Найти значение выражения sqrt(4-4*sin(2*a))*sqrt(2-2*cos(2*a))еслиa=-1/3 (квадратный корень из (4 минус 4 умножить на синус от (2 умножить на a)) умножить на квадратный корень из (2 минус 2 умножить на косинус от (2 умножить на a))еслиa равно минус 1 делить на 3) [Есть ответ!]
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sqrt(4-4*sin(2*a))*sqrt(2-2*cos(2*a))еслиa=-1/3 (упростите выражение)

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Примеры

Решение

Вы ввели [src]
  ________________   ________________
\/ 4 - 4*sin(2*a) *\/ 2 - 2*cos(2*a)
$$\sqrt{2 - 2 \cos{\left(2 a \right)}} \sqrt{4 - 4 \sin{\left(2 a \right)}}$$
Подстановка условия [src]
sqrt(4 - 4*sin(2*a))*sqrt(2 - 2*cos(2*a)) при a = -1/3
подставляем
  ________________   ________________
\/ 4 - 4*sin(2*a) *\/ 2 - 2*cos(2*a)
$$\sqrt{2 - 2 \cos{\left(2 a \right)}} \sqrt{4 - 4 \sin{\left(2 a \right)}}$$
     _________                 
    /    2       ______________
4*\/  sin (a) *\/ 1 - sin(2*a)
$$4 \sqrt{1 - \sin{\left(2 a \right)}} \sqrt{\sin^{2}{\left(a \right)}}$$
переменные
a = -1/3
$$a = - \frac{1}{3}$$
     ______________                      
    /    2            ___________________
4*\/  sin ((-1/3)) *\/ 1 - sin(2*(-1/3))
$$4 \sqrt{1 - \sin{\left(2 (-1/3) \right)}} \sqrt{\sin^{2}{\left((-1/3) \right)}}$$
     ____________                    
    /    2          _________________
4*\/  sin (-1/3) *\/ 1 - sin(2*-1/3)
$$4 \sqrt{1 - \sin{\left(2 \left(- \frac{1}{3}\right) \right)}} \sqrt{\sin^{2}{\left(- \frac{1}{3} \right)}}$$
    ______________         
4*\/ 1 + sin(2/3) *sin(1/3)
$$4 \sqrt{\sin{\left(\frac{2}{3} \right)} + 1} \sin{\left(\frac{1}{3} \right)}$$
Степени [src]
  ___________________________________
\/ (2 - 2*cos(2*a))*(4 - 4*sin(2*a))
$$\sqrt{\left(2 - 2 \cos{\left(2 a \right)}\right) \left(4 - 4 \sin{\left(2 a \right)}\right)}$$
   ______________________________    ______________________
  /         /   -2*I*a    2*I*a\    /      -2*I*a    2*I*a 
\/  4 + 2*I*\- e       + e     / *\/  2 - e       - e
$$\sqrt{2 i \left(e^{2 i a} - e^{- 2 i a}\right) + 4} \sqrt{- e^{2 i a} + 2 - e^{- 2 i a}}$$
Численный ответ [src]
2.82842712474619*(1 - cos(2*a))^0.5*(1 - sin(2*a))^0.5
Рациональный знаменатель [src]
  ________________   ________________
\/ 2 - 2*cos(2*a) *\/ 4 - 4*sin(2*a)
$$\sqrt{- 4 \sin{\left (2 a \right )} + 4} \sqrt{- 2 \cos{\left (2 a \right )} + 2}$$
Объединение рациональных выражений [src]
    ___   ______________   ______________
2*\/ 2 *\/ 1 - cos(2*a) *\/ 1 - sin(2*a)
$$2 \sqrt{2} \sqrt{1 - \sin{\left(2 a \right)}} \sqrt{1 - \cos{\left(2 a \right)}}$$
Общее упрощение [src]
     _________                 
    /    2       ______________
4*\/  sin (a) *\/ 1 - sin(2*a)
$$4 \sqrt{1 - \sin{\left(2 a \right)}} \sqrt{\sin^{2}{\left(a \right)}}$$
Собрать выражение [src]
  ________________   ________________
\/ 2 - 2*cos(2*a) *\/ 4 - 4*sin(2*a)
$$\sqrt{- 4 \sin{\left (2 a \right )} + 4} \sqrt{- 2 \cos{\left (2 a \right )} + 2}$$
Комбинаторика [src]
    ___   ______________   ______________
2*\/ 2 *\/ 1 - cos(2*a) *\/ 1 - sin(2*a)
$$2 \sqrt{2} \sqrt{1 - \sin{\left(2 a \right)}} \sqrt{1 - \cos{\left(2 a \right)}}$$
Общий знаменатель [src]
    ___   ______________   ______________
2*\/ 2 *\/ 1 - cos(2*a) *\/ 1 - sin(2*a)
$$2 \sqrt{2} \sqrt{1 - \sin{\left(2 a \right)}} \sqrt{1 - \cos{\left(2 a \right)}}$$
Тригонометрическая часть [src]
        __________________________________________________________                                                              
       /       //      1        for And(im(a) = 0, a mod pi = 0)\        _______________________________________________________
      /        ||                                               |       /       //   0      for And(im(a) = 0, 2*a mod pi = 0)\ 
     /         ||      1                                        |      /        ||                                            | 
    /    2 - 2*|<-------------             otherwise            | *   /   4 - 4*|<   1                                        | 
   /           ||   /pi      \                                  |    /          ||--------              otherwise             | 
  /            ||csc|-- - 2*a|                                  |  \/           \\csc(2*a)                                    / 
\/             \\   \2       /                                  /
$$\left(\sqrt{2 - \left(2 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right)}\right) \left(\sqrt{4 - \left(4 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\frac{1}{\csc{\left(2 a \right)}} & \text{otherwise} \end{cases}\right)\right)}\right)$$
      __________________________________________________________                                                            
     /       //      1        for And(im(a) = 0, a mod pi = 0)\      _______________________________________________________
    /        ||                                               |     /       //   0      for And(im(a) = 0, 2*a mod pi = 0)\ 
   /   2 - 2*|<   /pi      \                                  | *  /  4 - 4*|<                                            | 
  /          ||sin|-- + 2*a|             otherwise            |  \/         \\sin(2*a)              otherwise             / 
