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Simplificación de expresiones/ Descomposición en fracciones simples/ cos(2*pi/(3-a))+cos(pi/3+a)

¿Cómo vas a descomponer esta cos(2*pi/(3-a))+cos(pi/3+a) expresión en fracciones?

Expresión a simplificar:

Solución

Ha introducido [src] 
   / 2*pi\      /pi    \
cos|-----| + cos|-- + a|
   \3 - a/      \3     /
cos⁡(2π3−a)+cos⁡(a+π3)\cos{\left(\frac{2 \pi}{3 - a} \right)} + \cos{\left(a + \frac{\pi}{3} \right)} 
cos((2*pi)/(3 - a)) + cos(pi/3 + a)
Simplificación general [src] 
   / 2*pi \      /    pi\
cos|------| + cos|a + --|
   \-3 + a/      \    3 /
cos⁡(2πa−3)+cos⁡(a+π3)\cos{\left(\frac{2 \pi}{a - 3} \right)} + \cos{\left(a + \frac{\pi}{3} \right)} 
cos(2*pi/(-3 + a)) + cos(a + pi/3)
Potencias [src] 
   /    pi\      /     pi\    -2*pi*I    2*pi*I
 I*|a + --|    I*|-a - --|    -------    ------
   \    3 /      \     3 /     3 - a     3 - a 
e             e              e          e      
----------- + ------------ + -------- + -------
     2             2            2          2
ei(−a−π3)2+ei(a+π3)2+e2iπ3−a2+e−2iπ3−a2\frac{e^{i \left(- a - \frac{\pi}{3}\right)}}{2} + \frac{e^{i \left(a + \frac{\pi}{3}\right)}}{2} + \frac{e^{\frac{2 i \pi}{3 - a}}}{2} + \frac{e^{- \frac{2 i \pi}{3 - a}}}{2} 
exp(i*(a + pi/3))/2 + exp(i*(-a - pi/3))/2 + exp(-2*pi*i/(3 - a))/2 + exp(2*pi*i/(3 - a))/2
Unión de expresiones racionales [src] 
   /pi + 3*a\      / 2*pi\
cos|--------| + cos|-----|
   \   3    /      \3 - a/
cos⁡(2π3−a)+cos⁡(3a+π3)\cos{\left(\frac{2 \pi}{3 - a} \right)} + \cos{\left(\frac{3 a + \pi}{3} \right)} 
cos((pi + 3*a)/3) + cos(2*pi/(3 - a))
Denominador racional [src] 
   / 2*pi \      /    pi\
cos|------| + cos|a + --|
   \-3 + a/      \    3 /
cos⁡(2πa−3)+cos⁡(a+π3)\cos{\left(\frac{2 \pi}{a - 3} \right)} + \cos{\left(a + \frac{\pi}{3} \right)} 
cos(2*pi/(-3 + a)) + cos(a + pi/3)
Respuesta numérica [src] 
cos((2*pi)/(3 - a)) + cos(pi/3 + a)
cos((2*pi)/(3 - a)) + cos(pi/3 + a)
Parte trigonométrica [src] 
   /    5*pi\      /pi    2*pi \
sin|a + ----| + sin|-- + ------|
   \     6  /      \2    -3 + a/
sin⁡(π2+2πa−3)+sin⁡(a+5π6)\sin{\left(\frac{\pi}{2} + \frac{2 \pi}{a - 3} \right)} + \sin{\left(a + \frac{5 \pi}{6} \right)} 
       2/  pi \          2/a   pi\
1 - tan |-----|   1 - tan |- + --|
        \3 - a/           \2   6 /
--------------- + ----------------
       2/  pi \          2/a   pi\
1 + tan |-----|   1 + tan |- + --|
        \3 - a/           \2   6 /
1−tan⁡2(π3−a)tan⁡2(π3−a)+1+1−tan⁡2(a2+π6)tan⁡2(a2+π6)+1\frac{1 - \tan^{2}{\left(\frac{\pi}{3 - a} \right)}}{\tan^{2}{\left(\frac{\pi}{3 - a} \right)} + 1} + \frac{1 - \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{6} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{6} \right)} + 1} 
         4/a   pi\                   
    4*sin |- + --|                   
          \2   6 /                   
1 - --------------                   
        2/    pi\                    
     sin |a + --|                    
         \    3 /       /pi    2*pi \
------------------ + sin|-- + ------|
         4/a   pi\      \2    -3 + a/
    4*sin |- + --|                   
          \2   6 /                   
1 + --------------                   
