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Simplificación de expresiones/ Descomposición en fracciones simples/ sin(14*a-sin(10*a))/(cos(3*a)-cos(7*a))

¿Cómo vas a descomponer esta sin(14a-sin(10a))/(cos(3a)-cos(7a)) expresión en fracciones?

Solución

Ha introducido [src]

sin(14*a - sin(10*a))
---------------------
 cos(3*a) - cos(7*a)

$$\frac{\sin{\left(14 a - \sin{\left(10 a \right)} \right)}}{\cos{\left(3 a \right)} - \cos{\left(7 a \right)}}$$

sin(14*a - sin(10*a))/(cos(3*a) - cos(7*a))

Respuesta numérica [src]

sin(14*a - sin(10*a))/(-cos(7*a) + cos(3*a))

sin(14*a - sin(10*a))/(-cos(7*a) + cos(3*a))

Potencias [src]

   /     /          /   -10*I*a    10*I*a\\      /         /   -10*I*a    10*I*a\\\ 
   |     |        I*\- e        + e      /|      |       I*\- e        + e      /|| 
   |   I*|-14*a - ------------------------|    I*|14*a + ------------------------|| 
   |     \                   2            /      \                  2            /| 
-I*\- e                                     + e                                   / 
------------------------------------------------------------------------------------
                        / -3*I*a    3*I*a    -7*I*a    7*I*a\                       
                        |e         e        e         e     |                       
                      2*|------- + ------ - ------- - ------|                       
                        \   2        2         2        2   /

$$- \frac{i \left(- e^{i \left(- 14 a - \frac{i \left(e^{10 i a} - e^{- 10 i a}\right)}{2}\right)} + e^{i \left(14 a + \frac{i \left(e^{10 i a} - e^{- 10 i a}\right)}{2}\right)}\right)}{2 \left(- \frac{e^{7 i a}}{2} + \frac{e^{3 i a}}{2} + \frac{e^{- 3 i a}}{2} - \frac{e^{- 7 i a}}{2}\right)}$$

-i*(-exp(i*(-14*a - i*(-exp(-10*i*a) + exp(10*i*a))/2)) + exp(i*(14*a + i*(-exp(-10*i*a) + exp(10*i*a))/2)))/(2*(exp(-3*i*a)/2 + exp(3*i*a)/2 - exp(-7*i*a)/2 - exp(7*i*a)/2))

Parte trigonométrica [src]

                        /         tan(5*a)  \                  
                   2*tan|7*a - -------------|                  
                        |             2     |                  
                        \      1 + tan (5*a)/                  
---------------------------------------------------------------
                                /       2/3*a\          2/7*a\\
                                |1 - tan |---|   1 - tan |---||
/       2/         tan(5*a)  \\ |        \ 2 /           \ 2 /|
|1 + tan |7*a - -------------||*|------------- - -------------|
|        |             2     || |       2/3*a\          2/7*a\|
\        \      1 + tan (5*a)// |1 + tan |---|   1 + tan |---||
                                \        \ 2 /           \ 2 //

$$\frac{2 \tan{\left(7 a - \frac{\tan{\left(5 a \right)}}{\tan^{2}{\left(5 a \right)} + 1} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{3 a}{2} \right)}}{\tan^{2}{\left(\frac{3 a}{2} \right)} + 1} - \frac{1 - \tan^{2}{\left(\frac{7 a}{2} \right)}}{\tan^{2}{\left(\frac{7 a}{2} \right)} + 1}\right) \left(\tan^{2}{\left(7 a - \frac{\tan{\left(5 a \right)}}{\tan^{2}{\left(5 a \right)} + 1} \right)} + 1\right)}$$

                                /      sin(10*a)\    /      sin(10*a)\
(1 + cos(-sin(10*a) + 14*a))*sec|7*a - ---------|*sin|7*a - ---------|
                                \          2    /    \          2    /
----------------------------------------------------------------------
                         -cos(7*a) + cos(3*a)

$$\frac{\left(\cos{\left(14 a - \sin{\left(10 a \right)} \right)} + 1\right) \sin{\left(7 a - \frac{\sin{\left(10 a \right)}}{2} \right)} \sec{\left(7 a - \frac{\sin{\left(10 a \right)}}{2} \right)}}{\cos{\left(3 a \right)} - \cos{\left(7 a \right)}}$$

