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Simplificación de expresiones/ Descomposición en fracciones simples/ cos(6*a)/(cos(3*a)+sin(3*a))

¿Cómo vas a descomponer esta cos(6*a)/(cos(3*a)+sin(3*a)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
      cos(6*a)     
-------------------
cos(3*a) + sin(3*a)
cos⁡(6a)sin⁡(3a)+cos⁡(3a)\frac{\cos{\left(6 a \right)}}{\sin{\left(3 a \right)} + \cos{\left(3 a \right)}} 
cos(6*a)/(cos(3*a) + sin(3*a))
Simplificación general [src] 
   ___         
 \/ 2 *cos(6*a)
---------------
     /      pi\
2*sin|3*a + --|
     \      4 /
2cos⁡(6a)2sin⁡(3a+π4)\frac{\sqrt{2} \cos{\left(6 a \right)}}{2 \sin{\left(3 a + \frac{\pi}{4} \right)}} 
sqrt(2)*cos(6*a)/(2*sin(3*a + pi/4))
Potencias [src] 
              -6*I*a    6*I*a            
             e         e                 
             ------- + ------            
                2        2               
-----------------------------------------
 -3*I*a    3*I*a     /   -3*I*a    3*I*a\
e         e        I*\- e       + e     /
------- + ------ - ----------------------
   2        2                2
e6ia2+e−6ia2−i(e3ia−e−3ia)2+e3ia2+e−3ia2\frac{\frac{e^{6 i a}}{2} + \frac{e^{- 6 i a}}{2}}{- \frac{i \left(e^{3 i a} - e^{- 3 i a}\right)}{2} + \frac{e^{3 i a}}{2} + \frac{e^{- 3 i a}}{2}} 
(exp(-6*i*a)/2 + exp(6*i*a)/2)/(exp(-3*i*a)/2 + exp(3*i*a)/2 - i*(-exp(-3*i*a) + exp(3*i*a))/2)
Abrimos la expresión [src] 
                                                                          4                                               2                                               6                    
                        1                                           48*cos (a)                                      18*cos (a)                                      32*cos (a)                 
- --------------------------------------------- - --------------------------------------------- + --------------------------------------------- + ---------------------------------------------
         3                                 3             3                                 3             3                                 3             3                                 3   
  - 4*sin (a) - 3*cos(a) + 3*sin(a) + 4*cos (a)   - 4*sin (a) - 3*cos(a) + 3*sin(a) + 4*cos (a)   - 4*sin (a) - 3*cos(a) + 3*sin(a) + 4*cos (a)   - 4*sin (a) - 3*cos(a) + 3*sin(a) + 4*cos (a)
32cos⁡6(a)−4sin⁡3(a)+3sin⁡(a)+4cos⁡3(a)−3cos⁡(a)−48cos⁡4(a)−4sin⁡3(a)+3sin⁡(a)+4cos⁡3(a)−3cos⁡(a)+18cos⁡2(a)−4sin⁡3(a)+3sin⁡(a)+4cos⁡3(a)−3cos⁡(a)−1−4sin⁡3(a)+3sin⁡(a)+4cos⁡3(a)−3cos⁡(a)\frac{32 \cos^{6}{\left(a \right)}}{- 4 \sin^{3}{\left(a \right)} + 3 \sin{\left(a \right)} + 4 \cos^{3}{\left(a \right)} - 3 \cos{\left(a \right)}} - \frac{48 \cos^{4}{\left(a \right)}}{- 4 \sin^{3}{\left(a \right)} + 3 \sin{\left(a \right)} + 4 \cos^{3}{\left(a \right)} - 3 \cos{\left(a \right)}} + \frac{18 \cos^{2}{\left(a \right)}}{- 4 \sin^{3}{\left(a \right)} + 3 \sin{\left(a \right)} + 4 \cos^{3}{\left(a \right)} - 3 \cos{\left(a \right)}} - \frac{1}{- 4 \sin^{3}{\left(a \right)} + 3 \sin{\left(a \right)} + 4 \cos^{3}{\left(a \right)} - 3 \cos{\left(a \right)}} 
