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Simplificación de expresiones/ Descomposición en fracciones simples/ sin(a+b+c)/((sin(a)*sin(b)*sin(c)))

¿Cómo vas a descomponer esta sin(a+b+c)/((sin(a)*sin(b)*sin(c))) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
   sin(a + b + c)   
--------------------
sin(a)*sin(b)*sin(c)
$$\frac{\sin{\left(c + \left(a + b\right) \right)}}{\sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)}}$$ 
sin(a + b + c)/(((sin(a)*sin(b))*sin(c)))
Simplificación general [src] 
   sin(a + b + c)   
--------------------
sin(a)*sin(b)*sin(c)
$$\frac{\sin{\left(a + b + c \right)}}{\sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)}}$$ 
sin(a + b + c)/(sin(a)*sin(b)*sin(c))
Potencias [src] 
        /   I*(-a - b - c)    I*(a + b + c)\      
     -4*\- e               + e             /      
--------------------------------------------------
/   -I*a    I*a\ /   -I*b    I*b\ /   -I*c    I*c\
\- e     + e   /*\- e     + e   /*\- e     + e   /
$$- \frac{4 \left(- e^{i \left(- a - b - c\right)} + e^{i \left(a + b + c\right)}\right)}{\left(e^{i a} - e^{- i a}\right) \left(e^{i b} - e^{- i b}\right) \left(e^{i c} - e^{- i c}\right)}$$ 
   sin(a + b + c)   
--------------------
sin(a)*sin(b)*sin(c)
$$\frac{\sin{\left(a + b + c \right)}}{\sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)}}$$ 
sin(a + b + c)/(sin(a)*sin(b)*sin(c))
Unión de expresiones racionales [src] 
   sin(a + b + c)   
--------------------
sin(a)*sin(b)*sin(c)
$$\frac{\sin{\left(a + b + c \right)}}{\sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)}}$$ 
sin(a + b + c)/(sin(a)*sin(b)*sin(c))
Denominador racional [src] 
   sin(a + b + c)   
--------------------
sin(a)*sin(b)*sin(c)
$$\frac{\sin{\left(a + b + c \right)}}{\sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)}}$$ 
sin(a + b + c)/(sin(a)*sin(b)*sin(c))
Denominador común [src] 
   sin(a + b + c)   
--------------------
sin(a)*sin(b)*sin(c)
$$\frac{\sin{\left(a + b + c \right)}}{\sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)}}$$ 
sin(a + b + c)/(sin(a)*sin(b)*sin(c))
Respuesta numérica [src] 
sin(a + b + c)/(sin(a)*sin(b)*sin(c))
sin(a + b + c)/(sin(a)*sin(b)*sin(c))
Abrimos la expresión [src] 
     cos(a)*cos(b)   cos(a)*cos(c)   cos(b)*cos(c)
-1 + ------------- + ------------- + -------------
     sin(a)*sin(b)   sin(a)*sin(c)   sin(b)*sin(c)
$$-1 + \frac{\cos{\left(b \right)} \cos{\left(c \right)}}{\sin{\left(b \right)} \sin{\left(c \right)}} + \frac{\cos{\left(a \right)} \cos{\left(c \right)}}{\sin{\left(a \right)} \sin{\left(c \right)}} + \frac{\cos{\left(a \right)} \cos{\left(b \right)}}{\sin{\left(a \right)} \sin{\left(b \right)}}$$ 
-1 + cos(a)*cos(b)/(sin(a)*sin(b)) + cos(a)*cos(c)/(sin(a)*sin(c)) + cos(b)*cos(c)/(sin(b)*sin(c))
Combinatoria [src] 
   sin(a + b + c)   
--------------------
sin(a)*sin(b)*sin(c)
$$\frac{\sin{\left(a + b + c \right)}}{\sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)}}$$ 
sin(a + b + c)/(sin(a)*sin(b)*sin(c))
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