                           1                           
-------------------------------------------------------
/   1          1    \    /        1                 pi\
|-------- - --------|*sec|- -------------- + 14*a - --|
\sec(3*a)   sec(7*a)/    |     /       pi\          2 |
                         |  sec|10*a - --|            |
                         \     \       2 /            /

$$\frac{1}{\left(- \frac{1}{\sec{\left(7 a \right)}} + \frac{1}{\sec{\left(3 a \right)}}\right) \sec{\left(14 a - \frac{\pi}{2} - \frac{1}{\sec{\left(10 a - \frac{\pi}{2} \right)}} \right)}}$$

                           1                           
-------------------------------------------------------
/      1               1      \    /      1           \
|------------- - -------------|*csc|- --------- + 14*a|
|   /pi      \      /pi      \|    \  csc(10*a)       /
|csc|-- - 3*a|   csc|-- - 7*a||                        
\   \2       /      \2       //

$$\frac{1}{\left(\frac{1}{\csc{\left(- 3 a + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(- 7 a + \frac{\pi}{2} \right)}}\right) \csc{\left(14 a - \frac{1}{\csc{\left(10 a \right)}} \right)}}$$

                   /      sin(10*a)\              
              2*tan|7*a - ---------|              
                   \          2    /              
--------------------------------------------------
/       2/      sin(10*a)\\                       
|1 + tan |7*a - ---------||*(-cos(7*a) + cos(3*a))
\        \          2    //

$$\frac{2 \tan{\left(7 a - \frac{\sin{\left(10 a \right)}}{2} \right)}}{\left(\cos{\left(3 a \right)} - \cos{\left(7 a \right)}\right) \left(\tan^{2}{\left(7 a - \frac{\sin{\left(10 a \right)}}{2} \right)} + 1\right)}$$

   /     /       pi\          pi\
cos|- cos|10*a - --| + 14*a - --|
   \     \       2 /          2 /
---------------------------------
       -cos(7*a) + cos(3*a)

$$\frac{\cos{\left(14 a - \cos{\left(10 a - \frac{\pi}{2} \right)} - \frac{\pi}{2} \right)}}{\cos{\left(3 a \right)} - \cos{\left(7 a \right)}}$$

                         /         cot(5*a)  \                   
                    2*cot|7*a - -------------|                   
                         |             2     |                   
                         \      1 + cot (5*a)/                   
-----------------------------------------------------------------
                                /        2/3*a\           2/7*a\\
                                |-1 + cot |---|   -1 + cot |---||
/       2/         cot(5*a)  \\ |         \ 2 /            \ 2 /|
|1 + cot |7*a - -------------||*|-------------- - --------------|
|        |             2     || |       2/3*a\           2/7*a\ |
\        \      1 + cot (5*a)// |1 + cot |---|    1 + cot |---| |
                                \        \ 2 /            \ 2 / /

$$\frac{2 \cot{\left(7 a - \frac{\cot{\left(5 a \right)}}{\cot^{2}{\left(5 a \right)} + 1} \right)}}{\left(\frac{\cot^{2}{\left(\frac{3 a}{2} \right)} - 1}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} - \frac{\cot^{2}{\left(\frac{7 a}{2} \right)} - 1}{\cot^{2}{\left(\frac{7 a}{2} \right)} + 1}\right) \left(\cot^{2}{\left(7 a - \frac{\cot{\left(5 a \right)}}{\cot^{2}{\left(5 a \right)} + 1} \right)} + 1\right)}$$

                                   /pi        1           \                              
                              2*csc|-- + ----------- - 7*a|                              
                                   \2    2*csc(10*a)      /                              
-----------------------------------------------------------------------------------------
/       2/pi        1           \\                                                       
|    csc |-- + ----------- - 7*a||                                                       
|        \2    2*csc(10*a)      /| /      1               1      \    /           1     \
|1 + ----------------------------|*|------------- - -------------|*csc|7*a - -----------|
|         2/           1     \   | |   /pi      \      /pi      \|    \      2*csc(10*a)/
|      csc |7*a - -----------|   | |csc|-- - 3*a|   csc|-- - 7*a||                       
\          \      2*csc(10*a)/   / \   \2       /      \2       //

$$\frac{2 \csc{\left(- 7 a + \frac{\pi}{2} + \frac{1}{2 \csc{\left(10 a \right)}} \right)}}{\left(1 + \frac{\csc^{2}{\left(- 7 a + \frac{\pi}{2} + \frac{1}{2 \csc{\left(10 a \right)}} \right)}}{\csc^{2}{\left(7 a - \frac{1}{2 \csc{\left(10 a \right)}} \right)}}\right) \left(\frac{1}{\csc{\left(- 3 a + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(- 7 a + \frac{\pi}{2} \right)}}\right) \csc{\left(7 a - \frac{1}{2 \csc{\left(10 a \right)}} \right)}}$$