-1/(-4*sin(a)^3 - 3*cos(a) + 3*sin(a) + 4*cos(a)^3) - 48*cos(a)^4/(-4*sin(a)^3 - 3*cos(a) + 3*sin(a) + 4*cos(a)^3) + 18*cos(a)^2/(-4*sin(a)^3 - 3*cos(a) + 3*sin(a) + 4*cos(a)^3) + 32*cos(a)^6/(-4*sin(a)^3 - 3*cos(a) + 3*sin(a) + 4*cos(a)^3)
Respuesta numérica [src] 
cos(6*a)/(cos(3*a) + sin(3*a))
cos(6*a)/(cos(3*a) + sin(3*a))
Parte trigonométrica [src] 
                        2                      
                 1 - tan (3*a)                 
-----------------------------------------------
                /       2/3*a\          /3*a\ \
                |1 - tan |---|     2*tan|---| |
/       2     \ |        \ 2 /          \ 2 / |
\1 + tan (3*a)/*|------------- + -------------|
                |       2/3*a\          2/3*a\|
                |1 + tan |---|   1 + tan |---||
                \        \ 2 /           \ 2 //
1−tan⁡2(3a)(1−tan⁡2(3a2)tan⁡2(3a2)+1+2tan⁡(3a2)tan⁡2(3a2)+1)(tan⁡2(3a)+1)\frac{1 - \tan^{2}{\left(3 a \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{3 a}{2} \right)}}{\tan^{2}{\left(\frac{3 a}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{3 a}{2} \right)}}{\tan^{2}{\left(\frac{3 a}{2} \right)} + 1}\right) \left(\tan^{2}{\left(3 a \right)} + 1\right)} 
  ___             /      pi\
\/ 2 *cos(6*a)*csc|3*a + --|
                  \      4 /
----------------------------
             2
2cos⁡(6a)csc⁡(3a+π4)2\frac{\sqrt{2} \cos{\left(6 a \right)} \csc{\left(3 a + \frac{\pi}{4} \right)}}{2} 
      ___     
    \/ 2      
--------------
   /       pi\
csc|-3*a + --|
   \       4 /
2csc⁡(−3a+π4)\frac{\sqrt{2}}{\csc{\left(- 3 a + \frac{\pi}{4} \right)}} 
                   1                    
----------------------------------------
/   1             1      \    /pi      \
|-------- + -------------|*csc|-- - 6*a|
|csc(3*a)      /pi      \|    \2       /
|           csc|-- - 3*a||              
\              \2       //
1(1csc⁡(−3a+π2)+1csc⁡(3a))csc⁡(−6a+π2)\frac{1}{\left(\frac{1}{\csc{\left(- 3 a + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(3 a \right)}}\right) \csc{\left(- 6 a + \frac{\pi}{2} \right)}} 
  ___    2/pi + 12*a\ /        2/pi   3*a\\
\/ 2 *sin |---------|*|-1 + cot |-- + ---||
          \    8    / \         \8     2 //
2(cot⁡2(3a2+π8)−1)sin⁡2(12a+π8)\sqrt{2} \left(\cot^{2}{\left(\frac{3 a}{2} + \frac{\pi}{8} \right)} - 1\right) \sin^{2}{\left(\frac{12 a + \pi}{8} \right)} 
  ___    2      /       2/pi   3*a\\ /       2     \
\/ 2 *cos (3*a)*|1 + tan |-- + ---||*\1 - tan (3*a)/
                \        \8     2 //                
----------------------------------------------------
                       /pi   3*a\                   
                  4*tan|-- + ---|                   
                       \8     2 /
2(1−tan⁡2(3a))(tan⁡2(3a2+π8)+1)cos⁡2(3a)4tan⁡(3a2+π8)\frac{\sqrt{2} \left(1 - \tan^{2}{\left(3 a \right)}\right) \left(\tan^{2}{\left(\frac{3 a}{2} + \frac{\pi}{8} \right)} + 1\right) \cos^{2}{\left(3 a \right)}}{4 \tan{\left(\frac{3 a}{2} + \frac{\pi}{8} \right)}} 
  ___    2      /        2     \    /pi   3*a\
\/ 2 *sin (3*a)*\-1 + cot (3*a)/*tan|-- + ---|
                                    \8     2 /
----------------------------------------------
