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Simplificación de expresiones/ Descomposición en fracciones simples/ sin(14*x)/((2*cos(7*x)))

¿Cómo vas a descomponer esta sin(14*x)/((2*cos(7*x))) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
sin(14*x) 
----------
2*cos(7*x)
$$\frac{\sin{\left(14 x \right)}}{2 \cos{\left(7 x \right)}}$$ 
sin(14*x)/((2*cos(7*x)))
Simplificación general [src] 
sin(7*x)
$$\sin{\left(7 x \right)}$$ 
sin(7*x)
Potencias [src] 
   /   -14*I*x    14*I*x\ 
-I*\- e        + e      / 
--------------------------
     / -7*I*x    7*I*x\   
   2*\e       + e     /
$$- \frac{i \left(e^{14 i x} - e^{- 14 i x}\right)}{2 \left(e^{7 i x} + e^{- 7 i x}\right)}$$ 
-i*(-exp(-14*i*x) + exp(14*i*x))/(2*(exp(-7*i*x) + exp(7*i*x)))
Abrimos la expresión [src] 
                         11                                                   7                                                    3                                                                                                         5                                                    13                                                    9                         
                12288*sin  (x)*cos(x)                                 7680*sin (x)*cos(x)                                   224*sin (x)*cos(x)                                    7*cos(x)*sin(x)                                    2016*sin (x)*cos(x)                                  4096*sin  (x)*cos(x)                                 14080*sin (x)*cos(x)               
- -------------------------------------------------- - -------------------------------------------------- - -------------------------------------------------- + -------------------------------------------------- + -------------------------------------------------- + -------------------------------------------------- + --------------------------------------------------
           5                       3            7               5                       3            7               5                       3            7               5                       3            7               5                       3            7               5                       3            7               5                       3            7   
  - 112*cos (x) - 7*cos(x) + 56*cos (x) + 64*cos (x)   - 112*cos (x) - 7*cos(x) + 56*cos (x) + 64*cos (x)   - 112*cos (x) - 7*cos(x) + 56*cos (x) + 64*cos (x)   - 112*cos (x) - 7*cos(x) + 56*cos (x) + 64*cos (x)   - 112*cos (x) - 7*cos(x) + 56*cos (x) + 64*cos (x)   - 112*cos (x) - 7*cos(x) + 56*cos (x) + 64*cos (x)   - 112*cos (x) - 7*cos(x) + 56*cos (x) + 64*cos (x)
$$\frac{4096 \sin^{13}{\left(x \right)} \cos{\left(x \right)}}{64 \cos^{7}{\left(x \right)} - 112 \cos^{5}{\left(x \right)} + 56 \cos^{3}{\left(x \right)} - 7 \cos{\left(x \right)}} - \frac{12288 \sin^{11}{\left(x \right)} \cos{\left(x \right)}}{64 \cos^{7}{\left(x \right)} - 112 \cos^{5}{\left(x \right)} + 56 \cos^{3}{\left(x \right)} - 7 \cos{\left(x \right)}} + \frac{14080 \sin^{9}{\left(x \right)} \cos{\left(x \right)}}{64 \cos^{7}{\left(x \right)} - 112 \cos^{5}{\left(x \right)} + 56 \cos^{3}{\left(x \right)} - 7 \cos{\left(x \right)}} - \frac{7680 \sin^{7}{\left(x \right)} \cos{\left(x \right)}}{64 \cos^{7}{\left(x \right)} - 112 \cos^{5}{\left(x \right)} + 56 \cos^{3}{\left(x \right)} - 7 \cos{\left(x \right)}} + \frac{2016 \sin^{5}{\left(x \right)} \cos{\left(x \right)}}{64 \cos^{7}{\left(x \right)} - 112 \cos^{5}{\left(x \right)} + 56 \cos^{3}{\left(x \right)} - 7 \cos{\left(x \right)}} - \frac{224 \sin^{3}{\left(x \right)} \cos{\left(x \right)}}{64 \cos^{7}{\left(x \right)} - 112 \cos^{5}{\left(x \right)} + 56 \cos^{3}{\left(x \right)} - 7 \cos{\left(x \right)}} + \frac{7 \sin{\left(x \right)} \cos{\left(x \right)}}{64 \cos^{7}{\left(x \right)} - 112 \cos^{5}{\left(x \right)} + 56 \cos^{3}{\left(x \right)} - 7 \cos{\left(x \right)}}$$ 
