Sr Examen

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Simplificación de expresiones/ Descomposición en fracciones simples/ tan(((3*pi)/2)-a)+((sin(3*pi-a))/(1-cos(pi+a)))

¿Cómo vas a descomponer esta tan(((3*pi)/2)-a)+((sin(3*pi-a))/(1-cos(pi+a))) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
   /3*pi    \    sin(3*pi - a) 
tan|---- - a| + ---------------
   \ 2      /   1 - cos(pi + a)
tan⁡(−a+3π2)+sin⁡(−a+3π)1−cos⁡(a+π)\tan{\left(- a + \frac{3 \pi}{2} \right)} + \frac{\sin{\left(- a + 3 \pi \right)}}{1 - \cos{\left(a + \pi \right)}} 
tan((3*pi)/2 - a) + sin(3*pi - a)/(1 - cos(pi + a))
Simplificación general [src] 
  1   
------
sin(a)
1sin⁡(a)\frac{1}{\sin{\left(a \right)}} 
1/sin(a)
Combinatoria [src] 
cos(a)*cot(a) + cot(a) + sin(a)
-------------------------------
           1 + cos(a)
sin⁡(a)+cos⁡(a)cot⁡(a)+cot⁡(a)cos⁡(a)+1\frac{\sin{\left(a \right)} + \cos{\left(a \right)} \cot{\left(a \right)} + \cot{\left(a \right)}}{\cos{\left(a \right)} + 1} 
(cos(a)*cot(a) + cot(a) + sin(a))/(1 + cos(a))
Denominador racional [src] 
-cot(a) - sin(a) - cos(a)*cot(a)
--------------------------------
        -1 + cos(pi + a)
−sin⁡(a)−cos⁡(a)cot⁡(a)−cot⁡(a)cos⁡(a+π)−1\frac{- \sin{\left(a \right)} - \cos{\left(a \right)} \cot{\left(a \right)} - \cot{\left(a \right)}}{\cos{\left(a + \pi \right)} - 1} 
(-cot(a) - sin(a) - cos(a)*cot(a))/(-1 + cos(pi + a))
Denominador común [src] 
       sin(a)              
- ---------------- + cot(a)
  -1 + cos(pi + a)
cot⁡(a)−sin⁡(a)cos⁡(a+π)−1\cot{\left(a \right)} - \frac{\sin{\left(a \right)}}{\cos{\left(a + \pi \right)} - 1} 
-sin(a)/(-1 + cos(pi + a)) + cot(a)
Unión de expresiones racionales [src] 
                /-2*a + 3*pi\         
(1 + cos(a))*tan|-----------| + sin(a)
                \     2     /         
--------------------------------------
              1 + cos(a)
(cos⁡(a)+1)tan⁡(−2a+3π2)+sin⁡(a)cos⁡(a)+1\frac{\left(\cos{\left(a \right)} + 1\right) \tan{\left(\frac{- 2 a + 3 \pi}{2} \right)} + \sin{\left(a \right)}}{\cos{\left(a \right)} + 1} 
((1 + cos(a))*tan((-2*a + 3*pi)/2) + sin(a))/(1 + cos(a))
Abrimos la expresión [src] 
  sin(a)           
---------- + cot(a)
1 + cos(a)
cot⁡(a)+sin⁡(a)cos⁡(a)+1\cot{\left(a \right)} + \frac{\sin{\left(a \right)}}{\cos{\left(a \right)} + 1} 
sin(a)/(1 + cos(a)) + cot(a)
Parte trigonométrica [src] 
     1     
-----------
   /    pi\
cos|a - --|
   \    2 /
1cos⁡(a−π2)\frac{1}{\cos{\left(a - \frac{\pi}{2} \right)}} 
csc(a)
csc⁡(a)\csc{\left(a \right)} 
   /    pi\              
cos|a - --|              
   \    2 /      cos(a)  
----------- + -----------
 1 + cos(a)      /    pi\
              cos|a - --|
                 \    2 /
cos⁡(a)cos⁡(a−π2)+cos⁡(a−π2)cos⁡(a)+1\frac{\cos{\left(a \right)}}{\cos{\left(a - \frac{\pi}{2} \right)}} + \frac{\cos{\left(a - \frac{\pi}{2} \right)}}{\cos{\left(a \right)} + 1} 
nan
NaN\text{NaN} 
                                 /    pi\
                              sec|a - --|
             1                   \    2 /
--------------------------- + -----------
/      1   \    /     5*pi\      sec(a)  
|1 + ------|*sec|-a + ----|              
\    sec(a)/    \      2  /