                                /      sin(10*a)\
(1 + cos(-sin(10*a) + 14*a))*sin|7*a - ---------|
                                \          2    /
-------------------------------------------------
                             /      sin(10*a)\   
   (-cos(7*a) + cos(3*a))*cos|7*a - ---------|   
                             \          2    /

$$\frac{\left(\cos{\left(14 a - \sin{\left(10 a \right)} \right)} + 1\right) \sin{\left(7 a - \frac{\sin{\left(10 a \right)}}{2} \right)}}{\left(\cos{\left(3 a \right)} - \cos{\left(7 a \right)}\right) \cos{\left(7 a - \frac{\sin{\left(10 a \right)}}{2} \right)}}$$

                                /              /       pi\\                           
                                |           cos|10*a - --||                           
                                |      pi      \       2 /|                           
                           2*cos|7*a - -- - --------------|                           
                                \      2          2       /                           
--------------------------------------------------------------------------------------
/        /              /       pi\\\                                                 
|        |           cos|10*a - --|||                                                 
|       2|      pi      \       2 /||                           /         /       pi\\
|    cos |7*a - -- - --------------||                           |      cos|10*a - --||
|        \      2          2       /|                           |         \       2 /|
|1 + -------------------------------|*(-cos(7*a) + cos(3*a))*cos|7*a - --------------|
|           /         /       pi\\  |                           \            2       /
|           |      cos|10*a - --||  |                                                 
|          2|         \       2 /|  |                                                 
|       cos |7*a - --------------|  |                                                 
\           \            2       /  /

$$\frac{2 \cos{\left(7 a - \frac{\cos{\left(10 a - \frac{\pi}{2} \right)}}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(7 a - \frac{\cos{\left(10 a - \frac{\pi}{2} \right)}}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(7 a - \frac{\cos{\left(10 a - \frac{\pi}{2} \right)}}{2} \right)}}\right) \left(\cos{\left(3 a \right)} - \cos{\left(7 a \right)}\right) \cos{\left(7 a - \frac{\cos{\left(10 a - \frac{\pi}{2} \right)}}{2} \right)}}$$

                               2/      sin(10*a)\                          
                          4*sin |7*a - ---------|                          
                                \          2    /                          
---------------------------------------------------------------------------
/         4/      sin(10*a)\\                                              
|    4*sin |7*a - ---------||                                              
|          \          2    /|                                              
|1 + -----------------------|*(-cos(7*a) + cos(3*a))*sin(-sin(10*a) + 14*a)
|       2                   |                                              
\    sin (-sin(10*a) + 14*a)/

$$\frac{4 \sin^{2}{\left(7 a - \frac{\sin{\left(10 a \right)}}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(7 a - \frac{\sin{\left(10 a \right)}}{2} \right)}}{\sin^{2}{\left(14 a - \sin{\left(10 a \right)} \right)}} + 1\right) \left(\cos{\left(3 a \right)} - \cos{\left(7 a \right)}\right) \sin{\left(14 a - \sin{\left(10 a \right)} \right)}}$$

                                            2                                             
------------------------------------------------------------------------------------------
                                /        2/3*a\           2/7*a\\                         
                                |-1 + cot |---|   -1 + cot |---||                         
/                1            \ |         \ 2 /            \ 2 /|    /         cot(5*a)  \
|1 + -------------------------|*|-------------- - --------------|*cot|7*a - -------------|
|       2/         cot(5*a)  \| |       2/3*a\           2/7*a\ |    |             2     |
|    cot |7*a - -------------|| |1 + cot |---|    1 + cot |---| |    \      1 + cot (5*a)/
|        |             2     || \        \ 2 /            \ 2 / /                         
\        \      1 + cot (5*a)//

$$\frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(7 a - \frac{\cot{\left(5 a \right)}}{\cot^{2}{\left(5 a \right)} + 1} \right)}}\right) \left(\frac{\cot^{2}{\left(\frac{3 a}{2} \right)} - 1}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} - \frac{\cot^{2}{\left(\frac{7 a}{2} \right)} - 1}{\cot^{2}{\left(\frac{7 a}{2} \right)} + 1}\right) \cot{\left(7 a - \frac{\cot{\left(5 a \right)}}{\cot^{2}{\left(5 a \right)} + 1} \right)}}$$

