Найти значение выражения 2*sin(11*a)*cos(5*a)-sin(6*a)еслиa=1/4 (2 умножить на синус от (11 умножить на a) умножить на косинус от (5 умножить на a) минус синус от (6 умножить на a)еслиa равно 1 делить на 4) [Есть ответ!]
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2*sin(11*a)*cos(5*a)-sin(6*a)еслиa=1/4 (упростите выражение)

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Примеры

Решение

Вы ввели [src]
2*sin(11*a)*cos(5*a) - sin(6*a)
$$- \sin{\left(6 a \right)} + 2 \sin{\left(11 a \right)} \cos{\left(5 a \right)}$$
Подстановка условия [src]
2*sin(11*a)*cos(5*a) - sin(6*a) при a = 1/4
подставляем
2*sin(11*a)*cos(5*a) - sin(6*a)
$$- \sin{\left(6 a \right)} + 2 \sin{\left(11 a \right)} \cos{\left(5 a \right)}$$
sin(16*a)
$$\sin{\left(16 a \right)}$$
переменные
a = 1/4
$$a = \frac{1}{4}$$
sin(16*(1/4))
$$\sin{\left(16 (1/4) \right)}$$
sin(4)
$$\sin{\left(4 \right)}$$
Степени [src]
  /   -6*I*a    6*I*a\     / -5*I*a    5*I*a\                       
I*\- e       + e     /     |e         e     | /   -11*I*a    11*I*a\
---------------------- - I*|------- + ------|*\- e        + e      /
          2                \   2        2   /
$$- i \left(\frac{e^{5 i a}}{2} + \frac{e^{- 5 i a}}{2}\right) \left(e^{11 i a} - e^{- 11 i a}\right) + \frac{i \left(e^{6 i a} - e^{- 6 i a}\right)}{2}$$
Численный ответ [src]
-sin(6*a) + 2.0*cos(5*a)*sin(11*a)
Общее упрощение [src]
sin(16*a)
$$\sin{\left(16 a \right)}$$
Собрать выражение [src]
sin(16*a)
$$\sin{\left (16 a \right )}$$
Тригонометрическая часть [src]
     1               2         
- -------- + ------------------
  csc(6*a)   csc(11*a)*sec(5*a)
$$\frac{2}{\csc{\left(11 a \right)} \sec{\left(5 a \right)}} - \frac{1}{\csc{\left(6 a \right)}}$$
  //      0        for And(im(a) = 0, 6*a mod pi = 0)\     //      0         for And(im(a) = 0, 11*a mod pi = 0)\                                                  
  ||                                                 |     ||                                                   | //   1      for And(im(a) = 0, 5*a mod 2*pi = 0)\
  ||      1                                          |     ||      1                                            | ||                                              |
- |<-------------              otherwise             | + 2*|<--------------               otherwise             |*|<   1                                          |
  ||   /      pi\                                    |     ||   /       pi\                                     | ||--------               otherwise              |
  ||sec|6*a - --|                                    |     ||sec|11*a - --|                                     | \\sec(5*a)                                      /
  \\   \      2 /                                    /     \\   \       2 /                                     /
$$\left(- \begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\frac{1}{\sec{\left(6 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 11 a \bmod \pi = 0 \\\frac{1}{\sec{\left(11 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 5 a \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(5 a \right)}} & \text{otherwise} \end{cases}\right)\right)$$
/    0      for And(im(a) = 0, 16*a mod pi = 0)
<                                              
\sin(16*a)               otherwise
$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 16 a \bmod \pi = 0 \\\sin{\left(16 a \right)} & \text{otherwise} \end{cases}$$
     1                  2           
- -------- + -----------------------
  csc(6*a)                /pi      \
             csc(11*a)*csc|-- - 5*a|
                          \2       /
$$\frac{2}{\csc{\left(11 a \right)} \csc{\left(- 5 a + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(6 a \right)}}$$
  2*tan(8*a) 
-------------
       2     
1 + tan (8*a)
$$\frac{2 \tan{\left(8 a \right)}}{\tan^{2}{\left(8 a \right)} + 1}$$
                                                                                                        //      1        for And(im(a) = 0, 5*a mod 2*pi = 0)\
  //   0      for And(im(a) = 0, 6*a mod pi = 0)\     //    0      for And(im(a) = 0, 11*a mod pi = 0)\ ||                                                   |
  ||                                            |     ||                                              | ||      1                                            |
- |<   1                                        | + 2*|<    1                                         |*|<-------------               otherwise              |
  ||--------              otherwise             |     ||---------               otherwise             | ||   /pi      \                                      |
  \\csc(6*a)                                    /     \\csc(11*a)                                     / ||csc|-- - 5*a|                                      |
                                                                                                        \\   \2       /                                      /
$$\left(- \begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\frac{1}{\csc{\left(6 a \right)}} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 11 a \bmod \pi = 0 \\\frac{1}{\csc{\left(11 a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 5 a \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- 5 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right)$$