sec⁡(a−π2)sec⁡(a)+1(1+1sec⁡(a))sec⁡(−a+5π2)\frac{\sec{\left(a - \frac{\pi}{2} \right)}}{\sec{\left(a \right)}} + \frac{1}{\left(1 + \frac{1}{\sec{\left(a \right)}}\right) \sec{\left(- a + \frac{5 \pi}{2} \right)}} 
  sin(a)        cos(a)  
---------- + -----------
1 + cos(a)      /    pi\
             cos|a - --|
                \    2 /
cos⁡(a)cos⁡(a−π2)+sin⁡(a)cos⁡(a)+1\frac{\cos{\left(a \right)}}{\cos{\left(a - \frac{\pi}{2} \right)}} + \frac{\sin{\left(a \right)}}{\cos{\left(a \right)} + 1} 
  1   
------
sin(a)
1sin⁡(a)\frac{1}{\sin{\left(a \right)}} 
                          /a\            
                     2*tan|-|            
  1                       \2/            
------ + --------------------------------
tan(a)                 /            2/a\\
                       |    -1 + tan |-||
         /       2/a\\ |             \2/|
         |1 + tan |-||*|1 - ------------|
         \        \2// |           2/a\ |
                       |    1 + tan |-| |
                       \            \2/ /
1tan⁡(a)+2tan⁡(a2)(−tan⁡2(a2)−1tan⁡2(a2)+1+1)(tan⁡2(a2)+1)\frac{1}{\tan{\left(a \right)}} + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\left(- \frac{\tan^{2}{\left(\frac{a}{2} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)} 
                 /a\                    
            2*cot|-|                    
                 \2/                    
------------------------------- + cot(a)
              /           2/a\\         
              |    1 - cot |-||         
/       2/a\\ |            \2/|         
|1 + cot |-||*|1 - -----------|         
\        \2// |           2/a\|         
              |    1 + cot |-||         
              \            \2//
cot⁡(a)+2cot⁡(a2)(−1−cot⁡2(a2)cot⁡2(a2)+1+1)(cot⁡2(a2)+1)\cot{\left(a \right)} + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\left(- \frac{1 - \cot^{2}{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)} 
     sin(a)        sin(2*a)
--------------- + ---------
       /    pi\        2   
1 + sin|a + --|   2*sin (a)
       \    2 /
sin⁡(2a)2sin⁡2(a)+sin⁡(a)sin⁡(a+π2)+1\frac{\sin{\left(2 a \right)}}{2 \sin^{2}{\left(a \right)}} + \frac{\sin{\left(a \right)}}{\sin{\left(a + \frac{\pi}{2} \right)} + 1} 
   /    pi\
sec|a - --|
   \    2 /
sec⁡(a−π2)\sec{\left(a - \frac{\pi}{2} \right)} 
  sin(a)           
---------- + cot(a)
1 + cos(a)
cot⁡(a)+sin⁡(a)cos⁡(a)+1\cot{\left(a \right)} + \frac{\sin{\left(a \right)}}{\cos{\left(a \right)} + 1} 
                  2/    pi\
             2*sin |a + --|
  sin(a)           \    2 /
---------- + --------------
1 + cos(a)      sin(2*a)
2sin⁡2(a+π2)sin⁡(2a)+sin⁡(a)cos⁡(a)+1\frac{2 \sin^{2}{\left(a + \frac{\pi}{2} \right)}}{\sin{\left(2 a \right)}} + \frac{\sin{\left(a \right)}}{\cos{\left(a \right)} + 1} 
                   2                            
--------------------------------------- + cot(a)
              /            1   \                
              |    -1 + -------|                
              |            2/a\|                
              |         cot |-||                
/       1   \ |             \2/|    /a\         
|1 + -------|*|1 - ------------|*cot|-|         
|       2/a\| |           1    |    \2/         
|    cot |-|| |    1 + ------- |                
\        \2// |           2/a\ |                
