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Simplificación de expresiones/ Descomposición en fracciones simples/ sin(3*pi/2+a)/(3*cos(pi-a))еслиa=2*pi/3

¿Cómo vas a descomponer esta sin(3*pi/2+a)/(3*cos(pi-a))еслиa=2*pi/3 expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
   /3*pi    \
sin|---- + a|
   \ 2      /
-------------
3*cos(pi - a)
sin⁡(a+3π2)3cos⁡(π−a)\frac{\sin{\left(a + \frac{3 \pi}{2} \right)}}{3 \cos{\left(\pi - a \right)}} 
sin((3*pi)/2 + a)/((3*cos(pi - a)))
Simplificación general [src] 
1/3
13\frac{1}{3} 
1/3
Sustitución de una condición [src] 
sin((3*pi)/2 + a)/((3*cos(pi - a))) con a = (2*pi)/3
sin((3*pi)/2 + a)/((3*cos(pi - a)))
sin⁡(a+3π2)3cos⁡(π−a)\frac{\sin{\left(a + \frac{3 \pi}{2} \right)}}{3 \cos{\left(\pi - a \right)}} 
sin((3*pi)/2 + ((2*pi)/3^1))/((3*cos(pi - ((2*pi)/3^1))))
sin⁡(((2∗pi)/31)+3π2)3cos⁡(π−((2∗pi)/31))\frac{\sin{\left(((2*pi)/3^{1)} + \frac{3 \pi}{2} \right)}}{3 \cos{\left(\pi - ((2*pi)/3^{1)} \right)}} 
sin((3*pi)/2 + (2*pi)/3)/((3*cos(pi - 2*pi/3)))
sin⁡(2π3+3π2)3cos⁡(π−2π3)\frac{\sin{\left(\frac{2 \pi}{3} + \frac{3 \pi}{2} \right)}}{3 \cos{\left(\pi - \frac{2 \pi}{3} \right)}} 
1/3
13\frac{1}{3} 
1/3
Respuesta numérica [src] 
0.333333333333333*sin((3*pi)/2 + a)/cos(pi - a)
0.333333333333333*sin((3*pi)/2 + a)/cos(pi - a)
Combinatoria [src] 
1/3
13\frac{1}{3} 
1/3
Denominador racional [src] 
1/3
13\frac{1}{3} 
1/3
Potencias [src] 
1/3
13\frac{1}{3} 
   /     /     3*pi\      /    3*pi\\ 
   |   I*|-a - ----|    I*|a + ----|| 
   |     \      2  /      \     2  /| 
-I*\- e              + e            / 
--------------------------------------
    /   I*(pi - a)      I*(a - pi)\   
    |3*e             3*e          |   
  2*|------------- + -------------|   
    \      2               2      /
−i(−ei(−a−3π2)+ei(a+3π2))2(3ei(π−a)2+3ei(a−π)2)- \frac{i \left(- e^{i \left(- a - \frac{3 \pi}{2}\right)} + e^{i \left(a + \frac{3 \pi}{2}\right)}\right)}{2 \left(\frac{3 e^{i \left(\pi - a\right)}}{2} + \frac{3 e^{i \left(a - \pi\right)}}{2}\right)} 
-i*(-exp(i*(-a - 3*pi/2)) + exp(i*(a + 3*pi/2)))/(2*(3*exp(i*(pi - a))/2 + 3*exp(i*(a - pi))/2))
Denominador común [src] 
   -cos(a)   
-------------
3*cos(pi - a)
−cos⁡(a)3cos⁡(π−a)- \frac{\cos{\left(a \right)}}{3 \cos{\left(\pi - a \right)}} 
-cos(a)/(3*cos(pi - a))
Unión de expresiones racionales [src] 
    /2*a + 3*pi\ 
-sin|----------| 
    \    2     / 
-----------------
     3*cos(a)
−sin⁡(2a+3π2)3cos⁡(a)- \frac{\sin{\left(\frac{2 a + 3 \pi}{2} \right)}}{3 \cos{\left(a \right)}} 
-sin((2*a + 3*pi)/2)/(3*cos(a))
Abrimos la expresión [src] 
1/3
13\frac{1}{3} 
1/3
Parte trigonométrica [src] 
   -sec(a)   
-------------
3*sec(pi + a)
−sec⁡(a)3sec⁡(a+π)- \frac{\sec{\left(a \right)}}{3 \sec{\left(a + \pi \right)}} 
                /           /a\ \             
              2 |      2*cot|-| |             
 /       2/a\\  |           \2/ |    /a   pi\ 
-|1 + cot |-|| *|1 + -----------|*cot|- + --| 
 \        \4//  |           2/a\|    \2   4 / 
                |    1 + cot |-||             
                \            \2//             
----------------------------------------------
              /       2/a\\    2/a\           
           12*|1 - cot |-||*cot |-|           
              \        \2//     \4/
−(1+2cot⁡(a2)cot⁡2(a2)+1)(cot⁡2(a4)+1)2cot⁡(a2+π4)12(1−cot⁡2(a2))cot⁡2(a4)- \frac{\left(1 + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2} \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{12 \left(1 - \cot^{2}{\left(\frac{a}{2} \right)}\right) \cot^{2}{\left(\frac{a}{4} \right)}} 
