 0)\ //                       1                         for And(im(x) = 0, x mod pi = 0)\
||                                                                                    | ||                                                                                 |
||/     0       for And(im(x) = 0, 2*x mod pi = 0)                                    | ||/     1        for And(im(x) = 0, x mod pi = 0)                                  |
|||                                                                                   | |||                                                                                |
|<|  2*cot(x)                                                                         |*|<|        2                                                                       |
||<-----------              otherwise                           otherwise             | ||<-1 + cot (x)                                               otherwise            |
|||       2                                                                           | |||------------             otherwise                                              |
|||1 + cot (x)                                                                        | |||       2                                                                        |
\\\                                                                                   / \\\1 + cot (x)                                                                     /
({0for im⁡(x)=0∧2x mod π=0{0for im⁡(x)=0∧2x mod π=02cot⁡(x)cot⁡2(x)+1otherwiseotherwise)({1for im⁡(x)=0∧x mod π=0{1for im⁡(x)=0∧x mod π=0cot⁡2(x)−1cot⁡2(x)+1otherwiseotherwise)\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\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} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)
  /       2   \       
2*\1 - tan (x)/*tan(x)
----------------------
                 2    
    /       2   \     
    \1 + tan (x)/
2⋅(1−tan⁡2(x))tan⁡(x)(tan⁡2(x)+1)2\frac{2 \cdot \left(1 - \tan^{2}{\left(x \right)}\right) \tan{\left(x \right)}}{\left(\tan^{2}{\left(x \right)} + 1\right)^{2}}
                                                //      1        for And(im(x) = 0, x mod pi = 0)\
//   0      for And(im(x) = 0, 2*x mod pi = 0)\ ||                                               |
||                                            | ||      1                                        |
|<   1                                        |*|<-------------             otherwise            |
||--------              otherwise             | ||   /pi      \                                  |
\\csc(2*x)                                    / ||csc|-- - 2*x|                                  |
                                                \\   \2       /                                  /
({0for im⁡(x)=0∧2x mod π=01csc⁡(2x)otherwise)({1for im⁡(x)=0∧x mod π=01csc⁡(−2x+π2)otherwise)\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\frac{1}{\csc{\left(2 x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)
   tan(2*x)  
-------------
       2     
1 + tan (2*x)
tan⁡(2x)tan⁡2(2x)+1\frac{\tan{\left(2 x \right)}}{\tan^{2}{\left(2 x \right)} + 1}
//                      0                        for And(im(x) = 0, 2*x mod pi = 0)\ //                     1                       for And(im(x) = 0, x mod pi = 0)\
||                                                                                 | ||                                                                             |
|
({0for im⁡(x)=0∧2x mod π=0{0for im⁡(x)=0∧2x mod π=0sin⁡(2x)otherwiseotherwise)({1for im⁡(x)=0∧x mod π=0{1for im⁡(x)=0∧x mod π=0cos⁡(2x)otherwiseotherwise)\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\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} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)
//      0        for And(im(x) = 0, 2*x mod pi = 0)\                                              
||                                                 | //   1      for And(im(x) = 0, x mod pi = 0)\
|<   /      pi\                                    |*|<                                          |
||cos|2*x - --|              otherwise             | \\cos(2*x)             otherwise            /
\\   \      2 /                                    /
({0for im⁡(x)=0∧2x mod π=0cos⁡(2x−π2)otherwise)({1for im⁡(x)=0∧x mod π=0cos⁡(2x)otherwise)\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\cos{\left(2 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 \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases}\right)
//     0       for And(im(x) = 0, 2*x mod pi = 0)\ //     1       for And(im(x) = 0, x mod pi = 0)\
||                                               | ||                                             |
||  2*tan(x)                                     | ||       2                                     |
|<-----------              otherwise             |*|<1 - tan (x)                                  |
||       2                                       | ||-----------             otherwise            |
||1 + tan (x)                                    | ||       2                                     |
\\                                               / \\1 + tan (x)                                  /
({0for im⁡(x)=0∧2x mod π=02tan⁡(x)tan⁡2(x)+1otherwise)({1for im⁡(x)=0∧x mod π=01−tan⁡2(x)tan⁡2(x)+1otherwise)\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 2 x \bmod \pi = 0 \\\frac{2 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge x \bmod \pi = 0 \\\frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right)
sin(4*x)
--------
   2
sin⁡(4x)2\frac{\sin{\left(4 x \right)}}{2}
/   0      for And(im(x) = 0, 4*x mod pi = 0)
<                                            
\sin(4*x)              otherwise             
---------------------------------------------
                      2
{0for im⁡(x)=0∧4x mod π=0sin⁡(4x)otherwise2\frac{\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(x\right)} = 0 \wedge 4 x \bmod \pi = 0 \\\sin{\left(4 x \right)} & \text{otherwise} \end{cases}}{2}
    1     
----------
2*csc(4*x)
12csc⁡(4x)\frac{1}{2 \csc{\left(4 x \right)}}
          1           
----------------------
            /      pi\
sec(2*x)*sec|2*x - --|
            \      2 /
1sec⁡(2x)sec⁡(2x−π2)\frac{1}{\sec{\left(2 x \right)} \sec{\left(2 x - \frac{\pi}{2} \right)}}
          1           
----------------------
            /pi      \
csc(2*x)*csc|-- - 2*x|
            \2       /
1csc⁡(2x)csc⁡(−2x+π2)\frac{1}{\csc{\left(2 x \right)} \csc{\left(- 2 x + \frac{\pi}{2} \right)}}
       1       
---------------
     /      pi\
2*sec|4*x - --|
     \      2 /
12sec⁡(4x−π2)\frac{1}{2 \sec{\left(4 x - \frac{\pi}{2} \right)}}
Раскрыть выражение [src]
                        3          
-2*cos(x)*sin(x) + 4*cos (x)*sin(x)
4sin⁡(x)cos⁡3(x)−2sin⁡(x)cos⁡(x)4 \sin{\left(x \right)} \cos^{3}{\left(x \right)} - 2 \sin{\left(x \right)} \cos{\left(x \right)}