Найти значение выражения 1-cos(2*a)+sin(2*a)еслиa=3 (1 минус косинус от (2 умножить на a) плюс синус от (2 умножить на a)еслиa равно 3) [Есть ответ!]

1-cos(2*a)+sin(2*a)еслиa=3 (упростите выражение)

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Решение

Вы ввели [src]
1 - cos(2*a) + sin(2*a)
$$\sin{\left(2 a \right)} - \cos{\left(2 a \right)} + 1$$
Подстановка условия [src]
1 - cos(2*a) + sin(2*a) при a = 3
подставляем
1 - cos(2*a) + sin(2*a)
$$\sin{\left(2 a \right)} - \cos{\left(2 a \right)} + 1$$
      ___    /      pi\
1 - \/ 2 *cos|2*a + --|
             \      4 /
$$- \sqrt{2} \cos{\left(2 a + \frac{\pi}{4} \right)} + 1$$
переменные
a = 3
$$a = 3$$
      ___    /        pi\
1 - \/ 2 *cos|2*(3) + --|
             \        4 /
$$- \sqrt{2} \cos{\left(2 (3) + \frac{\pi}{4} \right)} + 1$$
      ___    /      pi\
1 - \/ 2 *cos|2*3 + --|
             \      4 /
$$- \sqrt{2} \cos{\left(\frac{\pi}{4} + 2 \cdot 3 \right)} + 1$$
      ___    /    pi\
1 - \/ 2 *cos|6 + --|
             \    4 /
$$- \sqrt{2} \cos{\left(\frac{\pi}{4} + 6 \right)} + 1$$
Степени [src]
     -2*I*a    2*I*a     /   -2*I*a    2*I*a\
    e         e        I*\- e       + e     /
1 - ------- - ------ - ----------------------
       2        2                2           
$$- \frac{i \left(e^{2 i a} - e^{- 2 i a}\right)}{2} - \frac{e^{2 i a}}{2} + 1 - \frac{e^{- 2 i a}}{2}$$
Численный ответ [src]
1.0 - cos(2*a) + sin(2*a)
Рациональный знаменатель [src]
1 - cos(2*a) + sin(2*a)
$$\sin{\left (2 a \right )} - \cos{\left (2 a \right )} + 1$$
Объединение рациональных выражений [src]
1 - cos(2*a) + sin(2*a)
$$\sin{\left (2 a \right )} - \cos{\left (2 a \right )} + 1$$
Общее упрощение [src]
      ___    /      pi\
1 - \/ 2 *cos|2*a + --|
             \      4 /
$$- \sqrt{2} \cos{\left(2 a + \frac{\pi}{4} \right)} + 1$$
Собрать выражение [src]
1 - cos(2*a) + sin(2*a)
$$\sin{\left (2 a \right )} - \cos{\left (2 a \right )} + 1$$
Комбинаторика [src]
1 - cos(2*a) + sin(2*a)
$$\sin{\left (2 a \right )} - \cos{\left (2 a \right )} + 1$$
Общий знаменатель [src]
1 - cos(2*a) + sin(2*a)
$$\sin{\left (2 a \right )} - \cos{\left (2 a \right )} + 1$$
Тригонометрическая часть [src]
    //     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)                                     |
1 - |<-1 + cot (a)                                  | + |<-----------              otherwise             |
    ||------------             otherwise            |   ||       2                                       |
    ||       2                                      |   ||1 + cot (a)                                    |
    \\1 + cot (a)                                   /   \\                                               /
$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) + 1$$
                                                    //      0        for And(im(a) = 0, 2*a mod pi = 0)\
    //   1      for And(im(a) = 0, a mod pi = 0)\   ||                                                 |
    ||                                          |   ||      1                                          |
1 - |<   1                                      | + |<-------------              otherwise             |
    ||--------             otherwise            |   ||   /      pi\                                    |
    \\sec(2*a)                                  /   ||sec|2*a - --|                                    |
                                                    \\   \      2 /                                    /
$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\frac{1}{\sec{\left(2 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1}{\sec{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) + 1$$
          //                                         /           /      pi\             \\
          ||               1                  for And|im(a) = 0, |2*a + --| mod 2*pi = 0||
      ___ ||                                         \           \      4 /             /|
1 - \/ 2 *|<                                                                             |
          ||   2/    pi\ /        2/    pi\\                                             |
          ||sin |a + --|*|-1 + cot |a + --||                   otherwise                 |
          \\    \    8 / \         \    8 //                                             /
$$\left(- \sqrt{2} \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(2 a + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\left(\cot^{2}{\left(a + \frac{\pi}{8} \right)} - 1\right) \sin^{2}{\left(a + \frac{\pi}{8} \right)} & \text{otherwise} \end{cases}\right)\right) + 1$$
          ___     
        \/ 2      
1 - --------------
       /       pi\
    csc|-2*a + --|
       \       4 /
$$1 - \frac{\sqrt{2}}{\csc{\left(- 2 a + \frac{\pi}{4} \right)}}$$
    //                       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)                                    |
    |||                                                                                |   |||                                                                                   |
1 - |<|        2                                                                       | + |<|  2*cot(a)                                                                         |
    ||<-1 + cot (a)                                               otherwise            |   ||<-----------              otherwise                           otherwise             |
    |||------------             otherwise                                              |   |||       2                                                                           |
    |||       2                                                                        |   |||1 + cot (a)                                                                        |
    \\\1 + cot (a)                                                                     /   \\\                                                                                   /
$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1$$
          1            1    
1 + ------------- - --------
       /      pi\   sec(2*a)
    sec|2*a - --|           
       \      2 /           
$$1 + \frac{1}{\sec{\left(2 a - \frac{\pi}{2} \right)}} - \frac{1}{\sec{\left(2 a \right)}}$$
                  /      pi\
1 - cos(2*a) + cos|2*a - --|
                  \      2 /
$$- \cos{\left(2 a \right)} + \cos{\left(2 a - \frac{\pi}{2} \right)} + 1$$
           2                 
    1 - tan (a)     2*tan(a) 
1 - ----------- + -----------
           2             2   
