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(10*cos(a)-2*sin(a)+10)/( ... sin(a)-5*cos(a)+5)еслиa=2 (упростите выражение)

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Примеры

Решение

Вы ввели [src]
10*cos(a) - 2*sin(a) + 10
-------------------------
  sin(a) - 5*cos(a) + 5
−2sin⁡(a)+10cos⁡(a)+10sin⁡(a)−5cos⁡(a)+5\frac{- 2 \sin{\left(a \right)} + 10 \cos{\left(a \right)} + 10}{\sin{\left(a \right)} - 5 \cos{\left(a \right)} + 5}
Подстановка условия [src]
(10*cos(a) - 2*sin(a) + 10)/(sin(a) - 5*cos(a) + 5) при a = 2
подставляем
10*cos(a) - 2*sin(a) + 10
-------------------------
  sin(a) - 5*cos(a) + 5
−2sin⁡(a)+10cos⁡(a)+10sin⁡(a)−5cos⁡(a)+5\frac{- 2 \sin{\left(a \right)} + 10 \cos{\left(a \right)} + 10}{\sin{\left(a \right)} - 5 \cos{\left(a \right)} + 5}
2*(5 - sin(a) + 5*cos(a))
-------------------------
  5 - 5*cos(a) + sin(a)
2(−sin⁡(a)+5cos⁡(a)+5)sin⁡(a)−5cos⁡(a)+5\frac{2 \left(- \sin{\left(a \right)} + 5 \cos{\left(a \right)} + 5\right)}{\sin{\left(a \right)} - 5 \cos{\left(a \right)} + 5}
переменные
a = 2
a=2a = 2
2*(5 - sin((2)) + 5*cos((2)))
-----------------------------
  5 - 5*cos((2)) + sin((2))
2(−sin⁡((2))+5cos⁡((2))+5)sin⁡((2))−5cos⁡((2))+5\frac{2 \left(- \sin{\left((2) \right)} + 5 \cos{\left((2) \right)} + 5\right)}{\sin{\left((2) \right)} - 5 \cos{\left((2) \right)} + 5}
2*(5 - sin(2) + 5*cos(2))
-------------------------
  5 - 5*cos(2) + sin(2)
2⋅(5cos⁡(2)−sin⁡(2)+5)sin⁡(2)−5cos⁡(2)+5\frac{2 \cdot \left(5 \cos{\left(2 \right)} - \sin{\left(2 \right)} + 5\right)}{\sin{\left(2 \right)} - 5 \cos{\left(2 \right)} + 5}
Степени [src]
        I*a      -I*a     /   -I*a    I*a\
10 + 5*e    + 5*e     + I*\- e     + e   /
------------------------------------------
       I*a      -I*a     /   -I*a    I*a\ 
    5*e      5*e       I*\- e     + e   / 
5 - ------ - ------- - ------------------ 
      2         2              2
i(eia−e−ia)+5eia+10+5e−ia−i(eia−e−ia)2−5eia2+5−5e−ia2\frac{i \left(e^{i a} - e^{- i a}\right) + 5 e^{i a} + 10 + 5 e^{- i a}}{- \frac{i \left(e^{i a} - e^{- i a}\right)}{2} - \frac{5 e^{i a}}{2} + 5 - \frac{5 e^{- i a}}{2}}
Численный ответ [src]
(10.0 + 10.0*cos(a) - 2.0*sin(a))/(5.0 - 5.0*cos(a) + sin(a))
Рациональный знаменатель [src]
          10                   2*sin(a)               10*cos(a)      
--------------------- - --------------------- + ---------------------
5 - 5*cos(a) + sin(a)   5 - 5*cos(a) + sin(a)   5 - 5*cos(a) + sin(a)
−2sin⁡(a)sin⁡(a)−5cos⁡(a)+5+10cos⁡(a)sin⁡(a)−5cos⁡(a)+5+10sin⁡(a)−5cos⁡(a)+5- \frac{2 \sin{\left(a \right)}}{\sin{\left(a \right)} - 5 \cos{\left(a \right)} + 5} + \frac{10 \cos{\left(a \right)}}{\sin{\left(a \right)} - 5 \cos{\left(a \right)} + 5} + \frac{10}{\sin{\left(a \right)} - 5 \cos{\left(a \right)} + 5}
Объединение рациональных выражений [src]
2*(5 - sin(a) + 5*cos(a))
-------------------------
  5 - 5*cos(a) + sin(a)
2(−sin⁡(a)+5cos⁡(a)+5)sin⁡(a)−5cos⁡(a)+5\frac{2 \left(- \sin{\left(a \right)} + 5 \cos{\left(a \right)} + 5\right)}{\sin{\left(a \right)} - 5 \cos{\left(a \right)} + 5}
Общее упрощение [src]
2*(5 - sin(a) + 5*cos(a))
-------------------------
  5 - 5*cos(a) + sin(a)
2(−sin⁡(a)+5cos⁡(a)+5)sin⁡(a)−5cos⁡(a)+5\frac{2 \left(- \sin{\left(a \right)} + 5 \cos{\left(a \right)} + 5\right)}{\sin{\left(a \right)} - 5 \cos{\left(a \right)} + 5}
Собрать выражение [src]
          10                   2*sin(a)               10*cos(a)      
--------------------- - --------------------- + ---------------------
5 - 5*cos(a) + sin(a)   5 - 5*cos(a) + sin(a)   5 - 5*cos(a) + sin(a)
−2sin⁡(a)sin⁡(a)−5cos⁡(a)+5+10cos⁡(a)sin⁡(a)−5cos⁡(a)+5+10sin⁡(a)−5cos⁡(a)+5- \frac{2 \sin{\left (a \right )}}{\sin{\left (a \right )} - 5 \cos{\left (a \right )} + 5} + \frac{10 \cos{\left (a \right )}}{\sin{\left (a \right )} - 5 \cos{\left (a \right )} + 5} + \frac{10}{\sin{\left (a \right )} - 5 \cos{\left (a \right )} + 5}
Общий знаменатель [src]
               20          
-2 - ----------------------
     -5 - sin(a) + 5*cos(a)
−2−20−sin⁡(a)+5cos⁡(a)−5-2 - \frac{20}{- \sin{\left(a \right)} + 5 \cos{\left(a \right)} - 5}
Тригонометрическая часть [src]
                                                        //     1       for And(im(a) = 0, a mod 2*pi = 0)\
       //  0     for And(im(a) = 0, a mod pi = 0)\      ||                                               |
10 - 2*|<                                        | + 10*|<   /    pi\                                    |
       \\sin(a)             otherwise            /      ||sin|a + --|              otherwise             |
                                                        \\   \    2 /                                    /