\/           \\   \2       /                                  /
$$\left(\sqrt{2 - \left(2 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin{\left(2 a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right)}\right) \left(\sqrt{4 - \left(4 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases}\right)\right)}\right)$$
                                                                     //             0               for And(im(a) = 0, a mod pi = 0)\
          __________________________________________________________ ||                                                             |
         /       //     0       for And(im(a) = 0, 2*a mod pi = 0)\  ||           ________________                                  |
        /        ||                                               |  ||          /       2/a\                                       |
       /         ||  2*cot(a)                                     |  ||         /     cot |-|                                       |
2*    /    4 - 4*|<-----------              otherwise             | *|<        /          \2/                                       |
     /           ||       2                                       |  ||2*     /    --------------              otherwise            |
    /            ||1 + cot (a)                                    |  ||      /                  2                                   |
  \/             \\                                               /  ||     /      /       2/a\\                                    |
                                                                     ||    /       |1 + cot |-||                                    |
                                                                     \\  \/        \        \2//                                    /
$$2 \left(\sqrt{4 - \left(4 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)\right)}\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\2 \sqrt{\frac{\cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}} & \text{otherwise} \end{cases}\right)$$
    ______________     ______________
   /        2         /        4     
  /  2 - -------- *  /  4 - -------- 
\/       sec(2*a)  \/       csc(2*a)
$$\sqrt{2 - \frac{2}{\sec{\left(2 a \right)}}} \sqrt{4 - \frac{4}{\csc{\left(2 a \right)}}}$$
        ________________________________________________________         __________________________________________________________
       /       //     1       for And(im(a) = 0, a mod pi = 0)\         /       //     0       for And(im(a) = 0, 2*a mod pi = 0)\ 
      /        ||                                             |        /        ||                                               | 
     /         ||       2                                     |       /         ||  2*tan(a)                                     | 
    /    2 - 2*|<1 - tan (a)                                  | *    /    4 - 4*|<-----------              otherwise             | 
   /           ||-----------             otherwise            |     /           ||       2                                       | 
  /            ||       2                                     |    /            ||1 + tan (a)                                    | 
\/             \\1 + tan (a)                                  /  \/             \\                                               /
$$\left(\sqrt{2 - \left(2 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1 - \tan^{2}{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)\right)}\right) \left(\sqrt{4 - \left(4 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\frac{2 \tan{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)\right)}\right)$$
    ______________       ___________________
   /        2           /           4       
  /  2 - -------- *    /  4 - ------------- 
\/       sec(2*a)     /          /      pi\ 
                     /        sec|2*a - --| 
                   \/            \      2 /
$$\sqrt{2 - \frac{2}{\sec{\left(2 a \right)}}} \sqrt{4 - \frac{4}{\sec{\left(2 a - \frac{\pi}{2} \right)}}}$$
           ________________                       
          /       2/a\                            
         /     tan |-|           _________________
        /          \2/          /       8*tan(a)  
4*     /    -------------- *   /  4 - ----------- 
      /                  2    /              2    
     /      /       2/a\\   \/        1 + tan (a) 
    /       |1 + tan |-||                         
  \/        \        \2//
$$4 \sqrt{\frac{\tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}} \sqrt{4 - \frac{8 \tan{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1}}$$
      ___________________     ______________
     /           2           /        4     
    /  2 - ------------- *  /  4 - -------- 
   /          /pi      \  \/       csc(2*a) 
  /        csc|-- - 2*a|                    
\/            \2       /
$$\sqrt{2 - \frac{2}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}}} \sqrt{4 - \frac{4}{\csc{\left(2 a \right)}}}$$
          ____________________________________________________________________________________________           _______________________________________________________________________________________________
