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Simplificación de expresiones/ Descomposición en fracciones simples/ sin(6*x)/((2*sin(3*x)))

¿Cómo vas a descomponer esta sin(6*x)/((2*sin(3*x))) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
 sin(6*x) 
----------
2*sin(3*x)
$$\frac{\sin{\left(6 x \right)}}{2 \sin{\left(3 x \right)}}$$ 
sin(6*x)/((2*sin(3*x)))
Simplificación general [src] 
cos(3*x)
$$\cos{\left(3 x \right)}$$ 
cos(3*x)
Respuesta numérica [src] 
0.5*sin(6*x)/sin(3*x)
0.5*sin(6*x)/sin(3*x)
Potencias [src] 
     -6*I*x    6*I*x  
  - e       + e       
----------------------
  /   -3*I*x    3*I*x\
2*\- e       + e     /
$$\frac{e^{6 i x} - e^{- 6 i x}}{2 \left(e^{3 i x} - e^{- 3 i x}\right)}$$ 
(-exp(-6*i*x) + exp(6*i*x))/(2*(-exp(-3*i*x) + exp(3*i*x)))
Abrimos la expresión [src] 
          3                                                 5             
    16*sin (x)*cos(x)         3*cos(x)*sin(x)         16*sin (x)*cos(x)   
- ---------------------- + ---------------------- + ----------------------
         3                        3                        3              
  - 4*sin (x) + 3*sin(x)   - 4*sin (x) + 3*sin(x)   - 4*sin (x) + 3*sin(x)
$$\frac{16 \sin^{5}{\left(x \right)} \cos{\left(x \right)}}{- 4 \sin^{3}{\left(x \right)} + 3 \sin{\left(x \right)}} - \frac{16 \sin^{3}{\left(x \right)} \cos{\left(x \right)}}{- 4 \sin^{3}{\left(x \right)} + 3 \sin{\left(x \right)}} + \frac{3 \sin{\left(x \right)} \cos{\left(x \right)}}{- 4 \sin^{3}{\left(x \right)} + 3 \sin{\left(x \right)}}$$ 
-16*sin(x)^3*cos(x)/(-4*sin(x)^3 + 3*sin(x)) + 3*cos(x)*sin(x)/(-4*sin(x)^3 + 3*sin(x)) + 16*sin(x)^5*cos(x)/(-4*sin(x)^3 + 3*sin(x))
Parte trigonométrica [src] 
        2/3*x\
-1 + cot |---|
         \ 2 /
--------------
       2/3*x\ 
1 + cot |---| 
        \ 2 /
$$\frac{\cot^{2}{\left(\frac{3 x}{2} \right)} - 1}{\cot^{2}{\left(\frac{3 x}{2} \right)} + 1}$$ 
 /       2/3*x\\          
 |1 + cot |---||*cot(3*x) 
 \        \ 2 //          
--------------------------
  /       2     \    /3*x\
2*\1 + cot (3*x)/*cot|---|
                     \ 2 /
$$\frac{\left(\cot^{2}{\left(\frac{3 x}{2} \right)} + 1\right) \cot{\left(3 x \right)}}{2 \left(\cot^{2}{\left(3 x \right)} + 1\right) \cot{\left(\frac{3 x}{2} \right)}}$$ 
 csc(3*x) 
----------
2*csc(6*x)
$$\frac{\csc{\left(3 x \right)}}{2 \csc{\left(6 x \right)}}$$ 
    /      pi\ 
 sec|3*x - --| 
    \      2 / 
---------------
     /      pi\
2*sec|6*x - --|
     \      2 /
$$\frac{\sec{\left(3 x - \frac{\pi}{2} \right)}}{2 \sec{\left(6 x - \frac{\pi}{2} \right)}}$$ 
       2/3*x\
1 - tan |---|
        \ 2 /
-------------
       2/3*x\
1 + tan |---|
        \ 2 /
$$\frac{1 - \tan^{2}{\left(\frac{3 x}{2} \right)}}{\tan^{2}{\left(\frac{3 x}{2} \right)} + 1}$$ 
   /pi      \
sin|-- + 3*x|
   \2       /
$$\sin{\left(3 x + \frac{\pi}{2} \right)}$$ 
   1    
--------
sec(3*x)
$$\frac{1}{\sec{\left(3 x \right)}}$$ 
 /       2/3*x\\          
 |1 + tan |---||*tan(3*x) 
 \        \ 2 //          
--------------------------
  /       2     \    /3*x\
2*\1 + tan (3*x)/*tan|---|
                     \ 2 /
$$\frac{\left(\tan^{2}{\left(\frac{3 x}{2} \right)} + 1\right) \tan{\left(3 x \right)}}{2 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(\frac{3 x}{2} \right)}}$$ 
cos(3*x)
$$\cos{\left(3 x \right)}$$ 
    /      pi\ 
 cos|6*x - --| 
    \      2 / 
---------------
     /      pi\
2*cos|3*x - --|
     \      2 /
$$\frac{\cos{\left(6 x - \frac{\pi}{2} \right)}}{2 \cos{\left(3 x - \frac{\pi}{2} \right)}}$$ 
      1      
-------------
   /pi      \
csc|-- - 3*x|
   \2       /
$$\frac{1}{\csc{\left(- 3 x + \frac{\pi}{2} \right)}}$$ 
1/csc(pi/2 - 3*x)
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