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

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

Expresión a simplificar:

Solución

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