    2/a\ /      1   \    /a   pi\ 
-csc |-|*|1 + ------|*csc|- + --| 
     \2/ \    csc(a)/    \2   4 / 
----------------------------------
  /         2/a\   \              
  |      csc |-|   |              
  |          \2/   |    /  a   pi\
3*|1 - ------------|*csc|- - + --|
  |       2/pi   a\|    \  2   4 /
  |    csc |-- - -||              
  \        \2    2//
−(1+1csc⁡(a))csc⁡2(a2)csc⁡(a2+π4)3(−csc⁡2(a2)csc⁡2(−a2+π2)+1)csc⁡(−a2+π4)- \frac{\left(1 + \frac{1}{\csc{\left(a \right)}}\right) \csc^{2}{\left(\frac{a}{2} \right)} \csc{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{3 \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}{4} \right)}} 
   /pi    \
csc|-- - a|
   \2     /
-----------
  3*sec(a)
csc⁡(−a+π2)3sec⁡(a)\frac{\csc{\left(- a + \frac{\pi}{2} \right)}}{3 \sec{\left(a \right)}} 
    sec(a)   
-------------
     /pi    \
3*csc|-- - a|
     \2     /
sec⁡(a)3csc⁡(−a+π2)\frac{\sec{\left(a \right)}}{3 \csc{\left(- a + \frac{\pi}{2} \right)}} 
       /       2/a\\    /a   pi\   
    -2*|1 + tan |-||*tan|- + --|   
       \        \2//    \2   4 /   
-----------------------------------
  /       2/a   pi\\ /        2/a\\
3*|1 + tan |- + --||*|-1 + tan |-||
  \        \2   4 // \         \2//
−2(tan⁡2(a2)+1)tan⁡(a2+π4)3(tan⁡2(a2)−1)(tan⁡2(a2+π4)+1)- \frac{2 \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{3 \left(\tan^{2}{\left(\frac{a}{2} \right)} - 1\right) \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)} 
-cos(pi + a) 
-------------
   3*cos(a)
−cos⁡(a+π)3cos⁡(a)- \frac{\cos{\left(a + \pi \right)}}{3 \cos{\left(a \right)}} 
    2/a\                 /a   pi\ 
-tan |-|*(1 + sin(a))*cot|- + --| 
     \4/                 \2   4 / 
----------------------------------
        /       2/a\\    4/a\     
     12*|1 - cot |-||*sin |-|     
        \        \2//     \4/
−(sin⁡(a)+1)tan⁡2(a4)cot⁡(a2+π4)12(1−cot⁡2(a2))sin⁡4(a4)- \frac{\left(\sin{\left(a \right)} + 1\right) \tan^{2}{\left(\frac{a}{4} \right)} \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{12 \left(1 - \cot^{2}{\left(\frac{a}{2} \right)}\right) \sin^{4}{\left(\frac{a}{4} \right)}} 
    2/a\ /      1   \                 /a   pi\    /a   pi\ 
-csc |-|*|1 - ------|*(1 + sin(a))*cos|- + --|*csc|- + --| 
     \2/ \    cos(a)/                 \2   4 /    \2   4 / 
-----------------------------------------------------------
                             6
−(1−1cos⁡(a))(sin⁡(a)+1)cos⁡(a2+π4)csc⁡2(a2)csc⁡(a2+π4)6- \frac{\left(1 - \frac{1}{\cos{\left(a \right)}}\right) \left(\sin{\left(a \right)} + 1\right) \cos{\left(\frac{a}{2} + \frac{\pi}{4} \right)} \csc^{2}{\left(\frac{a}{2} \right)} \csc{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{6} 
                  /       1   \               
               -2*|1 + -------|               
                  |       2/a\|               
                  |    tan |-||               
                  \        \2//               
----------------------------------------------
  /         1      \ /       1   \    /a   pi\
3*|1 + ------------|*|1 - -------|*tan|- + --|
  |       2/a   pi\| |       2/a\|    \2   4 /
  |    tan |- + --|| |    tan |-||            
  \        \2   4 // \        \2//
−2(1+1tan⁡2(a2))3(1−1tan⁡2(a2))(1+1tan⁡2(a2+π4))tan⁡(a2+π4)- \frac{2 \left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)}{3 \left(1 - \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}\right) \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} 
        /       /    pi\\    /a   pi\        
       -|1 + cos|a - --||*cos|- + --|        