        2/    pi\                    
     sin |a + --|                    
         \    3 /
−4sin⁡4(a2+π6)sin⁡2(a+π3)+14sin⁡4(a2+π6)sin⁡2(a+π3)+1+sin⁡(π2+2πa−3)\frac{- \frac{4 \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{6} \right)}}{\sin^{2}{\left(a + \frac{\pi}{3} \right)}} + 1}{\frac{4 \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{6} \right)}}{\sin^{2}{\left(a + \frac{\pi}{3} \right)}} + 1} + \sin{\left(\frac{\pi}{2} + \frac{2 \pi}{a - 3} \right)} 
         4/a   pi\              
    4*sin |- + --|              
          \2   6 /              
1 - --------------              
        2/    pi\               
     sin |a + --|               
         \    3 /       / 2*pi \
------------------ + cos|------|
         4/a   pi\      \-3 + a/
    4*sin |- + --|              
          \2   6 /              
1 + --------------              
        2/    pi\               
     sin |a + --|               
         \    3 /
−4sin⁡4(a2+π6)sin⁡2(a+π3)+14sin⁡4(a2+π6)sin⁡2(a+π3)+1+cos⁡(2πa−3)\frac{- \frac{4 \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{6} \right)}}{\sin^{2}{\left(a + \frac{\pi}{3} \right)}} + 1}{\frac{4 \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{6} \right)}}{\sin^{2}{\left(a + \frac{\pi}{3} \right)}} + 1} + \cos{\left(\frac{2 \pi}{a - 3} \right)} 
    1             1     
---------- + -----------
   / 2*pi\      /    pi\
sec|-----|   sec|a + --|
   \3 - a/      \    3 /
1sec⁡(a+π3)+1sec⁡(2π3−a)\frac{1}{\sec{\left(a + \frac{\pi}{3} \right)}} + \frac{1}{\sec{\left(\frac{2 \pi}{3 - a} \right)}} 
        2/  pi \           2/a   pi\
-1 + cot |-----|   -1 + cot |- + --|
         \3 - a/            \2   6 /
---------------- + -----------------
       2/  pi \            2/a   pi\
1 + cot |-----|     1 + cot |- + --|
        \3 - a/             \2   6 /
cot⁡2(π3−a)−1cot⁡2(π3−a)+1+cot⁡2(a2+π6)−1cot⁡2(a2+π6)+1\frac{\cot^{2}{\left(\frac{\pi}{3 - a} \right)} - 1}{\cot^{2}{\left(\frac{\pi}{3 - a} \right)} + 1} + \frac{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{6} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{6} \right)} + 1} 
       2/a   pi\              
1 - tan |- + --|              
        \2   6 /      / 2*pi \
---------------- + cos|------|
       2/a   pi\      \-3 + a/
1 + tan |- + --|              
        \2   6 /
1−tan⁡2(a2+π6)tan⁡2(a2+π6)+1+cos⁡(2πa−3)\frac{1 - \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{6} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{6} \right)} + 1} + \cos{\left(\frac{2 \pi}{a - 3} \right)} 
   / 2*pi \      /    pi\
cos|------| + cos|a + --|
   \-3 + a/      \    3 /
cos⁡(2πa−3)+cos⁡(a+π3)\cos{\left(\frac{2 \pi}{a - 3} \right)} + \cos{\left(a + \frac{\pi}{3} \right)} 
          2/a   pi\              
     2*cos |- - --|              
           \2   3 /              
1 - ---------------              
           /    pi\              
    1 + cos|a + --|              
           \    3 /      / 2*pi \
------------------- + cos|------|
          2/a   pi\      \-3 + a/
     2*cos |- - --|              
           \2   3 /              
1 + ---------------              
           /    pi\              
    1 + cos|a + --|              
