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Simplificación de expresiones/ Descomposición en fracciones simples/ sin(a+b+c)/((sin(a)*sin(b)*sin(c)))-3(tan(a)+tan(b)+tan(c))/(tan(a)tan(b)tan(c))

¿Cómo vas a descomponer esta sin(a+b+c)/((sin(a)*sin(b)*sin(c)))-3(tan(a)+tan(b)+tan(c))/(tan(a)tan(b)tan(c)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
   sin(a + b + c)      3*(tan(a) + tan(b) + tan(c))
-------------------- - ----------------------------
sin(a)*sin(b)*sin(c)       tan(a)*tan(b)*tan(c)
$$- \frac{3 \left(\left(\tan{\left(a \right)} + \tan{\left(b \right)}\right) + \tan{\left(c \right)}\right)}{\tan{\left(a \right)} \tan{\left(b \right)} \tan{\left(c \right)}} + \frac{\sin{\left(c + \left(a + b\right) \right)}}{\sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)}}$$ 
sin(a + b + c)/(((sin(a)*sin(b))*sin(c))) - 3*(tan(a) + tan(b) + tan(c))/((tan(a)*tan(b))*tan(c))
Simplificación general [src] 
           2               2               2      
-1 - ------------- - ------------- - -------------
     tan(a)*tan(b)   tan(a)*tan(c)   tan(b)*tan(c)
$$-1 - \frac{2}{\tan{\left(b \right)} \tan{\left(c \right)}} - \frac{2}{\tan{\left(a \right)} \tan{\left(c \right)}} - \frac{2}{\tan{\left(a \right)} \tan{\left(b \right)}}$$ 
-1 - 2/(tan(a)*tan(b)) - 2/(tan(a)*tan(c)) - 2/(tan(b)*tan(c))
Potencias [src] 
   sin(a + b + c)      3*tan(a) + 3*tan(b) + 3*tan(c)
-------------------- - ------------------------------
sin(a)*sin(b)*sin(c)        tan(a)*tan(b)*tan(c)
$$- \frac{3 \tan{\left(a \right)} + 3 \tan{\left(b \right)} + 3 \tan{\left(c \right)}}{\tan{\left(a \right)} \tan{\left(b \right)} \tan{\left(c \right)}} + \frac{\sin{\left(a + b + c \right)}}{\sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)}}$$ 
                                                                                                      /    /   I*a    -I*a\       /   I*b    -I*b\       /   I*c    -I*c\\
                                                         / I*a    -I*a\ / I*b    -I*b\ / I*c    -I*c\ |3*I*\- e    + e    /   3*I*\- e    + e    /   3*I*\- e    + e    /|
                                                       I*\e    + e    /*\e    + e    /*\e    + e    /*|-------------------- + -------------------- + --------------------|
          /   I*(-a - b - c)    I*(a + b + c)\                                                        |     I*a    -I*a            I*b    -I*b            I*c    -I*c    |
        4*\- e               + e             /                                                        \    e    + e               e    + e               e    + e        /
- -------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------
  /   -I*a    I*a\ /   -I*b    I*b\ /   -I*c    I*c\                                    /   I*a    -I*a\ /   I*b    -I*b\ /   I*c    -I*c\                                
  \- e     + e   /*\- e     + e   /*\- e     + e   /                                    \- e    + e    /*\- e    + e    /*\- e    + e    /
