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Simplificación de expresiones/ Descomposición en fracciones simples/ cos((7pi/24)+pi/8)/(√6*sin(pi/10+(3pi)/20))

¿Cómo vas a descomponer esta cos((7pi/24)+pi/8)/(√6*sin(pi/10+(3pi)/20)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
      /7*pi   pi\   
   cos|---- + --|   
      \ 24    8 /   
--------------------
  ___    /pi   3*pi\
\/ 6 *sin|-- + ----|
         \10    20 /
cos⁡(π8+7π24)6sin⁡(π10+3π20)\frac{\cos{\left(\frac{\pi}{8} + \frac{7 \pi}{24} \right)}}{\sqrt{6} \sin{\left(\frac{\pi}{10} + \frac{3 \pi}{20} \right)}} 
cos((7*pi)/24 + pi/8)/((sqrt(6)*sin(pi/10 + (3*pi)/20)))
Simplificación general [src] 
    ___     ___
  \/ 6    \/ 2 
- ----- + -----
    12      4
−612+24- \frac{\sqrt{6}}{12} + \frac{\sqrt{2}}{4} 
-sqrt(6)/12 + sqrt(2)/4
Respuesta numérica [src] 
0.149429245361342
0.149429245361342
Denominador racional [src] 
  ___    /7*pi   pi\
\/ 6 *cos|---- + --|
         \ 24    8 /
--------------------
       /3*pi   pi\  
  6*sin|---- + --|  
       \ 20    10/
6cos⁡(π8+7π24)6sin⁡(π10+3π20)\frac{\sqrt{6} \cos{\left(\frac{\pi}{8} + \frac{7 \pi}{24} \right)}}{6 \sin{\left(\frac{\pi}{10} + \frac{3 \pi}{20} \right)}} 
sqrt(6)*cos(7*pi/24 + pi/8)/(6*sin(3*pi/20 + pi/10))
Unión de expresiones racionales [src] 
      /    ___     ___\
  ___ |  \/ 2    \/ 6 |
\/ 3 *|- ----- + -----|
      \    4       4  /
-----------------------
           3
3(−24+64)3\frac{\sqrt{3} \left(- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)}{3} 
sqrt(3)*(-sqrt(2)/4 + sqrt(6)/4)/3
Denominador común [src] 
    ___     ___
  \/ 6    \/ 2 
- ----- + -----
    12      4
−612+24- \frac{\sqrt{6}}{12} + \frac{\sqrt{2}}{4} 
-sqrt(6)/12 + sqrt(2)/4
Combinatoria [src] 
   ___ /  ___     ___\ 
-\/ 3 *\\/ 2  - \/ 6 / 
-----------------------
           12
−3(−6+2)12- \frac{\sqrt{3} \left(- \sqrt{6} + \sqrt{2}\right)}{12} 
-sqrt(3)*(sqrt(2) - sqrt(6))/12
Potencias [src] 
      /    ___     ___\
  ___ |  \/ 2    \/ 6 |
\/ 3 *|- ----- + -----|
      \    4       4  /
-----------------------
           3
3(−24+64)3\frac{\sqrt{3} \left(- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)}{3} 
      /    ___     ___\
  ___ |  \/ 2    \/ 6 |
\/ 3 *|- ----- + -----|
      \    12      12 /
3(−212+612)\sqrt{3} \left(- \frac{\sqrt{2}}{12} + \frac{\sqrt{6}}{12}\right) 
        / -5*pi*I    5*pi*I\
        | -------    ------|
        |    12        12  |
    ___ |e          e      |
I*\/ 6 *|-------- + -------|
        \   2          2   /
----------------------------
     /   -pi*I     pi*I\    
     |   ------    ----|    
     |     4        4  |    
   3*\- e       + e    /
6i(e−5iπ122+e5iπ122)3(−e−iπ4+eiπ4)\frac{\sqrt{6} i \left(\frac{e^{- \frac{5 i \pi}{12}}}{2} + \frac{e^{\frac{5 i \pi}{12}}}{2}\right)}{3 \left(- e^{- \frac{i \pi}{4}} + e^{\frac{i \pi}{4}}\right)} 
i*sqrt(6)*(exp(-5*pi*i/12)/2 + exp(5*pi*i/12)/2)/(3*(-exp(-pi*i/4) + exp(pi*i/4)))
Parte trigonométrica [src] 
                         /                                      2\
                         |     /                    ___     ___\ |
                         |     |                  \/ 2    \/ 6 | |
      /               2\ |     |                - ----- + -----| |
  ___ |    /      ___\ | |     |      1             4       4  | |
\/ 6 *\1 + \1 + \/ 2 / /*|-1 + |------------- + ---------------| |
                         |     |  ___     ___      ___     ___ | |
                         |     |\/ 2    \/ 6     \/ 2    \/ 6  | |
                         |     |----- + -----    ----- + ----- | |
                         \     \  4       4        4       4   / /
------------------------------------------------------------------
       /                                     2\                   
       |    /                    ___     ___\ |                   
       |    |                  \/ 2    \/ 6 | |                   
       |    |                - ----- + -----| |                   
       |    |      1             4       4  | | /        ___\     
     6*|1 + |------------- + ---------------| |*\2 + 2*\/ 2 /     
       |    |  ___     ___      ___     ___ | |                   
       |    |\/ 2    \/ 6     \/ 2    \/ 6  | |                   
       |    |----- + -----    ----- + ----- | |                   
       \    \  4       4        4       4   / /