         /       //                       1                         for And(im(a) = 0, a mod pi = 0)\           /       //                       0                          for And(im(a) = 0, 2*a mod pi = 0)\ 
        /        ||                                                                                 |          /        ||                                                                                    | 
       /         ||/     1        for And(im(a) = 0, a mod pi = 0)                                  |         /         ||/     0       for And(im(a) = 0, 2*a mod pi = 0)                                    | 
      /          |||                                                                                |        /          |||                                                                                   | 
     /     2 - 2*|<|        2                                                                       | *     /     4 - 4*|<|  2*cot(a)                                                                         | 
    /            ||<-1 + cot (a)                                               otherwise            |      /            ||<-----------              otherwise                           otherwise             | 
   /             |||------------             otherwise                                              |     /             |||       2                                                                           | 
  /              |||       2                                                                        |    /              |||1 + cot (a)                                                                        | 
\/               \\\1 + cot (a)                                                                     /  \/               \\\                                                                                   /
$$\left(\sqrt{2 - \left(2 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)}\right) \left(\sqrt{4 - \left(4 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)}\right)$$
      ______________     _____________________
     /    2/    pi\     /          /      pi\ 
2*  /  cos |a - --| *  /  4 - 4*cos|2*a - --| 
  \/       \    2 /  \/            \      2 /
$$2 \sqrt{4 - 4 \cos{\left(2 a - \frac{\pi}{2} \right)}} \sqrt{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}$$
      _____________________                       
     /       /       2   \       _________________
    /      2*\1 - tan (a)/      /       8*tan(a)  
   /   2 - --------------- *   /  4 - ----------- 
  /                 2         /              2    
\/           1 + tan (a)    \/        1 + tan (a)
$$\sqrt{4 - \frac{8 \tan{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1}} \sqrt{- \frac{2 \cdot \left(1 - \tan^{2}{\left(a \right)}\right)}{\tan^{2}{\left(a \right)} + 1} + 2}$$
  ___   ______________   ________________
\/ 2 *\/ 1 - cos(2*a) *\/ 4 - 4*sin(2*a)
$$\sqrt{2} \sqrt{1 - \cos{\left(2 a \right)}} \sqrt{4 - 4 \sin{\left(2 a \right)}}$$
        _________________________________________________________         __________________________________________________________
       /       //     1        for And(im(a) = 0, a mod pi = 0)\         /       //     0       for And(im(a) = 0, 2*a mod pi = 0)\ 
      /        ||                                              |        /        ||                                               | 
     /         ||        2                                     |       /         ||  2*cot(a)                                     | 
    /    2 - 2*|<-1 + cot (a)                                  | *    /    4 - 4*|<-----------              otherwise             | 
   /           ||------------             otherwise            |     /           ||       2                                       | 
  /            ||       2                                      |    /            ||1 + cot (a)                                    | 
\/             \\1 + cot (a)                                   /  \/             \\                                               /
$$\left(\sqrt{2 - \left(2 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)\right)}\right) \left(\sqrt{4 - \left(4 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right)\right)}\right)$$
      _______________________________________________________ //     0        for And(im(a) = 0, a mod pi = 0)\
     /       //   0      for And(im(a) = 0, 2*a mod pi = 0)\  ||                                              |
2*  /  4 - 4*|<                                            | *|<   _________                                  |
  \/         \\sin(2*a)              otherwise             /  ||  /    2                                      |
                                                              \\\/  sin (a)              otherwise            /
$$2 \left(\sqrt{4 - \left(4 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases}\right)\right)}\right) \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sqrt{\sin^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right)$$
    _____________________                   
   /          /pi      \    ________________
  /  2 - 2*sin|-- + 2*a| *\/ 4 - 4*sin(2*a) 
\/            \2       /