        \       \    2 //    \2   4 /        
---------------------------------------------
  /         2/a\   \                         
  |      cos |-|   |                         
  |          \2/   |    2/a   pi\    /a   pi\
3*|1 - ------------|*cos |- - --|*cos|- - --|
  |       2/a   pi\|     \2   2 /    \2   4 /
  |    cos |- - --||                         
  \        \2   2 //
−(cos⁡(a−π2)+1)cos⁡(a2+π4)3(−cos⁡2(a2)cos⁡2(a2−π2)+1)cos⁡2(a2−π2)cos⁡(a2−π4)- \frac{\left(\cos{\left(a - \frac{\pi}{2} \right)} + 1\right) \cos{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{3 \left(- \frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)} \cos{\left(\frac{a}{2} - \frac{\pi}{4} \right)}} 
        -(1 + sin(a))*cos(a)          
--------------------------------------
  /        2    \                     
  |     sin (a) |    2/a\    2/a   pi\
6*|1 - ---------|*sin |-|*sin |- + --|
  |         4/a\|     \2/     \2   4 /
  |    4*sin |-||                     
  \          \2//
−(sin⁡(a)+1)cos⁡(a)6(1−sin⁡2(a)4sin⁡4(a2))sin⁡2(a2)sin⁡2(a2+π4)- \frac{\left(\sin{\left(a \right)} + 1\right) \cos{\left(a \right)}}{6 \left(1 - \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)} \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} 
    2/a   pi\ /         1     \    /a   pi\ 
-sec |- - --|*|1 + -----------|*sec|- - --| 
     \2   2 / |       /    pi\|    \2   4 / 
              |    sec|a - --||             
              \       \    2 //             
--------------------------------------------
        /       2/a   pi\\                  
        |    sec |- - --||                  
        |        \2   2 /|    /a   pi\      
      3*|1 - ------------|*sec|- + --|      
        |         2/a\   |    \2   4 /      
        |      sec |-|   |                  
        \          \2/   /
−(1+1sec⁡(a−π2))sec⁡2(a2−π2)sec⁡(a2−π4)3(1−sec⁡2(a2−π2)sec⁡2(a2))sec⁡(a2+π4)- \frac{\left(1 + \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}\right) \sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)} \sec{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}{3 \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} + \frac{\pi}{4} \right)}} 
                   /       1   \               
                -2*|1 + -------|               
                   |       2/a\|               
                   |    cot |-||               
                   \        \2//               
-----------------------------------------------
  /         1      \ /        1   \    /a   pi\
3*|1 + ------------|*|-1 + -------|*cot|- + --|
  |       2/a   pi\| |        2/a\|    \2   4 /
  |    cot |- + --|| |     cot |-||            
  \        \2   4 // \         \2//
−2(1+1cot⁡2(a2))3(−1+1cot⁡2(a2))(1+1cot⁡2(a2+π4))cot⁡(a2+π4)- \frac{2 \left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)}{3 \left(-1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}\right) \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} 
      /       2/a\\    /a   pi\   
   -2*|1 + cot |-||*cot|- + --|   
      \        \2//    \2   4 /   
----------------------------------
  /       2/a   pi\\ /       2/a\\
3*|1 + cot |- + --||*|1 - cot |-||
  \        \2   4 // \        \2//
−2(cot⁡2(a2)+1)cot⁡(a2+π4)3(1−cot⁡2(a2))(cot⁡2(a2+π4)+1)- \frac{2 \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right) \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{3 \left(1 - \cot^{2}{\left(\frac{a}{2} \right)}\right) \left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)} 
          -2*cos(a)           