----------------------------------------------------------------------------------------------------------
        //     1       for And(im(a) = 0, a mod 2*pi = 0)\                                                
        ||                                               |   //  0     for And(im(a) = 0, a mod pi = 0)\  
  5 - 5*|<   /    pi\                                    | + |<                                        |  
        ||sin|a + --|              otherwise             |   \\sin(a)             otherwise            /  
        \\   \    2 /                                    /
(−2({0for im⁡(a)=0∧a mod π=0sin⁡(a)otherwise))+(10({1for im⁡(a)=0∧a mod 2π=0sin⁡(a+π2)otherwise))+10({0for im⁡(a)=0∧a mod π=0sin⁡(a)otherwise)−(5({1for im⁡(a)=0∧a mod 2π=0sin⁡(a+π2)otherwise))+5\frac{\left(- 2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(10 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\sin{\left(a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + 10}{\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\sin{\left(a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + 5}
                                                        //     1       for And(im(a) = 0, a mod 2*pi = 0)\
       //  0     for And(im(a) = 0, a mod pi = 0)\      ||                                               |
       ||                                        |      ||     1                                         |
10 - 2*|<  1                                     | + 10*|<-----------              otherwise             |
       ||------             otherwise            |      ||   /pi    \                                    |
       \\csc(a)                                  /      ||csc|-- - a|                                    |
                                                        \\   \2     /                                    /
----------------------------------------------------------------------------------------------------------
        //     1       for And(im(a) = 0, a mod 2*pi = 0)\                                                
        ||                                               |   //  0     for And(im(a) = 0, a mod pi = 0)\  
        ||     1                                         |   ||                                        |  
  5 - 5*|<-----------              otherwise             | + |<  1                                     |  
        ||   /pi    \                                    |   ||------             otherwise            |  
        ||csc|-- - a|                                    |   \\csc(a)                                  /  
        \\   \2     /                                    /
(−2({0for im⁡(a)=0∧a mod π=01csc⁡(a)otherwise))+(10({1for im⁡(a)=0∧a mod 2π=01csc⁡(−a+π2)otherwise))+10({0for im⁡(a)=0∧a mod π=01csc⁡(a)otherwise)−(5({1for im⁡(a)=0∧a mod 2π=01csc⁡(−a+π2)otherwise))+5\frac{\left(- 2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1}{\csc{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(10 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + 10}{\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1}{\csc{\left(a \right)}} & \text{otherwise} \end{cases}\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + 5}
          2          10  
10 - ----------- + ------
        /    pi\   sec(a)
     sec|a - --|         
        \    2 /         
-------------------------
          1          5   
 5 + ----------- - ------
        /    pi\   sec(a)
     sec|a - --|         
        \    2 /
10−2sec⁡(a−π2)+10sec⁡(a)5+1sec⁡(a−π2)−5sec⁡(a)\frac{10 - \frac{2}{\sec{\left(a - \frac{\pi}{2} \right)}} + \frac{10}{\sec{\left(a \right)}}}{5 + \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}} - \frac{5}{\sec{\left(a \right)}}}
       //                    0                      for And(im(a) = 0, a mod pi = 0)\      //                     1                       for And(im(a) = 0, a mod 2*pi = 0)\
       ||                                                                           |      ||                                                                               |
10 - 2*|
(−2({0for im⁡(a)=0∧a mod π=0{0for im⁡(a)=0∧a mod π=0sin⁡(a)otherwiseotherwise))+(10({1for im⁡(a)=0∧a mod 2π=0{1for im⁡(a)=0∧a mod 2π=0cos⁡(a)otherwiseotherwise))+10({0for im⁡(a)=0∧a mod π=0{0for im⁡(a)=0∧a mod π=0sin⁡(a)otherwiseotherwise)−(5({1for im⁡(a)=0∧a mod 2π=0{1for im⁡(a)=0∧a mod 2π=0cos⁡(a)otherwiseotherwise))+5\frac{\left(- 2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(10 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + 10}{\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + 5}
       2        10  
10 - ------ + ------
     csc(a)   sec(a)
--------------------
      1        5    