   /       pi\
cos|16*a - --|
   \       2 /
$$\cos{\left(16 a - \frac{\pi}{2} \right)}$$
  //      0        for And(im(a) = 0, 6*a mod pi = 0)\     //      0         for And(im(a) = 0, 11*a mod pi = 0)\                                                  
  ||                                                 |     ||                                                   | //   1      for And(im(a) = 0, 5*a mod 2*pi = 0)\
- |<   /      pi\                                    | + 2*|<   /       pi\                                     |*|<                                              |
  ||cos|6*a - --|              otherwise             |     ||cos|11*a - --|               otherwise             | \\cos(5*a)               otherwise              /
  \\   \      2 /                                    /     \\   \       2 /                                     /
$$\left(- \begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\cos{\left(6 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 11 a \bmod \pi = 0 \\\cos{\left(11 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 5 a \bmod 2 \pi = 0 \\\cos{\left(5 a \right)} & \text{otherwise} \end{cases}\right)\right)$$
                                                                                                //                         0                            for And(im(a) = 0, 11*a mod pi = 0)\ //                          1                            for And(im(a) = 0, 5*a mod 2*pi = 0)\
  //                        0                           for And(im(a) = 0, 6*a mod pi = 0)\     ||                                                                                         | ||                                                                                           |
  ||                                                                                      |     ||/      0         for And(im(a) = 0, 11*a mod pi = 0)                                     | ||/      1         for And(im(a) = 0, 5*a mod 2*pi = 0)                                      |
  ||/      0        for And(im(a) = 0, 6*a mod pi = 0)                                    |     |||                                                                                        | |||                                                                                          |
  |||                                                                                     |     |||      /11*a\                                                                            | |||        2/5*a\                                                                            |
- |<|  2*cot(3*a)                                                                         | + 2*|<| 2*cot|----|                                                                            |*|<|-1 + cot |---|                                                                            |
  ||<-------------              otherwise                           otherwise             |     ||<      \ 2  /                                                      otherwise             | ||<         \ 2 /                                                     otherwise              |
  |||       2                                                                             |     |||--------------               otherwise                                                  | |||--------------               otherwise                                                    |
  |||1 + cot (3*a)                                                                        |     |||       2/11*a\                                                                          | |||       2/5*a\                                                                             |
  \\\                                                                                     /     |||1 + cot |----|                                                                          | |||1 + cot |---|                                                                             |
                                                                                                \\\        \ 2  /                                                                          / \\\        \ 2 /                                                                             /
$$\left(- \begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\frac{2 \cot{\left(3 a \right)}}{\cot^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 11 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 11 a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{11 a}{2} \right)}}{\cot^{2}{\left(\frac{11 a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 5 a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 5 a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{5 a}{2} \right)} - 1}{\cot^{2}{\left(\frac{5 a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
  //                      0                        for And(im(a) = 0, 6*a mod pi = 0)\     //                       0                         for And(im(a) = 0, 11*a mod pi = 0)\ //                       1                         for And(im(a) = 0, 5*a mod 2*pi = 0)\
