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Simplificación de expresiones/ Descomposición en fracciones simples/ sin(7*pi/(11+a))*sin((a-4*pi)/11)+cos(4*pi/(11-a))*cos(7*pi/(11+a))

¿Cómo vas a descomponer esta sin(7*pi/(11+a))*sin((a-4*pi)/11)+cos(4*pi/(11-a))*cos(7*pi/(11+a)) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
   / 7*pi \    /a - 4*pi\      / 4*pi \    / 7*pi \
sin|------|*sin|--------| + cos|------|*cos|------|
   \11 + a/    \   11   /      \11 - a/    \11 + a/
$$\sin{\left(\frac{a - 4 \pi}{11} \right)} \sin{\left(\frac{7 \pi}{a + 11} \right)} + \cos{\left(\frac{4 \pi}{11 - a} \right)} \cos{\left(\frac{7 \pi}{a + 11} \right)}$$ 
sin((7*pi)/(11 + a))*sin((a - 4*pi)/11) + cos((4*pi)/(11 - a))*cos((7*pi)/(11 + a))
Simplificación general [src] 
   /  4*pi \    / 7*pi \      /a    3*pi\    / 7*pi \
cos|-------|*cos|------| - cos|-- + ----|*sin|------|
   \-11 + a/    \11 + a/      \11    22 /    \11 + a/
$$- \sin{\left(\frac{7 \pi}{a + 11} \right)} \cos{\left(\frac{a}{11} + \frac{3 \pi}{22} \right)} + \cos{\left(\frac{4 \pi}{a - 11} \right)} \cos{\left(\frac{7 \pi}{a + 11} \right)}$$ 
cos(4*pi/(-11 + a))*cos(7*pi/(11 + a)) - cos(a/11 + 3*pi/22)*sin(7*pi/(11 + a))
Respuesta numérica [src] 
cos((4*pi)/(11 - a))*cos((7*pi)/(11 + a)) + sin((7*pi)/(11 + a))*sin((a - 4*pi)/11)
cos((4*pi)/(11 - a))*cos((7*pi)/(11 + a)) + sin((7*pi)/(11 + a))*sin((a - 4*pi)/11)
Denominador racional [src] 
   /  4*pi \    / 7*pi \      /a    3*pi\    / 7*pi \
cos|-------|*cos|------| - cos|-- + ----|*sin|------|
   \-11 + a/    \11 + a/      \11    22 /    \11 + a/
$$- \sin{\left(\frac{7 \pi}{a + 11} \right)} \cos{\left(\frac{a}{11} + \frac{3 \pi}{22} \right)} + \cos{\left(\frac{4 \pi}{a - 11} \right)} \cos{\left(\frac{7 \pi}{a + 11} \right)}$$ 
cos(4*pi/(-11 + a))*cos(7*pi/(11 + a)) - cos(a/11 + 3*pi/22)*sin(7*pi/(11 + a))
Combinatoria [src] 
   / 4*pi \    / 7*pi \      /a    3*pi\    / 7*pi \
cos|------|*cos|------| - cos|-- + ----|*sin|------|
   \11 - a/    \11 + a/      \11    22 /    \11 + a/
$$- \sin{\left(\frac{7 \pi}{a + 11} \right)} \cos{\left(\frac{a}{11} + \frac{3 \pi}{22} \right)} + \cos{\left(\frac{4 \pi}{11 - a} \right)} \cos{\left(\frac{7 \pi}{a + 11} \right)}$$ 
cos(4*pi/(11 - a))*cos(7*pi/(11 + a)) - cos(a/11 + 3*pi/22)*sin(7*pi/(11 + a))
Unión de expresiones racionales [src] 
   / 4*pi \    / 7*pi \      /a    3*pi\    / 7*pi \
cos|------|*cos|------| - cos|-- + ----|*sin|------|
   \11 - a/    \11 + a/      \11    22 /    \11 + a/
$$- \sin{\left(\frac{7 \pi}{a + 11} \right)} \cos{\left(\frac{a}{11} + \frac{3 \pi}{22} \right)} + \cos{\left(\frac{4 \pi}{11 - a} \right)} \cos{\left(\frac{7 \pi}{a + 11} \right)}$$ 
cos(4*pi/(11 - a))*cos(7*pi/(11 + a)) - cos(a/11 + 3*pi/22)*sin(7*pi/(11 + a))
Denominador común [src] 
   / 4*pi \    / 7*pi \      /a    3*pi\    / 7*pi \
cos|------|*cos|------| - cos|-- + ----|*sin|------|