6(−1+(−24+6424+64+124+64)2)(1+(1+2)2)6(1+(−24+6424+64+124+64)2)(2+22)\frac{\sqrt{6} \left(-1 + \left(\frac{- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}} + \frac{1}{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}\right)^{2}\right) \left(1 + \left(1 + \sqrt{2}\right)^{2}\right)}{6 \left(1 + \left(\frac{- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}} + \frac{1}{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}\right)^{2}\right) \left(2 + 2 \sqrt{2}\right)} 
      /    ___     ___\          
  ___ |  \/ 2    \/ 6 |    /-pi \
\/ 6 *|- ----- + -----|*sec|----|
      \    4       4  /    \ 4  /
---------------------------------
                6
6(−24+64)sec⁡(−π4)6\frac{\sqrt{6} \left(- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right) \sec{\left(- \frac{\pi}{4} \right)}}{6} 
      /    ___     ___\
  ___ |  \/ 2    \/ 6 |
\/ 3 *|- ----- + -----|
      \    4       4  /
-----------------------
           3
3(−24+64)3\frac{\sqrt{3} \left(- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)}{3} 
  ___    /11*pi\
\/ 3 *sin|-----|
         \  12 /
----------------
       3
3sin⁡(11π12)3\frac{\sqrt{3} \sin{\left(\frac{11 \pi}{12} \right)}}{3} 
      /    ___     ___\
  ___ |  \/ 2    \/ 6 |
\/ 6 *|- ----- + -----|
      \    4       4  /
-----------------------
           /-pi \      
      6*cos|----|      
           \ 4  /
6(−24+64)6cos⁡(−π4)\frac{\sqrt{6} \left(- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\right)}{6 \cos{\left(- \frac{\pi}{4} \right)}} 
  ___ /  ___     ___\
\/ 3 *\\/ 6  - \/ 2 /
---------------------
          12
3(−2+6)12\frac{\sqrt{3} \left(- \sqrt{2} + \sqrt{6}\right)}{12} 
                          /                                     2\
                          |    /                    ___     ___\ |
                          |    |                  \/ 2    \/ 6 | |
      /                2\ |    |                - ----- + -----| |
  ___ |    /       ___\ | |    |      1             4       4  | |
\/ 6 *\1 + \-1 + \/ 2 / /*|1 - |------------- - ---------------| |
                          |    |  ___     ___      ___     ___ | |
                          |    |\/ 2    \/ 6     \/ 2    \/ 6  | |
                          |    |----- + -----    ----- + ----- | |
                          \    \  4       4        4       4   / /
------------------------------------------------------------------
      /                                     2\                    
      |    /                    ___     ___\ |                    
      |    |                  \/ 2    \/ 6 | |                    
      |    |                - ----- + -----| |                    
      |    |      1             4       4  | | /         ___\     
    6*|1 + |------------- - ---------------| |*\-2 + 2*\/ 2 /     
      |    |  ___     ___      ___     ___ | |                    
      |    |\/ 2    \/ 6     \/ 2    \/ 6  | |                    
      |    |----- + -----    ----- + ----- | |                    
      \    \  4       4        4       4   / /
6(1−(−−24+6424+64+124+64)2)((−1+2)2+1)6(−2+22)((−−24+6424+64+124+64)2+1)\frac{\sqrt{6} \left(1 - \left(- \frac{- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}} + \frac{1}{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}\right)^{2}\right) \left(\left(-1 + \sqrt{2}\right)^{2} + 1\right)}{6 \left(-2 + 2 \sqrt{2}\right) \left(\left(- \frac{- \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}} + \frac{1}{\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}}\right)^{2} + 1\right)} 
sqrt(6)*(1 + (-1 + sqrt(2))^2)*(1 - (1/(sqrt(2)/4 + sqrt(6)/4) - (-sqrt(2)/4 + sqrt(6)/4)/(sqrt(2)/4 + sqrt(6)/4))^2)/(6*(1 + (1/(sqrt(2)/4 + sqrt(6)/4) - (-sqrt(2)/4 + sqrt(6)/4)/(sqrt(2)/4 + sqrt(6)/4))^2)*(-2 + 2*sqrt(2)))
Abrimos la expresión [src] 
    ___     ___
  \/ 6    \/ 2 
- ----- + -----
    12      4
−612+24- \frac{\sqrt{6}}{12} + \frac{\sqrt{2}}{4} 
  ___    /7*pi   pi\
\/ 6 *cos|---- + --|
         \ 24    8 /
--------------------
       /pi   3*pi\  
  6*sin|-- + ----|  
       \10    20 /
6cos⁡(π8+7π24)6sin⁡(π10+3π20)\frac{\sqrt{6} \cos{\left(\frac{\pi}{8} + \frac{7 \pi}{24} \right)}}{6 \sin{\left(\frac{\pi}{10} + \frac{3 \pi}{20} \right)}} 
sqrt(6)*cos((7*pi)/24 + pi/8)/(6*sin(pi/10 + (3*pi)/20))
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