$$- \frac{4 \left(- e^{i \left(- a - b - c\right)} + e^{i \left(a + b + c\right)}\right)}{\left(e^{i a} - e^{- i a}\right) \left(e^{i b} - e^{- i b}\right) \left(e^{i c} - e^{- i c}\right)} - \frac{i \left(e^{i a} + e^{- i a}\right) \left(e^{i b} + e^{- i b}\right) \left(e^{i c} + e^{- i c}\right) \left(\frac{3 i \left(- e^{i a} + e^{- i a}\right)}{e^{i a} + e^{- i a}} + \frac{3 i \left(- e^{i b} + e^{- i b}\right)}{e^{i b} + e^{- i b}} + \frac{3 i \left(- e^{i c} + e^{- i c}\right)}{e^{i c} + e^{- i c}}\right)}{\left(- e^{i a} + e^{- i a}\right) \left(- e^{i b} + e^{- i b}\right) \left(- e^{i c} + e^{- i c}\right)}$$ 
   sin(a + b + c)      -3*tan(a) - 3*tan(b) - 3*tan(c)
-------------------- + -------------------------------
sin(a)*sin(b)*sin(c)         tan(a)*tan(b)*tan(c)
$$\frac{- 3 \tan{\left(a \right)} - 3 \tan{\left(b \right)} - 3 \tan{\left(c \right)}}{\tan{\left(a \right)} \tan{\left(b \right)} \tan{\left(c \right)}} + \frac{\sin{\left(a + b + c \right)}}{\sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)}}$$ 
sin(a + b + c)/(sin(a)*sin(b)*sin(c)) + (-3*tan(a) - 3*tan(b) - 3*tan(c))/(tan(a)*tan(b)*tan(c))
Combinatoria [src] 
sin(a + b + c)*tan(a)*tan(b)*tan(c) - 3*sin(a)*sin(b)*sin(c)*tan(a) - 3*sin(a)*sin(b)*sin(c)*tan(b) - 3*sin(a)*sin(b)*sin(c)*tan(c)
-----------------------------------------------------------------------------------------------------------------------------------
                                             sin(a)*sin(b)*sin(c)*tan(a)*tan(b)*tan(c)
$$\frac{- 3 \sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)} \tan{\left(a \right)} - 3 \sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)} \tan{\left(b \right)} - 3 \sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)} \tan{\left(c \right)} + \sin{\left(a + b + c \right)} \tan{\left(a \right)} \tan{\left(b \right)} \tan{\left(c \right)}}{\sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)} \tan{\left(a \right)} \tan{\left(b \right)} \tan{\left(c \right)}}$$ 
(sin(a + b + c)*tan(a)*tan(b)*tan(c) - 3*sin(a)*sin(b)*sin(c)*tan(a) - 3*sin(a)*sin(b)*sin(c)*tan(b) - 3*sin(a)*sin(b)*sin(c)*tan(c))/(sin(a)*sin(b)*sin(c)*tan(a)*tan(b)*tan(c))
Denominador común [src] 
sin(a + b + c)*tan(a)*tan(b)*tan(c) - 3*sin(a)*sin(b)*sin(c)*tan(a) - 3*sin(a)*sin(b)*sin(c)*tan(b) - 3*sin(a)*sin(b)*sin(c)*tan(c)
-----------------------------------------------------------------------------------------------------------------------------------
                                             sin(a)*sin(b)*sin(c)*tan(a)*tan(b)*tan(c)
$$\frac{- 3 \sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)} \tan{\left(a \right)} - 3 \sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)} \tan{\left(b \right)} - 3 \sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)} \tan{\left(c \right)} + \sin{\left(a + b + c \right)} \tan{\left(a \right)} \tan{\left(b \right)} \tan{\left(c \right)}}{\sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)} \tan{\left(a \right)} \tan{\left(b \right)} \tan{\left(c \right)}}$$ 
(sin(a + b + c)*tan(a)*tan(b)*tan(c) - 3*sin(a)*sin(b)*sin(c)*tan(a) - 3*sin(a)*sin(b)*sin(c)*tan(b) - 3*sin(a)*sin(b)*sin(c)*tan(c))/(sin(a)*sin(b)*sin(c)*tan(a)*tan(b)*tan(c))
Denominador racional [src] 
sin(a + b + c)*tan(a)*tan(b)*tan(c) - 3*sin(a)*sin(b)*sin(c)*tan(a) - 3*sin(a)*sin(b)*sin(c)*tan(b) - 3*sin(a)*sin(b)*sin(c)*tan(c)
-----------------------------------------------------------------------------------------------------------------------------------
                                             sin(a)*sin(b)*sin(c)*tan(a)*tan(b)*tan(c)