    1 + tan (a)   1 + tan (a)
$$- \frac{1 - \tan^{2}{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} + 1 + \frac{2 \tan{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1}$$
      ___ /       2/    pi\\
    \/ 2 *|1 - tan |a + --||
          \        \    8 //
1 - ------------------------
               2/    pi\    
        1 + tan |a + --|    
                \    8 /    
$$- \frac{\sqrt{2} \cdot \left(1 - \tan^{2}{\left(a + \frac{\pi}{8} \right)}\right)}{\tan^{2}{\left(a + \frac{\pi}{8} \right)} + 1} + 1$$
      ___    /      3*pi\
1 - \/ 2 *sin|2*a + ----|
             \       4  /
$$- \sqrt{2} \sin{\left(2 a + \frac{3 \pi}{4} \right)} + 1$$
    //      1        for And(im(a) = 0, a mod pi = 0)\                                                  
    ||                                               |   //   0      for And(im(a) = 0, 2*a mod pi = 0)\
    ||      1                                        |   ||                                            |
1 - |<-------------             otherwise            | + |<   1                                        |
    ||   /pi      \                                  |   ||--------              otherwise             |
    ||csc|-- - 2*a|                                  |   \\csc(2*a)                                    /
    \\   \2       /                                  /                                                  
$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\frac{1}{\csc{\left(2 a \right)}} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1$$
    //      1        for And(im(a) = 0, a mod pi = 0)\                                                  
    ||                                               |   //   0      for And(im(a) = 0, 2*a mod pi = 0)\
1 - |<   /pi      \                                  | + |<                                            |
    ||sin|-- + 2*a|             otherwise            |   \\sin(2*a)              otherwise             /
    \\   \2       /                                  /                                                  
$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin{\left(2 a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + 1$$
                                                    //      0        for And(im(a) = 0, 2*a mod pi = 0)\
    //   1      for And(im(a) = 0, a mod pi = 0)\   ||                                                 |
1 - |<                                          | + |<   /      pi\                                    |
    \\cos(2*a)             otherwise            /   ||cos|2*a - --|              otherwise             |
                                                    \\   \      2 /                                    /
$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\cos{\left(2 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + 1$$
    //   1      for And(im(a) = 0, a mod pi = 0)\   //   0      for And(im(a) = 0, 2*a mod pi = 0)\
1 - |<                                          | + |<                                            |
    \\cos(2*a)             otherwise            /   \\sin(2*a)              otherwise             /
$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}\right) + 1$$
      ___    /      pi\
1 - \/ 2 *cos|2*a + --|
             \      4 /
$$- \sqrt{2} \cos{\left(2 a + \frac{\pi}{4} \right)} + 1$$
          //                          /           /      pi\             \\
          ||        1          for And|im(a) = 0, |2*a + --| mod 2*pi = 0||
          ||                          \           \      4 /             /|
          ||                                                              |
      ___ ||        2/    pi\                                             |
1 - \/ 2 *|<-1 + cot |a + --|                                             |
          ||         \    8 /                                             |
          ||-----------------                   otherwise                 |
          ||        2/    pi\                                             |
          || 1 + cot |a + --|                                             |
          \\         \    8 /                                             /
$$\left(- \sqrt{2} \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge \left(2 a + \frac{\pi}{4}\right) \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(a + \frac{\pi}{8} \right)} - 1}{\cot^{2}{\left(a + \frac{\pi}{8} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 1$$
       /pi      \           
1 - sin|-- + 2*a| + sin(2*a)
       \2       /           
$$\sin{\left(2 a \right)} - \sin{\left(2 a + \frac{\pi}{2} \right)} + 1$$
       1          1    
1 + -------- - --------
    csc(2*a)   sec(2*a)
$$1 - \frac{1}{\sec{\left(2 a \right)}} + \frac{1}{\csc{\left(2 a \right)}}$$
       1             1      
1 + -------- - -------------
    csc(2*a)      /pi      \
               csc|-- - 2*a|
                  \2       /
$$1 - \frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(2 a \right)}}$$
    //                     1                       for And(im(a) = 0, a mod pi = 0)\   //                      0                        for And(im(a) = 0, 2*a mod pi = 0)\
    ||                                                                             |   ||                                                                                 |
1 - |
$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1$$
          ___    
        \/ 2     
1 - -------------
       /      pi\
    sec|2*a + --|
       \      4 /
$$1 - \frac{\sqrt{2}}{\sec{\left(2 a + \frac{\pi}{4} \right)}}$$
    //     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)                                     |
1 - |<1 - tan (a)                                  | + |<-----------              otherwise             |
    ||-----------             otherwise            |   ||       2                                       |
    ||       2                                     |   ||1 + tan (a)                                    |
    \\1 + tan (a)                                  /   \\                                               /
$$\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 2 a \bmod \pi = 0 \\\frac{2 \tan{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) - \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1 - \tan^{2}{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}\right) + 1$$
Раскрыть выражение [src]
         2                     
2 - 2*cos (a) + 2*cos(a)*sin(a)
$$2 \sin{\left(a \right)} \cos{\left(a \right)} - 2 \cos^{2}{\left(a \right)} + 2$$