5 + ------ - ------ 
    csc(a)   sec(a)
10+10sec⁡(a)−2csc⁡(a)5−5sec⁡(a)+1csc⁡(a)\frac{10 + \frac{10}{\sec{\left(a \right)}} - \frac{2}{\csc{\left(a \right)}}}{5 - \frac{5}{\sec{\left(a \right)}} + \frac{1}{\csc{\left(a \right)}}}
       //     0       for And(im(a) = 0, a mod pi = 0)\      //     1        for And(im(a) = 0, a mod 2*pi = 0)\
       ||                                             |      ||                                                |
       ||       /a\                                   |      ||        2/a\                                    |
       ||  2*cot|-|                                   |      ||-1 + cot |-|                                    |
10 - 2*|<       \2/                                   | + 10*|<         \2/                                    |
       ||-----------             otherwise            |      ||------------              otherwise             |
       ||       2/a\                                  |      ||       2/a\                                     |
       ||1 + cot |-|                                  |      ||1 + cot |-|                                     |
       \\        \2/                                  /      \\        \2/                                     /
----------------------------------------------------------------------------------------------------------------
        //     1        for And(im(a) = 0, a mod 2*pi = 0)\   //     0       for And(im(a) = 0, a mod pi = 0)\  
        ||                                                |   ||                                             |  
        ||        2/a\                                    |   ||       /a\                                   |  
        ||-1 + cot |-|                                    |   ||  2*cot|-|                                   |  
  5 - 5*|<         \2/                                    | + |<       \2/                                   |  
        ||------------              otherwise             |   ||-----------             otherwise            |  
        ||       2/a\                                     |   ||       2/a\                                  |  
        ||1 + cot |-|                                     |   ||1 + cot |-|                                  |  
        \\        \2/                                     /   \\        \2/                                  /
(−2({0for im⁡(a)=0∧a mod π=02cot⁡(a2)cot⁡2(a2)+1otherwise))+(10({1for im⁡(a)=0∧a mod 2π=0cot⁡2(a2)−1cot⁡2(a2)+1otherwise))+10({0for im⁡(a)=0∧a mod π=02cot⁡(a2)cot⁡2(a2)+1otherwise)−(5({1for im⁡(a)=0∧a mod 2π=0cot⁡2(a2)−1cot⁡2(a2)+1otherwise))+5\frac{\left(- 2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 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)\right) + \left(10 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 10}{\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 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) - \left(5 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 5}
            /a\       /       2/a\\
       4*tan|-|    10*|1 - tan |-||
            \2/       \        \2//
10 - ----------- + ----------------
            2/a\            2/a\   
     1 + tan |-|     1 + tan |-|   
             \2/             \2/   
-----------------------------------
       /       2/a\\          /a\  
     5*|1 - tan |-||     2*tan|-|  
       \        \2//          \2/  
 5 - --------------- + ----------- 
              2/a\            2/a\ 
       1 + tan |-|     1 + tan |-| 
               \2/             \2/
10⋅(1−tan⁡2(a2))tan⁡2(a2)+1+10−4tan⁡(a2)tan⁡2(a2)+1−5⋅(1−tan⁡2(a2))tan⁡2(a2)+1+5+2tan⁡(a2)tan⁡2(a2)+1\frac{\frac{10 \cdot \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + 10 - \frac{4 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}}{- \frac{5 \cdot \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + 5 + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}}
       //  0     for And(im(a) = 0, a mod pi = 0)\      //  1     for And(im(a) = 0, a mod 2*pi = 0)\
10 - 2*|<                                        | + 10*|<                                          |
       \\sin(a)             otherwise            /      \\cos(a)              otherwise             /
-----------------------------------------------------------------------------------------------------
        //  1     for And(im(a) = 0, a mod 2*pi = 0)\   //  0     for And(im(a) = 0, a mod pi = 0)\  
  5 - 5*|<                                          | + |<                                        |  
        \\cos(a)              otherwise             /   \\sin(a)             otherwise            /
