Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sin(a)*cos(a)/(sin(a^2)-cos(a^2))
¿Cómo vas a descomponer esta sin(a)*cos(a)/(sin(a^2)-cos(a^2)) expresión en fracciones?
Solución
Ha introducido
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sin(a)*cos(a) ----------------- / 2\ / 2\ sin\a / - cos\a /
$$\frac{\sin{\left(a \right)} \cos{\left(a \right)}}{\sin{\left(a^{2} \right)} - \cos{\left(a^{2} \right)}}$$
(sin(a)*cos(a))/(sin(a^2) - cos(a^2))
Simplificación general
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___
-\/ 2 *sin(2*a)
----------------
/ 2 pi\
4*cos|a + --|
\ 4 /
$$- \frac{\sqrt{2} \sin{\left(2 a \right)}}{4 \cos{\left(a^{2} + \frac{\pi}{4} \right)}}$$
-sqrt(2)*sin(2*a)/(4*cos(a^2 + pi/4))
Combinatoria
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-cos(a)*sin(a)
-------------------
/ 2\ / 2\
- sin\a / + cos\a /
$$- \frac{\sin{\left(a \right)} \cos{\left(a \right)}}{- \sin{\left(a^{2} \right)} + \cos{\left(a^{2} \right)}}$$
-cos(a)*sin(a)/(-sin(a^2) + cos(a^2))
Denominador común
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-cos(a)*sin(a)
-------------------
/ 2\ / 2\
- sin\a / + cos\a /
$$- \frac{\sin{\left(a \right)} \cos{\left(a \right)}}{- \sin{\left(a^{2} \right)} + \cos{\left(a^{2} \right)}}$$
-cos(a)*sin(a)/(-sin(a^2) + cos(a^2))
Potencias
[src]
/ I*a -I*a\
|e e | / -I*a I*a\
-I*|---- + -----|*\- e + e /
\ 2 2 /
-------------------------------------------
/ 2 2 / 2 2\\
| I*a -I*a | -I*a I*a ||
| e e I*\- e + e /|
2*|- ----- - ------ - --------------------|
\ 2 2 2 /
$$- \frac{i \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right) \left(e^{i a} - e^{- i a}\right)}{2 \left(- \frac{i \left(e^{i a^{2}} - e^{- i a^{2}}\right)}{2} - \frac{e^{i a^{2}}}{2} - \frac{e^{- i a^{2}}}{2}\right)}$$
-i*(exp(i*a)/2 + exp(-i*a)/2)*(-exp(-i*a) + exp(i*a))/(2*(-exp(i*a^2)/2 - exp(-i*a^2)/2 - i*(-exp(-i*a^2) + exp(i*a^2))/2))
Parte trigonométrica
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___ / 2 pi\
-\/ 2 *csc|- a + --|
\ 4 /
----------------------
4*csc(2*a)
$$- \frac{\sqrt{2} \csc{\left(- a^{2} + \frac{\pi}{4} \right)}}{4 \csc{\left(2 a \right)}}$$
___ / 2 pi\
-\/ 2 *sec|a + --|
\ 4 /
--------------------
/ pi\
4*sec|2*a - --|
\ 2 /
$$- \frac{\sqrt{2} \sec{\left(a^{2} + \frac{\pi}{4} \right)}}{4 \sec{\left(2 a - \frac{\pi}{2} \right)}}$$
/ / 2 \\
___ 2 | 2|a pi||
-\/ 2 *cos (a)*|1 + tan |-- + --||*tan(a)
\ \2 8 //
------------------------------------------
/ / 2 \\
| 2|a pi||
2*|1 - tan |-- + --||
\ \2 8 //
$$- \frac{\sqrt{2} \left(\tan^{2}{\left(\frac{a^{2}}{2} + \frac{\pi}{8} \right)} + 1\right) \cos^{2}{\left(a \right)} \tan{\left(a \right)}}{2 \left(1 - \tan^{2}{\left(\frac{a^{2}}{2} + \frac{\pi}{8} \right)}\right)}$$
___
-\/ 2 *sin(2*a)
----------------
/ 2 3*pi\
4*sin|a + ----|
\ 4 /
$$- \frac{\sqrt{2} \sin{\left(2 a \right)}}{4 \sin{\left(a^{2} + \frac{3 \pi}{4} \right)}}$$
/ 2/a\\ /a\
2*|-1 + cot |-||*cot|-|
\ \2// \2/
-----------------------------------------------
/ / 2\ / 2\ \
| 2|a | |a | |
2 | -1 + cot |--| 2*cot|--| |
/ 2/a\\ | \2 / \2 / |
|1 + cot |-|| *|- ------------- + ------------|
\ \2// | / 2\ / 2\|
| 2|a | 2|a ||
| 1 + cot |--| 1 + cot |--||
\ \2 / \2 //
$$\frac{2 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right) \cot{\left(\frac{a}{2} \right)}}{\left(- \frac{\cot^{2}{\left(\frac{a^{2}}{2} \right)} - 1}{\cot^{2}{\left(\frac{a^{2}}{2} \right)} + 1} + \frac{2 \cot{\left(\frac{a^{2}}{2} \right)}}{\cot^{2}{\left(\frac{a^{2}}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
___
-\/ 2 *sin(2*a)
----------------
/ 2 pi\
4*cos|a + --|
\ 4 /
$$- \frac{\sqrt{2} \sin{\left(2 a \right)}}{4 \cos{\left(a^{2} + \frac{\pi}{4} \right)}}$$
___
\/ 2 *sin(2*a)
-----------------------------------------------------------