$$\frac{- 3 \sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)} \tan{\left(a \right)} - 3 \sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)} \tan{\left(b \right)} - 3 \sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)} \tan{\left(c \right)} + \sin{\left(a + b + c \right)} \tan{\left(a \right)} \tan{\left(b \right)} \tan{\left(c \right)}}{\sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)} \tan{\left(a \right)} \tan{\left(b \right)} \tan{\left(c \right)}}$$ 
(sin(a + b + c)*tan(a)*tan(b)*tan(c) - 3*sin(a)*sin(b)*sin(c)*tan(a) - 3*sin(a)*sin(b)*sin(c)*tan(b) - 3*sin(a)*sin(b)*sin(c)*tan(c))/(sin(a)*sin(b)*sin(c)*tan(a)*tan(b)*tan(c))
Abrimos la expresión [src] 
           3               3               3         cos(a)*cos(b)   cos(a)*cos(c)   cos(b)*cos(c)
-1 - ------------- - ------------- - ------------- + ------------- + ------------- + -------------
     tan(a)*tan(b)   tan(a)*tan(c)   tan(b)*tan(c)   sin(a)*sin(b)   sin(a)*sin(c)   sin(b)*sin(c)
$$-1 - \frac{3}{\tan{\left(b \right)} \tan{\left(c \right)}} - \frac{3}{\tan{\left(a \right)} \tan{\left(c \right)}} - \frac{3}{\tan{\left(a \right)} \tan{\left(b \right)}} + \frac{\cos{\left(b \right)} \cos{\left(c \right)}}{\sin{\left(b \right)} \sin{\left(c \right)}} + \frac{\cos{\left(a \right)} \cos{\left(c \right)}}{\sin{\left(a \right)} \sin{\left(c \right)}} + \frac{\cos{\left(a \right)} \cos{\left(b \right)}}{\sin{\left(a \right)} \sin{\left(b \right)}}$$ 
-1 - 3/(tan(a)*tan(b)) - 3/(tan(a)*tan(c)) - 3/(tan(b)*tan(c)) + cos(a)*cos(b)/(sin(a)*sin(b)) + cos(a)*cos(c)/(sin(a)*sin(c)) + cos(b)*cos(c)/(sin(b)*sin(c))
Compilar la expresión [src] 
   sin(a + b + c)      3*tan(a) + 3*tan(b) + 3*tan(c)
-------------------- - ------------------------------
sin(a)*sin(b)*sin(c)        tan(a)*tan(b)*tan(c)
$$- \frac{3 \tan{\left(a \right)} + 3 \tan{\left(b \right)} + 3 \tan{\left(c \right)}}{\tan{\left(a \right)} \tan{\left(b \right)} \tan{\left(c \right)}} + \frac{\sin{\left(c + \left(a + b\right) \right)}}{\sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)}}$$ 
sin(a + b + c)/(sin(a)*sin(b)*sin(c)) - (3*tan(a) + 3*tan(b) + 3*tan(c))/(tan(a)*tan(b)*tan(c))
Unión de expresiones racionales [src] 
sin(a + b + c)*tan(a)*tan(b)*tan(c) - 3*(tan(a) + tan(b) + tan(c))*sin(a)*sin(b)*sin(c)
---------------------------------------------------------------------------------------
                       sin(a)*sin(b)*sin(c)*tan(a)*tan(b)*tan(c)
$$\frac{- 3 \left(\tan{\left(a \right)} + \tan{\left(b \right)} + \tan{\left(c \right)}\right) \sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)} + \sin{\left(a + b + c \right)} \tan{\left(a \right)} \tan{\left(b \right)} \tan{\left(c \right)}}{\sin{\left(a \right)} \sin{\left(b \right)} \sin{\left(c \right)} \tan{\left(a \right)} \tan{\left(b \right)} \tan{\left(c \right)}}$$ 
(sin(a + b + c)*tan(a)*tan(b)*tan(c) - 3*(tan(a) + tan(b) + tan(c))*sin(a)*sin(b)*sin(c))/(sin(a)*sin(b)*sin(c)*tan(a)*tan(b)*tan(c))
Respuesta numérica [src] 
sin(a + b + c)/(sin(a)*sin(b)*sin(c)) - (3.0*tan(a) + 3.0*tan(b) + 3.0*tan(c))/(tan(a)*tan(b)*tan(c))
sin(a + b + c)/(sin(a)*sin(b)*sin(c)) - (3.0*tan(a) + 3.0*tan(b) + 3.0*tan(c))/(tan(a)*tan(b)*tan(c))
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