(−2({0for im⁡(a)=0∧a mod π=0sin⁡(a)otherwise))+(10({1for im⁡(a)=0∧a mod 2π=0cos⁡(a)otherwise))+10({0for im⁡(a)=0∧a mod π=0sin⁡(a)otherwise)−(5({1for im⁡(a)=0∧a mod 2π=0cos⁡(a)otherwise))+5\frac{\left(- 2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(10 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + 10}{\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + 5}
       2           10    
10 - ------ + -----------
     csc(a)      /pi    \
              csc|-- - a|
                 \2     /
-------------------------
       1           5     
 5 + ------ - -----------
     csc(a)      /pi    \
              csc|-- - a|
                 \2     /
10+10csc⁡(−a+π2)−2csc⁡(a)5−5csc⁡(−a+π2)+1csc⁡(a)\frac{10 + \frac{10}{\csc{\left(- a + \frac{\pi}{2} \right)}} - \frac{2}{\csc{\left(a \right)}}}{5 - \frac{5}{\csc{\left(- a + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(a \right)}}}
       //     0       for And(im(a) = 0, a mod pi = 0)\                                                   
       ||                                             |      //  1     for And(im(a) = 0, a mod 2*pi = 0)\
10 - 2*|<   /    pi\                                  | + 10*|<                                          |
       ||cos|a - --|             otherwise            |      \\cos(a)              otherwise             /
       \\   \    2 /                                  /                                                   
----------------------------------------------------------------------------------------------------------
                                                        //     0       for And(im(a) = 0, a mod pi = 0)\  
        //  1     for And(im(a) = 0, a mod 2*pi = 0)\   ||                                             |  
  5 - 5*|<                                          | + |<   /    pi\                                  |  
        \\cos(a)              otherwise             /   ||cos|a - --|             otherwise            |  
                                                        \\   \    2 /                                  /
(−2({0for im⁡(a)=0∧a mod π=0cos⁡(a−π2)otherwise))+(10({1for im⁡(a)=0∧a mod 2π=0cos⁡(a)otherwise))+10({0for im⁡(a)=0∧a mod π=0cos⁡(a−π2)otherwise)−(5({1for im⁡(a)=0∧a mod 2π=0cos⁡(a)otherwise))+5\frac{\left(- 2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\cos{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(10 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + 10}{\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\cos{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + 5}
       //     0       for And(im(a) = 0, a mod pi = 0)\      //     1       for And(im(a) = 0, a mod 2*pi = 0)\
       ||                                             |      ||                                               |
       ||       /a\                                   |      ||       2/a\                                    |
       ||  2*tan|-|                                   |      ||1 - tan |-|                                    |
10 - 2*|<       \2/                                   | + 10*|<        \2/                                    |
       ||-----------             otherwise            |      ||-----------              otherwise             |
       ||       2/a\                                  |      ||       2/a\                                    |
       ||1 + tan |-|                                  |      ||1 + tan |-|                                    |
       \\        \2/                                  /      \\        \2/                                    /
---------------------------------------------------------------------------------------------------------------
        //     1       for And(im(a) = 0, a mod 2*pi = 0)\   //     0       for And(im(a) = 0, a mod pi = 0)\  
        ||                                               |   ||                                             |  
        ||       2/a\                                    |   ||       /a\                                   |  
        ||1 - tan |-|                                    |   ||  2*tan|-|                                   |  
  5 - 5*|<        \2/                                    | + |<       \2/                                   |  
        ||-----------              otherwise             |   ||-----------             otherwise            |  
        ||       2/a\                                    |   ||       2/a\                                  |  
        ||1 + tan |-|                                    |   ||1 + tan |-|                                  |  
        \\        \2/                                    /   \\        \2/                                  /