  ||                                                                                 |     ||                                                                                    | ||                                                                                     |
- |
$$\left(- \begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 11 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 11 a \bmod \pi = 0 \\\sin{\left(11 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 5 a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 5 a \bmod 2 \pi = 0 \\\cos{\left(5 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
/      0        for And(im(a) = 0, 16*a mod pi = 0)
|                                                  
|  2*cot(8*a)                                      
<-------------               otherwise             
|       2                                          
|1 + cot (8*a)                                     
\
$$\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 16 a \bmod \pi = 0 \\\frac{2 \cot{\left(8 a \right)}}{\cot^{2}{\left(8 a \right)} + 1} & \text{otherwise} \end{cases}$$
                           /pi      \
-sin(6*a) + 2*sin(11*a)*sin|-- + 5*a|
                           \2       /
$$- \sin{\left(6 a \right)} + 2 \sin{\left(11 a \right)} \sin{\left(5 a + \frac{\pi}{2} \right)}$$
      1       
--------------
   /       pi\
sec|16*a - --|
   \       2 /
$$\frac{1}{\sec{\left(16 a - \frac{\pi}{2} \right)}}$$
sin(16*a)
$$\sin{\left(16 a \right)}$$
        1                    2           
- ------------- + -----------------------
     /      pi\               /       pi\
  sec|6*a - --|   sec(5*a)*sec|11*a - --|
     \      2 /               \       2 /
$$- \frac{1}{\sec{\left(6 a - \frac{\pi}{2} \right)}} + \frac{2}{\sec{\left(5 a \right)} \sec{\left(11 a - \frac{\pi}{2} \right)}}$$
    1    
---------
csc(16*a)
$$\frac{1}{\csc{\left(16 a \right)}}$$
                      /       2/5*a\\    /11*a\   
                    4*|1 - tan |---||*tan|----|   
    2*tan(3*a)        \        \ 2 //    \ 2  /   
- ------------- + --------------------------------
         2        /       2/5*a\\ /       2/11*a\\
  1 + tan (3*a)   |1 + tan |---||*|1 + tan |----||
                  \        \ 2 // \        \ 2  //
$$\frac{4 \cdot \left(1 - \tan^{2}{\left(\frac{5 a}{2} \right)}\right) \tan{\left(\frac{11 a}{2} \right)}}{\left(\tan^{2}{\left(\frac{5 a}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{11 a}{2} \right)} + 1\right)} - \frac{2 \tan{\left(3 a \right)}}{\tan^{2}{\left(3 a \right)} + 1}$$
  //   0      for And(im(a) = 0, 6*a mod pi = 0)\     //    0      for And(im(a) = 0, 11*a mod pi = 0)\ //   1      for And(im(a) = 0, 5*a mod 2*pi = 0)\
- |<                                            | + 2*|<                                              |*|<                                              |
  \\sin(6*a)              otherwise             /     \\sin(11*a)               otherwise             / \\cos(5*a)               otherwise              /
$$\left(- \begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 11 a \bmod \pi = 0 \\\sin{\left(11 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 5 a \bmod 2 \pi = 0 \\\cos{\left(5 a \right)} & \text{otherwise} \end{cases}\right)\right)$$
     2              
2*cos (8*a)*tan(8*a)
$$2 \cos^{2}{\left(8 a \right)} \tan{\left(8 a \right)}$$
     /      pi\                 /       pi\
- cos|6*a - --| + 2*cos(5*a)*cos|11*a - --|
     \      2 /                 \       2 /
$$2 \cos{\left(5 a \right)} \cos{\left(11 a - \frac{\pi}{2} \right)} - \cos{\left(6 a - \frac{\pi}{2} \right)}$$
                                                                                                        //      1        for And(im(a) = 0, 5*a mod 2*pi = 0)\
  //   0      for And(im(a) = 0, 6*a mod pi = 0)\     //    0      for And(im(a) = 0, 11*a mod pi = 0)\ ||                                                   |
- |<                                            | + 2*|<                                              |*|<   /pi      \                                      |
  \\sin(6*a)              otherwise             /     \\sin(11*a)               otherwise             / ||sin|-- + 5*a|               otherwise              |
                                                                                                        \\   \2       /                                      /
$$\left(- \begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\sin{\left(6 a \right)} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 11 a \bmod \pi = 0 \\\sin{\left(11 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 5 a \bmod 2 \pi = 0 \\\sin{\left(5 a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right)$$
                                                           //      0         for And(im(a) = 0, 11*a mod pi = 0)\ //      1        for And(im(a) = 0, 5*a mod 2*pi = 0)\
  //      0        for And(im(a) = 0, 6*a mod pi = 0)\     ||                                                   | ||                                                   |
  ||                                                 |     ||      /11*a\                                       | ||       2/5*a\                                      |
  ||  2*tan(3*a)                                     |     || 2*tan|----|                                       | ||1 - tan |---|                                      |