           \    3 /
1−2cos⁡2(a2−π3)cos⁡(a+π3)+11+2cos⁡2(a2−π3)cos⁡(a+π3)+1+cos⁡(2πa−3)\frac{1 - \frac{2 \cos^{2}{\left(\frac{a}{2} - \frac{\pi}{3} \right)}}{\cos{\left(a + \frac{\pi}{3} \right)} + 1}}{1 + \frac{2 \cos^{2}{\left(\frac{a}{2} - \frac{\pi}{3} \right)}}{\cos{\left(a + \frac{\pi}{3} \right)} + 1}} + \cos{\left(\frac{2 \pi}{a - 3} \right)} 
                     2/a   pi\
                  sec |- + --|
                      \2   6 /
              1 - ------------
                     2/a   pi\
                  sec |- - --|
     1                \2   3 /
----------- + ----------------
   / 2*pi \          2/a   pi\
sec|------|       sec |- + --|
   \-3 + a/           \2   6 /
              1 + ------------
                     2/a   pi\
                  sec |- - --|
                      \2   3 /
1−sec⁡2(a2+π6)sec⁡2(a2−π3)1+sec⁡2(a2+π6)sec⁡2(a2−π3)+1sec⁡(2πa−3)\frac{1 - \frac{\sec^{2}{\left(\frac{a}{2} + \frac{\pi}{6} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{3} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{a}{2} + \frac{\pi}{6} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{3} \right)}}} + \frac{1}{\sec{\left(\frac{2 \pi}{a - 3} \right)}} 
     1             1     
----------- + -----------
   / 2*pi \      /    pi\
sec|------|   sec|a + --|
   \-3 + a/      \    3 /
1sec⁡(a+π3)+1sec⁡(2πa−3)\frac{1}{\sec{\left(a + \frac{\pi}{3} \right)}} + \frac{1}{\sec{\left(\frac{2 \pi}{a - 3} \right)}} 
                             1      
                    1 - ------------
        2/  pi  \          2/a   pi\
-1 + cot |------|       cot |- + --|
         \-3 + a/           \2   6 /
----------------- + ----------------
        2/  pi  \            1      
 1 + cot |------|   1 + ------------
         \-3 + a/          2/a   pi\
                        cot |- + --|
                            \2   6 /
1−1cot⁡2(a2+π6)1+1cot⁡2(a2+π6)+cot⁡2(πa−3)−1cot⁡2(πa−3)+1\frac{1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{6} \right)}}}{1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{6} \right)}}} + \frac{\cot^{2}{\left(\frac{\pi}{a - 3} \right)} - 1}{\cot^{2}{\left(\frac{\pi}{a - 3} \right)} + 1} 
       1               1      
--------------- + ------------
   /pi    2*pi\      /     pi\
csc|-- - -----|   csc|-a + --|
   \2    3 - a/      \     6 /
1csc⁡(−a+π6)+1csc⁡(π2−2π3−a)\frac{1}{\csc{\left(- a + \frac{\pi}{6} \right)}} + \frac{1}{\csc{\left(\frac{\pi}{2} - \frac{2 \pi}{3 - a} \right)}} 
        2/  pi  \           2/a   pi\
-1 + cot |------|   -1 + cot |- + --|
         \-3 + a/            \2   6 /
----------------- + -----------------
        2/  pi  \           2/a   pi\
 1 + cot |------|    1 + cot |- + --|
         \-3 + a/            \2   6 /
cot⁡2(πa−3)−1cot⁡2(πa−3)+1+cot⁡2(a2+π6)−1cot⁡2(a2+π6)+1\frac{\cot^{2}{\left(\frac{\pi}{a - 3} \right)} - 1}{\cot^{2}{\left(\frac{\pi}{a - 3} \right)} + 1} + \frac{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{6} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{6} \right)} + 1} 
                          2/  a   pi\
                       csc |- - + --|
                           \  2   3 /
                   1 - --------------
                           2/a   pi\ 