(−2({0for im⁡(a)=0∧a mod π=02tan⁡(a2)tan⁡2(a2)+1otherwise))+(10({1for im⁡(a)=0∧a mod 2π=01−tan⁡2(a2)tan⁡2(a2)+1otherwise))+10({0for im⁡(a)=0∧a mod π=02tan⁡(a2)tan⁡2(a2)+1otherwise)−(5({1for im⁡(a)=0∧a mod 2π=01−tan⁡2(a2)tan⁡2(a2)+1otherwise))+5\frac{\left(- 2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 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)\right) + \left(10 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{1 - \tan^{2}{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 10}{\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 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) - \left(5 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{1 - \tan^{2}{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 5}
       //     0       for And(im(a) = 0, a mod pi = 0)\                                                   
       ||                                             |      //  1     for And(im(a) = 0, a mod 2*pi = 0)\
       ||     1                                       |      ||                                          |
10 - 2*|<-----------             otherwise            | + 10*|<  1                                       |
       ||   /    pi\                                  |      ||------              otherwise             |
       ||sec|a - --|                                  |      \\sec(a)                                    /
       \\   \    2 /                                  /                                                   
----------------------------------------------------------------------------------------------------------
                                                        //     0       for And(im(a) = 0, a mod pi = 0)\  
        //  1     for And(im(a) = 0, a mod 2*pi = 0)\   ||                                             |  
        ||                                          |   ||     1                                       |  
  5 - 5*|<  1                                       | + |<-----------             otherwise            |  
        ||------              otherwise             |   ||   /    pi\                                  |  
        \\sec(a)                                    /   ||sec|a - --|                                  |  
                                                        \\   \    2 /                                  /
(−2({0for im⁡(a)=0∧a mod π=01sec⁡(a−π2)otherwise))+(10({1for im⁡(a)=0∧a mod 2π=01sec⁡(a)otherwise))+10({0for im⁡(a)=0∧a mod π=01sec⁡(a−π2)otherwise)−(5({1for im⁡(a)=0∧a mod 2π=01sec⁡(a)otherwise))+5\frac{\left(- 2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(10 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) + 10}{\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) + 5}
                      /    pi\
10 - 2*sin(a) + 10*sin|a + --|
                      \    2 /
------------------------------
           /    pi\           
  5 - 5*sin|a + --| + sin(a)  
           \    2 /
−2sin⁡(a)+10sin⁡(a+π2)+10sin⁡(a)−5sin⁡(a+π2)+5\frac{- 2 \sin{\left(a \right)} + 10 \sin{\left(a + \frac{\pi}{2} \right)} + 10}{\sin{\left(a \right)} - 5 \sin{\left(a + \frac{\pi}{2} \right)} + 5}
          /    pi\            
10 - 2*cos|a - --| + 10*cos(a)
          \    2 /            
------------------------------
                    /    pi\  
  5 - 5*cos(a) + cos|a - --|  
                    \    2 /
10cos⁡(a)−2cos⁡(a−π2)+10−5cos⁡(a)+cos⁡(a−π2)+5\frac{10 \cos{\left(a \right)} - 2 \cos{\left(a - \frac{\pi}{2} \right)} + 10}{- 5 \cos{\left(a \right)} + \cos{\left(a - \frac{\pi}{2} \right)} + 5}
       //                      0                         for And(im(a) = 0, a mod pi = 0)\      //                        1                          for And(im(a) = 0, a mod 2*pi = 0)\
       ||                                                                                |      ||                                                                                     |
       ||/     0       for And(im(a) = 0, a mod pi = 0)                                  |      ||/     1        for And(im(a) = 0, a mod 2*pi = 0)                                    |
       |||                                                                               |      |||                                                                                    |
       |||       /a\                                                                     |      |||        2/a\                                                                        |
10 - 2*|<|  2*cot|-|                                                                     | + 10*|<|-1 + cot |-|                                                                        |