              |        cot |-| |                
              \            \2/ /
cot⁡(a)+2(1+1cot⁡2(a2))(−−1+1cot⁡2(a2)1+1cot⁡2(a2)+1)cot⁡(a2)\cot{\left(a \right)} + \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \left(- \frac{-1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}} + 1\right) \cot{\left(\frac{a}{2} \right)}} 
                          /a\           
                     2*tan|-|           
  1                       \2/           
------ + -------------------------------
tan(a)                 /           2/a\\
                       |    1 - tan |-||
         /       2/a\\ |            \2/|
         |1 + tan |-||*|1 + -----------|
         \        \2// |           2/a\|
                       |    1 + tan |-||
                       \            \2//
1tan⁡(a)+2tan⁡(a2)(1−tan⁡2(a2)tan⁡2(a2)+1+1)(tan⁡2(a2)+1)\frac{1}{\tan{\left(a \right)}} + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)} 
       2/a\
1 + tan |-|
        \2/
-----------
       /a\ 
  2*tan|-| 
       \2/
tan⁡2(a2)+12tan⁡(a2)\frac{\tan^{2}{\left(\frac{a}{2} \right)} + 1}{2 \tan{\left(\frac{a}{2} \right)}} 
   /     5*pi\              
cos|-a + ----|              
   \      2  /   cos(pi - a)
-------------- - -----------
  1 + cos(a)        sin(a)
−cos⁡(π−a)sin⁡(a)+cos⁡(−a+5π2)cos⁡(a)+1- \frac{\cos{\left(\pi - a \right)}}{\sin{\left(a \right)}} + \frac{\cos{\left(- a + \frac{5 \pi}{2} \right)}}{\cos{\left(a \right)} + 1} 
                              /    pi\
                           sec|a - --|
           1                  \    2 /
------------------------ + -----------
/      1   \    /    pi\      sec(a)  
|1 + ------|*sec|a - --|              
\    sec(a)/    \    2 /
sec⁡(a−π2)sec⁡(a)+1(1+1sec⁡(a))sec⁡(a−π2)\frac{\sec{\left(a - \frac{\pi}{2} \right)}}{\sec{\left(a \right)}} + \frac{1}{\left(1 + \frac{1}{\sec{\left(a \right)}}\right) \sec{\left(a - \frac{\pi}{2} \right)}} 
           1                  csc(a)  
------------------------ + -----------
/         1     \             /pi    \
|1 + -----------|*csc(a)   csc|-- - a|
|       /pi    \|             \2     /
|    csc|-- - a||                     
\       \2     //
csc⁡(a)csc⁡(−a+π2)+1(1+1csc⁡(−a+π2))csc⁡(a)\frac{\csc{\left(a \right)}}{\csc{\left(- a + \frac{\pi}{2} \right)}} + \frac{1}{\left(1 + \frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}}\right) \csc{\left(a \right)}} 
                 /a\                     
            2*cot|-|                     
                 \2/                     
-------------------------------- + cot(a)
              /            2/a\\         
              |    -1 + cot |-||         
/       2/a\\ |             \2/|         
|1 + cot |-||*|1 + ------------|         
\        \2// |           2/a\ |         
              |    1 + cot |-| |         
              \            \2/ /
cot⁡(a)+2cot⁡(a2)(cot⁡2(a2)−1cot⁡2(a2)+1+1)(cot⁡2(a2)+1)\cot{\left(a \right)} + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)} 
                         2   
      sin(a)        2*cos (a)
----------------- + ---------
       /    3*pi\    sin(2*a)
1 - sin|a + ----|            
       \     2  /