-12288*sin(x)^11*cos(x)/(-112*cos(x)^5 - 7*cos(x) + 56*cos(x)^3 + 64*cos(x)^7) - 7680*sin(x)^7*cos(x)/(-112*cos(x)^5 - 7*cos(x) + 56*cos(x)^3 + 64*cos(x)^7) - 224*sin(x)^3*cos(x)/(-112*cos(x)^5 - 7*cos(x) + 56*cos(x)^3 + 64*cos(x)^7) + 7*cos(x)*sin(x)/(-112*cos(x)^5 - 7*cos(x) + 56*cos(x)^3 + 64*cos(x)^7) + 2016*sin(x)^5*cos(x)/(-112*cos(x)^5 - 7*cos(x) + 56*cos(x)^3 + 64*cos(x)^7) + 4096*sin(x)^13*cos(x)/(-112*cos(x)^5 - 7*cos(x) + 56*cos(x)^3 + 64*cos(x)^7) + 14080*sin(x)^9*cos(x)/(-112*cos(x)^5 - 7*cos(x) + 56*cos(x)^3 + 64*cos(x)^7)
Parte trigonométrica [src] 
   sin(14*x)   
---------------
     /pi      \
2*sin|-- + 7*x|
     \2       /
$$\frac{\sin{\left(14 x \right)}}{2 \sin{\left(7 x + \frac{\pi}{2} \right)}}$$ 
    /       2/7*x\\            
    |1 + tan |---||*tan(7*x)   
    \        \ 2 //            
-------------------------------
/       2     \ /       2/7*x\\
\1 + tan (7*x)/*|1 - tan |---||
                \        \ 2 //
$$\frac{\left(\tan^{2}{\left(\frac{7 x}{2} \right)} + 1\right) \tan{\left(7 x \right)}}{\left(1 - \tan^{2}{\left(\frac{7 x}{2} \right)}\right) \left(\tan^{2}{\left(7 x \right)} + 1\right)}$$ 
   /pi      \
csc|-- - 7*x|
   \2       /
-------------
 2*csc(14*x)
$$\frac{\csc{\left(- 7 x + \frac{\pi}{2} \right)}}{2 \csc{\left(14 x \right)}}$$ 
sin(7*x)
$$\sin{\left(7 x \right)}$$ 
    /       2/7*x\\             
    |1 + cot |---||*cot(7*x)    
    \        \ 2 //             
--------------------------------
/       2     \ /        2/7*x\\
\1 + cot (7*x)/*|-1 + cot |---||
                \         \ 2 //
$$\frac{\left(\cot^{2}{\left(\frac{7 x}{2} \right)} + 1\right) \cot{\left(7 x \right)}}{\left(\cot^{2}{\left(\frac{7 x}{2} \right)} - 1\right) \left(\cot^{2}{\left(7 x \right)} + 1\right)}$$ 
   1    
--------
csc(7*x)
$$\frac{1}{\csc{\left(7 x \right)}}$$ 
  sec(7*x) 
-----------
2*csc(14*x)
$$\frac{\sec{\left(7 x \right)}}{2 \csc{\left(14 x \right)}}$$ 
       /7*x\ 
  2*cot|---| 
       \ 2 / 
-------------
       2/7*x\
1 + cot |---|
        \ 2 /
$$\frac{2 \cot{\left(\frac{7 x}{2} \right)}}{\cot^{2}{\left(\frac{7 x}{2} \right)} + 1}$$ 
       /7*x\ 
  2*tan|---| 
       \ 2 / 
-------------
       2/7*x\
1 + tan |---|
        \ 2 /
$$\frac{2 \tan{\left(\frac{7 x}{2} \right)}}{\tan^{2}{\left(\frac{7 x}{2} \right)} + 1}$$ 
      1      
-------------
   /      pi\
sec|7*x - --|
   \      2 /
$$\frac{1}{\sec{\left(7 x - \frac{\pi}{2} \right)}}$$ 
    sec(7*x)    
----------------
     /       pi\
2*sec|14*x - --|
     \       2 /
$$\frac{\sec{\left(7 x \right)}}{2 \sec{\left(14 x - \frac{\pi}{2} \right)}}$$ 
   /       pi\
cos|14*x - --|
   \       2 /
--------------
  2*cos(7*x)
$$\frac{\cos{\left(14 x - \frac{\pi}{2} \right)}}{2 \cos{\left(7 x \right)}}$$ 
   /      pi\
cos|7*x - --|
   \      2 /
$$\cos{\left(7 x - \frac{\pi}{2} \right)}$$ 
cos(7*x - pi/2)
Respuesta numérica [src] 
0.5*sin(14*x)/cos(7*x)
0.5*sin(14*x)/cos(7*x)
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