------------------------------
               /        2    \
               |     sin (a) |
3*(1 - cos(a))*|1 - ---------|
               |         4/a\|
               |    4*sin |-||
               \          \2//
−2cos⁡(a)3(1−sin⁡2(a)4sin⁡4(a2))(1−cos⁡(a))- \frac{2 \cos{\left(a \right)}}{3 \left(1 - \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \left(1 - \cos{\left(a \right)}\right)} 
   /    pi\
sin|a + --|
   \    2 /
-----------
  3*cos(a)
sin⁡(a+π2)3cos⁡(a)\frac{\sin{\left(a + \frac{\pi}{2} \right)}}{3 \cos{\left(a \right)}} 
                 /           /a\ \  
               2 |      2*tan|-| |  
  /       2/a\\  |           \2/ |  
 -|1 + tan |-|| *|1 + -----------|  
  \        \4//  |           2/a\|  
                 |    1 + tan |-||  
                 \            \2//  
------------------------------------
   /       1   \    2/a\    /a   pi\
12*|1 - -------|*tan |-|*tan|- + --|
   |       2/a\|     \4/    \2   4 /
   |    tan |-||                    
   \        \2//
−(1+2tan⁡(a2)tan⁡2(a2)+1)(tan⁡2(a4)+1)212(1−1tan⁡2(a2))tan⁡2(a4)tan⁡(a2+π4)- \frac{\left(1 + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}}{12 \left(1 - \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \tan^{2}{\left(\frac{a}{4} \right)} \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} 
1/3
13\frac{1}{3} 
                 /a   pi\ 
-(1 + sin(a))*cot|- + --| 
                 \2   4 / 
--------------------------
   /       2/a\\    2/a\  
 3*|1 - cot |-||*sin |-|  
   \        \2//     \2/
−(sin⁡(a)+1)cot⁡(a2+π4)3(1−cot⁡2(a2))sin⁡2(a2)- \frac{\left(\sin{\left(a \right)} + 1\right) \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{3 \left(1 - \cot^{2}{\left(\frac{a}{2} \right)}\right) \sin^{2}{\left(\frac{a}{2} \right)}} 
              /      1   \              
             -|1 - ------|              
              \    cos(a)/              
----------------------------------------
  /         1      \    2/a\    /a   pi\
3*|1 + ------------|*sin |-|*tan|- + --|
  |       2/a   pi\|     \2/    \2   4 /
  |    tan |- + --||                    
  \        \2   4 //
−1−1cos⁡(a)3(1+1tan⁡2(a2+π4))sin⁡2(a2)tan⁡(a2+π4)- \frac{1 - \frac{1}{\cos{\left(a \right)}}}{3 \left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)} \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} 
/       /a\\ /  1            \
|1 - tan|-||*|------ + tan(a)|
\       \2// \cos(a)         /
------------------------------
          /       /a\\        
        3*|1 + tan|-||        
          \       \2//
(1−tan⁡(a2))(tan⁡(a)+1cos⁡(a))3(tan⁡(a2)+1)\frac{\left(1 - \tan{\left(\frac{a}{2} \right)}\right) \left(\tan{\left(a \right)} + \frac{1}{\cos{\left(a \right)}}\right)}{3 \left(\tan{\left(\frac{a}{2} \right)} + 1\right)} 
    -cos(a)     
----------------
     /     3*pi\
3*sin|-a + ----|
     \      2  /
−cos⁡(a)3sin⁡(−a+3π2)- \frac{\cos{\left(a \right)}}{3 \sin{\left(- a + \frac{3 \pi}{2} \right)}} 
  /      1   \                
 -|1 - ------|*(1 + sin(a))   
  \    cos(a)/                
------------------------------
      4/a\    2/a\    /a   pi\
24*cos |-|*tan |-|*tan|- + --|
       \4/     \4/    \2   4 /
−(1−1cos⁡(a))(sin⁡(a)+1)24cos⁡4(a4)tan⁡2(a4)tan⁡(a2+π4)- \frac{\left(1 - \frac{1}{\cos{\left(a \right)}}\right) \left(\sin{\left(a \right)} + 1\right)}{24 \cos^{4}{\left(\frac{a}{4} \right)} \tan^{2}{\left(\frac{a}{4} \right)} \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} 
-(1 - 1/cos(a))*(1 + sin(a))/(24*cos(a/4)^4*tan(a/4)^2*tan(a/2 + pi/4))
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