- |<-------------              otherwise             | + 2*|<      \ 2  /                                       |*|<        \ 2 /                                      |
  ||       2                                         |     ||--------------               otherwise             | ||-------------               otherwise              |
  ||1 + tan (3*a)                                    |     ||       2/11*a\                                     | ||       2/5*a\                                      |
  \\                                                 /     ||1 + tan |----|                                     | ||1 + tan |---|                                      |
                                                           \\        \ 2  /                                     / \\        \ 2 /                                      /
$$\left(- \begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\frac{2 \tan{\left(3 a \right)}}{\tan^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 11 a \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{11 a}{2} \right)}}{\tan^{2}{\left(\frac{11 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 5 a \bmod 2 \pi = 0 \\\frac{1 - \tan^{2}{\left(\frac{5 a}{2} \right)}}{\tan^{2}{\left(\frac{5 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
                                                           //      0         for And(im(a) = 0, 11*a mod pi = 0)\ //      1         for And(im(a) = 0, 5*a mod 2*pi = 0)\
  //      0        for And(im(a) = 0, 6*a mod pi = 0)\     ||                                                   | ||                                                    |
  ||                                                 |     ||      /11*a\                                       | ||        2/5*a\                                      |
  ||  2*cot(3*a)                                     |     || 2*cot|----|                                       | ||-1 + cot |---|                                      |
- |<-------------              otherwise             | + 2*|<      \ 2  /                                       |*|<         \ 2 /                                      |
  ||       2                                         |     ||--------------               otherwise             | ||--------------               otherwise              |
  ||1 + cot (3*a)                                    |     ||       2/11*a\                                     | ||       2/5*a\                                       |
  \\                                                 /     ||1 + cot |----|                                     | ||1 + cot |---|                                       |
                                                           \\        \ 2  /                                     / \\        \ 2 /                                       /
$$\left(- \begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 6 a \bmod \pi = 0 \\\frac{2 \cot{\left(3 a \right)}}{\cot^{2}{\left(3 a \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 11 a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{11 a}{2} \right)}}{\cot^{2}{\left(\frac{11 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 5 a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{5 a}{2} \right)} - 1}{\cot^{2}{\left(\frac{5 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
Раскрыть выражение [src]
            3       9               5       7               3       5               5       11               7                      11                     5       3              3                    3                                        5                     3       3               5                      9                      5       5               3       11               5       9                3       7   
- 112640*cos (a)*sin (a) - 90112*cos (a)*sin (a) - 49280*cos (a)*sin (a) - 32768*cos (a)*sin  (a) - 28160*sin (a)*cos(a) - 10240*sin  (a)*cos(a) - 7040*cos (a)*sin (a) - 2168*sin (a)*cos(a) - 440*cos (a)*sin(a) + 104*cos(a)*sin(a) + 352*cos (a)*sin(a) + 8800*cos (a)*sin (a) + 12288*sin (a)*cos(a) + 28160*sin (a)*cos(a) + 39424*cos (a)*sin (a) + 40960*cos (a)*sin  (a) + 90112*cos (a)*sin (a) + 112640*cos (a)*sin (a)
$$- 32768 \sin^{11}{\left(a \right)} \cos^{5}{\left(a \right)} + 40960 \sin^{11}{\left(a \right)} \cos^{3}{\left(a \right)} - 10240 \sin^{11}{\left(a \right)} \cos{\left(a \right)} + 90112 \sin^{9}{\left(a \right)} \cos^{5}{\left(a \right)} - 112640 \sin^{9}{\left(a \right)} \cos^{3}{\left(a \right)} + 28160 \sin^{9}{\left(a \right)} \cos{\left(a \right)} - 90112 \sin^{7}{\left(a \right)} \cos^{5}{\left(a \right)} + 112640 \sin^{7}{\left(a \right)} \cos^{3}{\left(a \right)} - 28160 \sin^{7}{\left(a \right)} \cos{\left(a \right)} + 39424 \sin^{5}{\left(a \right)} \cos^{5}{\left(a \right)} - 49280 \sin^{5}{\left(a \right)} \cos^{3}{\left(a \right)} + 12288 \sin^{5}{\left(a \right)} \cos{\left(a \right)} - 7040 \sin^{3}{\left(a \right)} \cos^{5}{\left(a \right)} + 8800 \sin^{3}{\left(a \right)} \cos^{3}{\left(a \right)} - 2168 \sin^{3}{\left(a \right)} \cos{\left(a \right)} + 352 \sin{\left(a \right)} \cos^{5}{\left(a \right)} - 440 \sin{\left(a \right)} \cos^{3}{\left(a \right)} + 104 \sin{\left(a \right)} \cos{\left(a \right)}$$