                                   /           1     \                              
                              2*sec|7*a - -----------|                              
                                   \      2*csc(10*a)/                              
------------------------------------------------------------------------------------
/         2/           1     \   \                                                  
|      sec |7*a - -----------|   |                                                  
|          \      2*csc(10*a)/   | /   1          1    \    /      pi        1     \
|1 + ----------------------------|*|-------- - --------|*sec|7*a - -- - -----------|
|       2/      pi        1     \| \sec(3*a)   sec(7*a)/    \      2    2*csc(10*a)/
|    sec |7*a - -- - -----------||                                                  
\        \      2    2*csc(10*a)//

$$\frac{2 \sec{\left(7 a - \frac{1}{2 \csc{\left(10 a \right)}} \right)}}{\left(\frac{\sec^{2}{\left(7 a - \frac{1}{2 \csc{\left(10 a \right)}} \right)}}{\sec^{2}{\left(7 a - \frac{\pi}{2} - \frac{1}{2 \csc{\left(10 a \right)}} \right)}} + 1\right) \left(- \frac{1}{\sec{\left(7 a \right)}} + \frac{1}{\sec{\left(3 a \right)}}\right) \sec{\left(7 a - \frac{\pi}{2} - \frac{1}{2 \csc{\left(10 a \right)}} \right)}}$$

     sin(-sin(10*a) + 14*a)    
-------------------------------
     /pi      \      /pi      \
- sin|-- + 7*a| + sin|-- + 3*a|
     \2       /      \2       /

$$\frac{\sin{\left(14 a - \sin{\left(10 a \right)} \right)}}{\sin{\left(3 a + \frac{\pi}{2} \right)} - \sin{\left(7 a + \frac{\pi}{2} \right)}}$$

                                    2/      sin(10*a)\                                
                               4*sin |7*a - ---------|                                
                                     \          2    /                                
--------------------------------------------------------------------------------------
/         4/      sin(10*a)\\                                                         
|    4*sin |7*a - ---------||                                                         
|          \          2    /| /     /pi      \      /pi      \\                       
|1 + -----------------------|*|- sin|-- + 7*a| + sin|-- + 3*a||*sin(-sin(10*a) + 14*a)
|       2                   | \     \2       /      \2       //                       
\    sin (-sin(10*a) + 14*a)/

$$\frac{4 \sin^{2}{\left(7 a - \frac{\sin{\left(10 a \right)}}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(7 a - \frac{\sin{\left(10 a \right)}}{2} \right)}}{\sin^{2}{\left(14 a - \sin{\left(10 a \right)} \right)}} + 1\right) \left(\sin{\left(3 a + \frac{\pi}{2} \right)} - \sin{\left(7 a + \frac{\pi}{2} \right)}\right) \sin{\left(14 a - \sin{\left(10 a \right)} \right)}}$$

                      1                      
---------------------------------------------
/   1          1    \    /      1           \
|-------- - --------|*csc|- --------- + 14*a|
\sec(3*a)   sec(7*a)/    \  csc(10*a)       /

$$\frac{1}{\left(- \frac{1}{\sec{\left(7 a \right)}} + \frac{1}{\sec{\left(3 a \right)}}\right) \csc{\left(14 a - \frac{1}{\csc{\left(10 a \right)}} \right)}}$$

1/((1/sec(3*a) - 1/sec(7*a))*csc(-1/csc(10*a) + 14*a))

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Expresiones con funciones

¿Cómo vas a descomponer esta sin(14*a-sin(10*a))/(cos(3*a)-cos(7*a)) expresión en fracciones?

Solución

¿Cómo vas a descomponer esta sin(14a-sin(10a))/(cos(3a)-cos(7a)) expresión en fracciones?