       ||<       \2/                                                otherwise            |      ||<         \2/                                                  otherwise             |
       |||-----------             otherwise                                              |      |||------------              otherwise                                                 |
       |||       2/a\                                                                    |      |||       2/a\                                                                         |
       |||1 + cot |-|                                                                    |      |||1 + cot |-|                                                                         |
       \\\        \2/                                                                    /      \\\        \2/                                                                         /
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
        //                        1                          for And(im(a) = 0, a mod 2*pi = 0)\   //                      0                         for And(im(a) = 0, a mod pi = 0)\  
        ||                                                                                     |   ||                                                                                |  
        ||/     1        for And(im(a) = 0, a mod 2*pi = 0)                                    |   ||/     0       for And(im(a) = 0, a mod pi = 0)                                  |  
        |||                                                                                    |   |||                                                                               |  
        |||        2/a\                                                                        |   |||       /a\                                                                     |  
  5 - 5*|<|-1 + cot |-|                                                                        | + |<|  2*cot|-|                                                                     |  
        ||<         \2/                                                  otherwise             |   ||<       \2/                                                otherwise            |  
        |||------------              otherwise                                                 |   |||-----------             otherwise                                              |  
        |||       2/a\                                                                         |   |||       2/a\                                                                    |  
        |||1 + cot |-|                                                                         |   |||1 + cot |-|                                                                    |  
        \\\        \2/                                                                         /   \\\        \2/                                                                    /
(−2({0for im⁡(a)=0∧a mod π=0{0for im⁡(a)=0∧a mod π=02cot⁡(a2)cot⁡2(a2)+1otherwiseotherwise))+(10({1for im⁡(a)=0∧a mod 2π=0{1for im⁡(a)=0∧a mod 2π=0cot⁡2(a2)−1cot⁡2(a2)+1otherwiseotherwise))+10({0for im⁡(a)=0∧a mod π=0{0for im⁡(a)=0∧a mod π=02cot⁡(a2)cot⁡2(a2)+1otherwiseotherwise)−(5({1for im⁡(a)=0∧a mod 2π=0{1for im⁡(a)=0∧a mod 2π=0cot⁡2(a2)−1cot⁡2(a2)+1otherwiseotherwise))+5\frac{\left(- 2 \left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 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)\right) + \left(10 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + 10}{\left(\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge 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) - \left(5 \left(\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \operatorname{im}{\left(a\right)} = 0 \wedge a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + 5}
Комбинаторика [src]
-2*(5 - sin(a) + 5*cos(a))
--------------------------
  -5 - sin(a) + 5*cos(a)
−2(−sin⁡(a)+5cos⁡(a)+5)−sin⁡(a)+5cos⁡(a)−5- \frac{2 \left(- \sin{\left(a \right)} + 5 \cos{\left(a \right)} + 5\right)}{- \sin{\left(a \right)} + 5 \cos{\left(a \right)} - 5}
Раскрыть выражение [src]
          10                   2*sin(a)               10*cos(a)      
--------------------- - --------------------- + ---------------------
5 - 5*cos(a) + sin(a)   5 - 5*cos(a) + sin(a)   5 - 5*cos(a) + sin(a)
−2sin⁡(a)sin⁡(a)−5cos⁡(a)+5+10cos⁡(a)sin⁡(a)−5cos⁡(a)+5+10sin⁡(a)−5cos⁡(a)+5- \frac{2 \sin{\left(a \right)}}{\sin{\left(a \right)} - 5 \cos{\left(a \right)} + 5} + \frac{10 \cos{\left(a \right)}}{\sin{\left(a \right)} - 5 \cos{\left(a \right)} + 5} + \frac{10}{\sin{\left(a \right)} - 5 \cos{\left(a \right)} + 5}