Господин Экзамен

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Упрощение выражений/ sin(pi+a)+cos(3*pi/2-a) если a=3

sin(pi+a)+cos(3*pi/2-a) если a=3

Выражение, которое надо упростить:

Решение

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