                   2/pi + 12*a\               
              4*sin |---------|               
                    \    8    /
2(cot⁡2(3a)−1)sin⁡2(3a)tan⁡(3a2+π8)4sin⁡2(12a+π8)\frac{\sqrt{2} \left(\cot^{2}{\left(3 a \right)} - 1\right) \sin^{2}{\left(3 a \right)} \tan{\left(\frac{3 a}{2} + \frac{\pi}{8} \right)}}{4 \sin^{2}{\left(\frac{12 a + \pi}{8} \right)}} 
  ___    /      pi\
\/ 2 *sec|3*a - --|
         \      4 /
-------------------
     2*sec(6*a)
2sec⁡(3a−π4)2sec⁡(6a)\frac{\sqrt{2} \sec{\left(3 a - \frac{\pi}{4} \right)}}{2 \sec{\left(6 a \right)}} 
        /pi      \      
     sin|-- + 6*a|      
        \2       /      
------------------------
              /pi      \
sin(3*a) + sin|-- + 3*a|
              \2       /
sin⁡(6a+π2)sin⁡(3a)+sin⁡(3a+π2)\frac{\sin{\left(6 a + \frac{\pi}{2} \right)}}{\sin{\left(3 a \right)} + \sin{\left(3 a + \frac{\pi}{2} \right)}} 
   ___         
 \/ 2 *cos(6*a)
---------------
     /      pi\
2*sin|3*a + --|
     \      4 /
2cos⁡(6a)2sin⁡(3a+π4)\frac{\sqrt{2} \cos{\left(6 a \right)}}{2 \sin{\left(3 a + \frac{\pi}{4} \right)}} 
        cos(6*a)        
------------------------
              /      pi\
cos(3*a) + cos|3*a - --|
              \      2 /
cos⁡(6a)cos⁡(3a)+cos⁡(3a−π2)\frac{\cos{\left(6 a \right)}}{\cos{\left(3 a \right)} + \cos{\left(3 a - \frac{\pi}{2} \right)}} 
  ___    /      pi\
\/ 2 *csc|3*a + --|
         \      4 /
-------------------
       /pi      \  
  2*csc|-- - 6*a|  
       \2       /
2csc⁡(3a+π4)2csc⁡(−6a+π2)\frac{\sqrt{2} \csc{\left(3 a + \frac{\pi}{4} \right)}}{2 \csc{\left(- 6 a + \frac{\pi}{2} \right)}} 
   ___         
 \/ 2 *cos(6*a)
---------------
     /      pi\
2*cos|3*a - --|
     \      4 /
2cos⁡(6a)2cos⁡(3a−π4)\frac{\sqrt{2} \cos{\left(6 a \right)}}{2 \cos{\left(3 a - \frac{\pi}{4} \right)}} 
  ___ /       2/pi   3*a\\
\/ 2 *|1 - tan |-- + ---||
      \        \8     2 //
--------------------------
           2/pi   3*a\    
    1 + tan |-- + ---|    
            \8     2 /
2(1−tan⁡2(3a2+π8))tan⁡2(3a2+π8)+1\frac{\sqrt{2} \left(1 - \tan^{2}{\left(\frac{3 a}{2} + \frac{\pi}{8} \right)}\right)}{\tan^{2}{\left(\frac{3 a}{2} + \frac{\pi}{8} \right)} + 1} 
  ___    /      3*pi\
\/ 2 *sin|3*a + ----|
         \       4  /
2sin⁡(3a+3π4)\sqrt{2} \sin{\left(3 a + \frac{3 \pi}{4} \right)} 
  ___             /      pi\
\/ 2 *cos(6*a)*sec|3*a - --|
                  \      4 /
----------------------------
             2
2cos⁡(6a)sec⁡(3a−π4)2\frac{\sqrt{2} \cos{\left(6 a \right)} \sec{\left(3 a - \frac{\pi}{4} \right)}}{2} 
              1               
------------------------------
/   1          1    \         
|-------- + --------|*sec(6*a)
\csc(3*a)   sec(3*a)/
1(1sec⁡(3a)+1csc⁡(3a))sec⁡(6a)\frac{1}{\left(\frac{1}{\sec{\left(3 a \right)}} + \frac{1}{\csc{\left(3 a \right)}}\right) \sec{\left(6 a \right)}} 
  ___ /       2/pi   3*a\\ /        2     \
\/ 2 *|1 + cot |-- + ---||*\-1 + cot (3*a)/
      \        \8     2 //                 
-------------------------------------------
        /       2     \    /pi   3*a\      
      4*\1 + cot (3*a)/*cot|-- + ---|      
                           \8     2 /