   \11 - a/    \11 + a/      \11    22 /    \11 + a/
$$- \sin{\left(\frac{7 \pi}{a + 11} \right)} \cos{\left(\frac{a}{11} + \frac{3 \pi}{22} \right)} + \cos{\left(\frac{4 \pi}{11 - a} \right)} \cos{\left(\frac{7 \pi}{a + 11} \right)}$$ 
cos(4*pi/(11 - a))*cos(7*pi/(11 + a)) - cos(a/11 + 3*pi/22)*sin(7*pi/(11 + a))
Potencias [src] 
/ -7*pi*I    7*pi*I\ / -4*pi*I    4*pi*I\   /     /  a    4*pi\      /  4*pi   a \\ /   -7*pi*I    7*pi*I\
| -------    ------| | -------    ------|   |   I*|- -- + ----|    I*|- ---- + --|| |   -------    ------|
|  11 + a    11 + a| |  11 - a    11 - a|   |     \  11    11 /      \   11    11/| |    11 + a    11 + a|
|e          e      | |e          e      |   \- e                + e               /*\- e        + e      /
|-------- + -------|*|-------- + -------| - --------------------------------------------------------------
\   2          2   / \   2          2   /                                 4
$$- \frac{\left(- e^{i \left(- \frac{a}{11} + \frac{4 \pi}{11}\right)} + e^{i \left(\frac{a}{11} - \frac{4 \pi}{11}\right)}\right) \left(e^{\frac{7 i \pi}{a + 11}} - e^{- \frac{7 i \pi}{a + 11}}\right)}{4} + \left(\frac{e^{\frac{4 i \pi}{11 - a}}}{2} + \frac{e^{- \frac{4 i \pi}{11 - a}}}{2}\right) \left(\frac{e^{\frac{7 i \pi}{a + 11}}}{2} + \frac{e^{- \frac{7 i \pi}{a + 11}}}{2}\right)$$ 
   / 4*pi \    / 7*pi \      /a    3*pi\    / 7*pi \
cos|------|*cos|------| - cos|-- + ----|*sin|------|
   \11 - a/    \11 + a/      \11    22 /    \11 + a/
$$- \sin{\left(\frac{7 \pi}{a + 11} \right)} \cos{\left(\frac{a}{11} + \frac{3 \pi}{22} \right)} + \cos{\left(\frac{4 \pi}{11 - a} \right)} \cos{\left(\frac{7 \pi}{a + 11} \right)}$$ 
cos(4*pi/(11 - a))*cos(7*pi/(11 + a)) - cos(a/11 + 3*pi/22)*sin(7*pi/(11 + a))
Abrimos la expresión [src] 
                                                                _______________                    
                                                               /        /3*pi\                     
                                                              /      cos|----|                     
   / 4*pi \    / 7*pi \      /4*pi\    /a \    / 7*pi \      /   1      \ 11 /     /a \    / 7*pi \
cos|------|*cos|------| + cos|----|*sin|--|*sin|------| -   /    - + --------- *cos|--|*sin|------|
   \11 - a/    \11 + a/      \ 11 /    \11/    \11 + a/   \/     2       2         \11/    \11 + a/
$$\sin{\left(\frac{a}{11} \right)} \sin{\left(\frac{7 \pi}{a + 11} \right)} \cos{\left(\frac{4 \pi}{11} \right)} - \sqrt{\frac{\cos{\left(\frac{3 \pi}{11} \right)}}{2} + \frac{1}{2}} \sin{\left(\frac{7 \pi}{a + 11} \right)} \cos{\left(\frac{a}{11} \right)} + \cos{\left(\frac{4 \pi}{11 - a} \right)} \cos{\left(\frac{7 \pi}{a + 11} \right)}$$ 
cos((4*pi)/(11 - a))*cos((7*pi)/(11 + a)) + cos(4*pi/11)*sin(a/11)*sin((7*pi)/(11 + a)) - sqrt(1/2 + cos(3*pi/11)/2)*cos(a/11)*sin((7*pi)/(11 + a))
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