2cos⁡2(a)sin⁡(2a)+sin⁡(a)1−sin⁡(a+3π2)\frac{2 \cos^{2}{\left(a \right)}}{\sin{\left(2 a \right)}} + \frac{\sin{\left(a \right)}}{1 - \sin{\left(a + \frac{3 \pi}{2} \right)}} 
  1                        2                   
------ + --------------------------------------
tan(a)                 /           1   \       
                       |    1 - -------|       
                       |           2/a\|       
                       |        tan |-||       
         /       1   \ |            \2/|    /a\
         |1 + -------|*|1 - -----------|*tan|-|
         |       2/a\| |           1   |    \2/
         |    tan |-|| |    1 + -------|       
         \        \2// |           2/a\|       
                       |        tan |-||       
                       \            \2//
1tan⁡(a)+2(1+1tan⁡2(a2))(−1−1tan⁡2(a2)1+1tan⁡2(a2)+1)tan⁡(a2)\frac{1}{\tan{\left(a \right)}} + \frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \left(- \frac{1 - \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}} + 1\right) \tan{\left(\frac{a}{2} \right)}} 
       2/a\
1 + cot |-|
        \2/
-----------
       /a\ 
  2*cot|-| 
       \2/
cot⁡2(a2)+12cot⁡(a2)\frac{\cot^{2}{\left(\frac{a}{2} \right)} + 1}{2 \cot{\left(\frac{a}{2} \right)}} 
         1            csc(a)
------------------- + ------
/      1   \          sec(a)
|1 + ------|*csc(a)         
\    sec(a)/
csc⁡(a)sec⁡(a)+1(1+1sec⁡(a))csc⁡(a)\frac{\csc{\left(a \right)}}{\sec{\left(a \right)}} + \frac{1}{\left(1 + \frac{1}{\sec{\left(a \right)}}\right) \csc{\left(a \right)}} 
1/((1 + 1/sec(a))*csc(a)) + csc(a)/sec(a)
Respuesta numérica [src] 
sin(3*pi - a)/(1.0 - cos(pi + a)) + tan((3*pi)/2 - a)
sin(3*pi - a)/(1.0 - cos(pi + a)) + tan((3*pi)/2 - a)
Potencias [src] 
  sin(a)           
---------- + cot(a)
1 + cos(a)
cot⁡(a)+sin⁡(a)cos⁡(a)+1\cot{\left(a \right)} + \frac{\sin{\left(a \right)}}{\cos{\left(a \right)} + 1} 
  /     /     3*pi\      /    3*pi\\                                       
  |   I*|-a + ----|    I*|a - ----||                                       
  |     \      2  /      \     2  /|     /   I*(a - 3*pi)    I*(-a + 3*pi)\
I*\- e              + e            /   I*\- e             + e             /
------------------------------------ - ------------------------------------
      /    3*pi\      /     3*pi\         /     I*(pi + a)    I*(-pi - a)\ 
    I*|a - ----|    I*|-a + ----|         |    e             e           | 
      \     2  /      \      2  /       2*|1 - ----------- - ------------| 
   e             + e                      \         2             2      /
i(−ei(−a+3π2)+ei(a−3π2))ei(−a+3π2)+ei(a−3π2)−i(ei(−a+3π)−ei(a−3π))2(−ei(−a−π)2−ei(a+π)2+1)\frac{i \left(- e^{i \left(- a + \frac{3 \pi}{2}\right)} + e^{i \left(a - \frac{3 \pi}{2}\right)}\right)}{e^{i \left(- a + \frac{3 \pi}{2}\right)} + e^{i \left(a - \frac{3 \pi}{2}\right)}} - \frac{i \left(e^{i \left(- a + 3 \pi\right)} - e^{i \left(a - 3 \pi\right)}\right)}{2 \left(- \frac{e^{i \left(- a - \pi\right)}}{2} - \frac{e^{i \left(a + \pi\right)}}{2} + 1\right)} 
i*(-exp(i*(-a + 3*pi/2)) + exp(i*(a - 3*pi/2)))/(exp(i*(a - 3*pi/2)) + exp(i*(-a + 3*pi/2))) - i*(-exp(i*(a - 3*pi)) + exp(i*(-a + 3*pi)))/(2*(1 - exp(i*(pi + a))/2 - exp(i*(-pi - a))/2))
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