                        csc |- + --| 
       1                    \2   6 / 
---------------- + ------------------
   /pi    2*pi \          2/  a   pi\
csc|-- - ------|       csc |- - + --|
   \2    -3 + a/           \  2   3 /
                   1 + --------------
                           2/a   pi\ 
                        csc |- + --| 
                            \2   6 /
−csc⁡2(−a2+π3)csc⁡2(a2+π6)+1csc⁡2(−a2+π3)csc⁡2(a2+π6)+1+1csc⁡(π2−2πa−3)\frac{- \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{3} \right)}}{\csc^{2}{\left(\frac{a}{2} + \frac{\pi}{6} \right)}} + 1}{\frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{3} \right)}}{\csc^{2}{\left(\frac{a}{2} + \frac{\pi}{6} \right)}} + 1} + \frac{1}{\csc{\left(\frac{\pi}{2} - \frac{2 \pi}{a - 3} \right)}} 
       1                1      
---------------- + ------------
   /pi    2*pi \      /     pi\
csc|-- - ------|   csc|-a + --|
   \2    -3 + a/      \     6 /
1csc⁡(−a+π6)+1csc⁡(π2−2πa−3)\frac{1}{\csc{\left(- a + \frac{\pi}{6} \right)}} + \frac{1}{\csc{\left(\frac{\pi}{2} - \frac{2 \pi}{a - 3} \right)}} 
       2/  pi  \          2/a   pi\
1 - tan |------|   1 - tan |- + --|
        \-3 + a/           \2   6 /
---------------- + ----------------
       2/  pi  \          2/a   pi\
1 + tan |------|   1 + tan |- + --|
        \-3 + a/           \2   6 /
1−tan⁡2(πa−3)tan⁡2(πa−3)+1+1−tan⁡2(a2+π6)tan⁡2(a2+π6)+1\frac{1 - \tan^{2}{\left(\frac{\pi}{a - 3} \right)}}{\tan^{2}{\left(\frac{\pi}{a - 3} \right)} + 1} + \frac{1 - \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{6} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{6} \right)} + 1} 
   /    5*pi\      /pi    2*pi\
sin|a + ----| + sin|-- + -----|
   \     6  /      \2    3 - a/
sin⁡(π2+2π3−a)+sin⁡(a+5π6)\sin{\left(\frac{\pi}{2} + \frac{2 \pi}{3 - a} \right)} + \sin{\left(a + \frac{5 \pi}{6} \right)} 
       2/a   pi\              
    cos |- - --|              
        \2   3 /              
1 - ------------              
       2/a   pi\              
    cos |- + --|              
        \2   6 /      / 2*pi \
---------------- + cos|------|
       2/a   pi\      \-3 + a/
    cos |- - --|              
        \2   3 /              
1 + ------------              
       2/a   pi\              
    cos |- + --|              
        \2   6 /
−cos⁡2(a2−π3)cos⁡2(a2+π6)+1cos⁡2(a2−π3)cos⁡2(a2+π6)+1+cos⁡(2πa−3)\frac{- \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{3} \right)}}{\cos^{2}{\left(\frac{a}{2} + \frac{\pi}{6} \right)}} + 1}{\frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{3} \right)}}{\cos^{2}{\left(\frac{a}{2} + \frac{\pi}{6} \right)}} + 1} + \cos{\left(\frac{2 \pi}{a - 3} \right)} 
(1 - cos(a/2 - pi/3)^2/cos(a/2 + pi/6)^2)/(1 + cos(a/2 - pi/3)^2/cos(a/2 + pi/6)^2) + cos(2*pi/(-3 + a))
Abrimos la expresión [src] 
           ___                    
cos(a)   \/ 3 *sin(a)      / 2*pi\
------ - ------------ + cos|-----|
  2           2            \3 - a/
−3sin⁡(a)2+cos⁡(a)2+cos⁡(2π3−a)- \frac{\sqrt{3} \sin{\left(a \right)}}{2} + \frac{\cos{\left(a \right)}}{2} + \cos{\left(\frac{2 \pi}{3 - a} \right)} 
cos(a)/2 - sqrt(3)*sin(a)/2 + cos((2*pi)/(3 - a))
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