/ ___________ ___________ \
| ___ / 2\ / ___ / ___ / 2\|
2*\- \/ 2 *cos\a / + \/ 2 + \/ 2 *\/ 2 - \/ 2 *sin\a //
$$\frac{\sqrt{2} \sin{\left(2 a \right)}}{2 \left(\sqrt{2 - \sqrt{2}} \sqrt{\sqrt{2} + 2} \sin{\left(a^{2} \right)} - \sqrt{2} \cos{\left(a^{2} \right)}\right)}$$
1 ------------------------------------------- / 1 1 \ /pi \ |------- - ------------|*csc(a)*csc|-- - a| | / 2\ /pi 2\| \2 / |csc\a / csc|-- - a || \ \2 //
$$\frac{1}{\left(- \frac{1}{\csc{\left(- a^{2} + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(a^{2} \right)}}\right) \csc{\left(a \right)} \csc{\left(- a + \frac{\pi}{2} \right)}}$$
/ / 2 \\
___ | 2|a pi||
-\/ 2 *|1 + tan |-- + --||*tan(a)
\ \2 8 //
-----------------------------------
/ / 2 \\
/ 2 \ | 2|a pi||
2*\1 + tan (a)/*|1 - tan |-- + --||
\ \2 8 //
$$- \frac{\sqrt{2} \left(\tan^{2}{\left(\frac{a^{2}}{2} + \frac{\pi}{8} \right)} + 1\right) \tan{\left(a \right)}}{2 \left(1 - \tan^{2}{\left(\frac{a^{2}}{2} + \frac{\pi}{8} \right)}\right) \left(\tan^{2}{\left(a \right)} + 1\right)}$$
1 --------------------------------- / 1 1 \ |------- - -------|*csc(a)*sec(a) | / 2\ / 2\| \csc\a / sec\a //
$$\frac{1}{\left(- \frac{1}{\sec{\left(a^{2} \right)}} + \frac{1}{\csc{\left(a^{2} \right)}}\right) \csc{\left(a \right)} \sec{\left(a \right)}}$$
___ / 2 pi\
-\/ 2 *csc|- a + --|*sin(2*a)
\ 4 /
-------------------------------
4
$$- \frac{\sqrt{2} \sin{\left(2 a \right)} \csc{\left(- a^{2} + \frac{\pi}{4} \right)}}{4}$$
1 ------------------------------------------- / 1 1 \ / pi\ |------------ - -------|*sec(a)*sec|a - --| | / 2 pi\ / 2\| \ 2 / |sec|a - --| sec\a /| \ \ 2 / /
$$\frac{1}{\left(\frac{1}{\sec{\left(a^{2} - \frac{\pi}{2} \right)}} - \frac{1}{\sec{\left(a^{2} \right)}}\right) \sec{\left(a \right)} \sec{\left(a - \frac{\pi}{2} \right)}}$$
___ / 2 pi\
-\/ 2 *sec|a + --|*sin(2*a)
\ 4 /
-----------------------------
4
$$- \frac{\sqrt{2} \sin{\left(2 a \right)} \sec{\left(a^{2} + \frac{\pi}{4} \right)}}{4}$$
/ / 2 \\
___ | 2|a pi||
-\/ 2 *|1 + cot |-- + --||*cot(a)
\ \2 8 //
------------------------------------
/ / 2 \\
/ 2 \ | 2|a pi||
2*\1 + cot (a)/*|-1 + cot |-- + --||
\ \2 8 //
$$- \frac{\sqrt{2} \left(\cot^{2}{\left(\frac{a^{2}}{2} + \frac{\pi}{8} \right)} + 1\right) \cot{\left(a \right)}}{2 \left(\cot^{2}{\left(a \right)} + 1\right) \left(\cot^{2}{\left(\frac{a^{2}}{2} + \frac{\pi}{8} \right)} - 1\right)}$$
/ 2/a\\ /a\
2*|1 - tan |-||*tan|-|
\ \2// \2/
----------------------------------------------
/ / 2\ / 2\ \
| 2|a | |a | |
2 | 1 - tan |--| 2*tan|--| |
/ 2/a\\ | \2 / \2 / |
|1 + tan |-|| *|- ------------ + ------------|
\ \2// | / 2\ / 2\|
| 2|a | 2|a ||
| 1 + tan |--| 1 + tan |--||
\ \2 / \2 //
$$\frac{2 \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right) \tan{\left(\frac{a}{2} \right)}}{\left(- \frac{1 - \tan^{2}{\left(\frac{a^{2}}{2} \right)}}{\tan^{2}{\left(\frac{a^{2}}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{a^{2}}{2} \right)}}{\tan^{2}{\left(\frac{a^{2}}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
/ pi\
sin(a)*sin|a + --|
\ 2 /
------------------------
/ 2 pi\ / 2\
- sin|a + --| + sin\a /
\ 2 /
$$\frac{\sin{\left(a \right)} \sin{\left(a + \frac{\pi}{2} \right)}}{\sin{\left(a^{2} \right)} - \sin{\left(a^{2} + \frac{\pi}{2} \right)}}$$
/ pi\
cos(a)*cos|a - --|
\ 2 /
------------------------
/ 2\ / 2 pi\
- cos\a / + cos|a - --|
\ 2 /
$$\frac{\cos{\left(a \right)} \cos{\left(a - \frac{\pi}{2} \right)}}{- \cos{\left(a^{2} \right)} + \cos{\left(a^{2} - \frac{\pi}{2} \right)}}$$
___ / pi\
-\/ 2 *cos|2*a - --|
\ 2 /
---------------------
/ 2 pi\
4*cos|a + --|
\ 4 /
$$- \frac{\sqrt{2} \cos{\left(2 a - \frac{\pi}{2} \right)}}{4 \cos{\left(a^{2} + \frac{\pi}{4} \right)}}$$
-sqrt(2)*cos(2*a - pi/2)/(4*cos(a^2 + pi/4))