$$\sqrt{2 - 2 \sin{\left(2 a + \frac{\pi}{2} \right)}} \sqrt{4 - 4 \sin{\left(2 a \right)}}$$
                       _____________________
  ________________    /          /      pi\ 
\/ 2 - 2*cos(2*a) *  /  4 - 4*cos|2*a - --| 
                   \/            \      2 /
$$\sqrt{2 - 2 \cos{\left(2 a \right)}} \sqrt{4 - 4 \cos{\left(2 a - \frac{\pi}{2} \right)}}$$
     _________                   
    /    2       ________________
2*\/  sin (a) *\/ 4 - 4*sin(2*a)
$$2 \sqrt{4 - 4 \sin{\left(2 a \right)}} \sqrt{\sin^{2}{\left(a \right)}}$$
      ________________________________________________________________________________________       ____________________________________________________________________________________________
     /       //                     1                       for And(im(a) = 0, a mod pi = 0)\       /       //                      0                        for And(im(a) = 0, 2*a mod pi = 0)\ 
    /        ||                                                                             |      /        ||                                                                                 | 
   /   2 - 2*|
$$\left(\sqrt{2 - \left(2 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)}\right) \left(\sqrt{4 - \left(4 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)}\right)$$
                                                                    ____________________________________________________________
      _____________________________________________________        /       //      0        for And(im(a) = 0, 2*a mod pi = 0)\ 
     /       //   1      for And(im(a) = 0, a mod pi = 0)\        /        ||                                                 | 
    /        ||                                          |       /         ||      1                                          | 
   /   2 - 2*|<   1                                      | *    /    4 - 4*|<-------------              otherwise             | 
  /          ||--------             otherwise            |     /           ||   /      pi\                                    | 
\/           \\sec(2*a)                                  /    /            ||sec|2*a - --|                                    | 
                                                            \/             \\   \      2 /                                    /
$$\left(\sqrt{2 - \left(2 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1}{\sec{\left(2 a \right)}} & \text{otherwise} \end{cases}\right)\right)}\right) \left(\sqrt{4 - \left(4 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\frac{1}{\sec{\left(2 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right)}\right)$$
       _________     ______________
      /    1        /        4     
2*   /  ------- *  /  4 - -------- 
    /      2     \/       csc(2*a) 
  \/    csc (a)
$$2 \sqrt{4 - \frac{4}{\csc{\left(2 a \right)}}} \sqrt{\frac{1}{\csc^{2}{\left(a \right)}}}$$
    _____________________________________________________     _______________________________________________________
   /       //   1      for And(im(a) = 0, a mod pi = 0)\     /       //   0      for And(im(a) = 0, 2*a mod pi = 0)\ 
  /  2 - 2*|<                                          | *  /  4 - 4*|<                                            | 
\/         \\cos(2*a)             otherwise            /  \/         \\sin(2*a)              otherwise             /
$$\left(\sqrt{2 - \left(2 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)\right)}\right) \left(\sqrt{4 - \left(4 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases}\right)\right)}\right)$$
        ______________       ___________________
       /      1             /           4       
2*    /  ------------ *    /  4 - ------------- 
     /      2/    pi\     /          /      pi\ 
    /    sec |a - --|    /        sec|2*a - --| 
  \/         \    2 /  \/            \      2 /
$$2 \sqrt{4 - \frac{4}{\sec{\left(2 a - \frac{\pi}{2} \right)}}} \sqrt{\frac{1}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}}$$
                                                                ____________________________________________________________
    _____________________________________________________      /       //      0        for And(im(a) = 0, 2*a mod pi = 0)\ 
   /       //   1      for And(im(a) = 0, a mod pi = 0)\      /        ||                                                 | 
  /  2 - 2*|<                                          | *   /   4 - 4*|<   /      pi\                                    | 
\/         \\cos(2*a)             otherwise            /    /          ||cos|2*a - --|              otherwise             | 
                                                          \/           \\   \      2 /                                    /
$$\left(\sqrt{2 - \left(2 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right)\right)}\right) \left(\sqrt{4 - \left(4 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\cos{\left(2 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right)}\right)$$
Раскрыть выражение [src]
   _______________                        
  /          2       _____________________
\/  4 - 4*cos (a) *\/ 4 - 8*cos(a)*sin(a)
$$\sqrt{4 - 4 \cos^{2}{\left(a \right)}} \sqrt{- 8 \sin{\left(a \right)} \cos{\left(a \right)} + 4}$$