2(cot⁡2(3a)−1)(cot⁡2(3a2+π8)+1)4(cot⁡2(3a)+1)cot⁡(3a2+π8)\frac{\sqrt{2} \left(\cot^{2}{\left(3 a \right)} - 1\right) \left(\cot^{2}{\left(\frac{3 a}{2} + \frac{\pi}{8} \right)} + 1\right)}{4 \left(\cot^{2}{\left(3 a \right)} + 1\right) \cot{\left(\frac{3 a}{2} + \frac{\pi}{8} \right)}} 
  ___    /      pi\
\/ 2 *cos|3*a + --|
         \      4 /
2cos⁡(3a+π4)\sqrt{2} \cos{\left(3 a + \frac{\pi}{4} \right)} 
                         2                      
                 -1 + cot (3*a)                 
------------------------------------------------
                /        2/3*a\          /3*a\ \
                |-1 + cot |---|     2*cot|---| |
/       2     \ |         \ 2 /          \ 2 / |
\1 + cot (3*a)/*|-------------- + -------------|
                |       2/3*a\           2/3*a\|
                |1 + cot |---|    1 + cot |---||
                \        \ 2 /            \ 2 //
cot⁡2(3a)−1(cot⁡2(3a2)−1cot⁡2(3a2)+1+2cot⁡(3a2)cot⁡2(3a2)+1)(cot⁡2(3a)+1)\frac{\cot^{2}{\left(3 a \right)} - 1}{\left(\frac{\cot^{2}{\left(\frac{3 a}{2} \right)} - 1}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} + \frac{2 \cot{\left(\frac{3 a}{2} \right)}}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1}\right) \left(\cot^{2}{\left(3 a \right)} + 1\right)} 
  ___    /pi      \
\/ 2 *sin|-- + 6*a|
         \2       /
-------------------
       /      pi\  
  2*sin|3*a + --|  
       \      4 /
2sin⁡(6a+π2)2sin⁡(3a+π4)\frac{\sqrt{2} \sin{\left(6 a + \frac{\pi}{2} \right)}}{2 \sin{\left(3 a + \frac{\pi}{4} \right)}} 
      ___    
    \/ 2     
-------------
   /      pi\
sec|3*a + --|
   \      4 /
2sec⁡(3a+π4)\frac{\sqrt{2}}{\sec{\left(3 a + \frac{\pi}{4} \right)}} 
  ___ /       2/pi   3*a\\ /       2     \
\/ 2 *|1 + tan |-- + ---||*\1 - tan (3*a)/
      \        \8     2 //                
------------------------------------------
       /       2     \    /pi   3*a\      
     4*\1 + tan (3*a)/*tan|-- + ---|      
                          \8     2 /
2(1−tan⁡2(3a))(tan⁡2(3a2+π8)+1)4(tan⁡2(3a)+1)tan⁡(3a2+π8)\frac{\sqrt{2} \left(1 - \tan^{2}{\left(3 a \right)}\right) \left(\tan^{2}{\left(\frac{3 a}{2} + \frac{\pi}{8} \right)} + 1\right)}{4 \left(\tan^{2}{\left(3 a \right)} + 1\right) \tan{\left(\frac{3 a}{2} + \frac{\pi}{8} \right)}} 
                 1                 
-----------------------------------
/   1             1      \         
|-------- + -------------|*sec(6*a)
|sec(3*a)      /      pi\|         
|           sec|3*a - --||         
\              \      2 //
1(1sec⁡(3a−π2)+1sec⁡(3a))sec⁡(6a)\frac{1}{\left(\frac{1}{\sec{\left(3 a - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(3 a \right)}}\right) \sec{\left(6 a \right)}} 
  ___ /        2/pi   3*a\\
\/ 2 *|-1 + cot |-- + ---||
      \         \8     2 //
---------------------------
            2/pi   3*a\    
     1 + cot |-- + ---|    
             \8     2 /
2(cot⁡2(3a2+π8)−1)cot⁡2(3a2+π8)+1\frac{\sqrt{2} \left(\cot^{2}{\left(\frac{3 a}{2} + \frac{\pi}{8} \right)} - 1\right)}{\cot^{2}{\left(\frac{3 a}{2} + \frac{\pi}{8} \right)} + 1} 
sqrt(2)*(-1 + cot(pi/8 + 3*a/2)^2)/(1 + cot(pi/8 + 3*a/2)^2)
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