Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sin(a)/((sin(a)-1)*(cos(a)-((sin(a)*cos(a))/(sin(a)-1))))
¿Cómo vas a descomponer esta sin(a)/((sin(a)-1)*(cos(a)-((sin(a)*cos(a))/(sin(a)-1)))) expresión en fracciones?
Solución
Ha introducido
[src]
sin(a)
-------------------------------------
/ sin(a)*cos(a)\
(sin(a) - 1)*|cos(a) - -------------|
\ sin(a) - 1 /
$$\frac{\sin{\left(a \right)}}{\left(- \frac{\sin{\left(a \right)} \cos{\left(a \right)}}{\sin{\left(a \right)} - 1} + \cos{\left(a \right)}\right) \left(\sin{\left(a \right)} - 1\right)}$$
sin(a)/(((sin(a) - 1)*(cos(a) - sin(a)*cos(a)/(sin(a) - 1))))
Denominador racional
[src]
-sin(a) -------- cos(a)
$$- \frac{\sin{\left(a \right)}}{\cos{\left(a \right)}}$$
-sin(a)/cos(a)
Respuesta numérica
[src]
sin(a)/((-1.0 + sin(a))*(-cos(a)*sin(a)/(-1.0 + sin(a)) + cos(a)))
sin(a)/((-1.0 + sin(a))*(-cos(a)*sin(a)/(-1.0 + sin(a)) + cos(a)))
Combinatoria
[src]
-sin(a) -------- cos(a)
$$- \frac{\sin{\left(a \right)}}{\cos{\left(a \right)}}$$
-sin(a)/cos(a)
Unión de expresiones racionales
[src]
-sin(a) -------- cos(a)
$$- \frac{\sin{\left(a \right)}}{\cos{\left(a \right)}}$$
-sin(a)/cos(a)
Potencias
[src]
/ -I*a I*a\
-I*\- e + e /
------------------------------------------------------------------------------
/ / I*a -I*a\ \
| |e e | / -I*a I*a\|
/ / -I*a I*a\\ | I*a -I*a I*|---- + -----|*\- e + e /|
| I*\- e + e /| |e e \ 2 2 / |
2*|-1 - ------------------|*|---- + ----- + ---------------------------------|
\ 2 / | 2 2 / / -I*a I*a\\ |
| | I*\- e + e /| |
| 2*|-1 - ------------------| |
\ \ 2 / /
$$- \frac{i \left(e^{i a} - e^{- i a}\right)}{2 \left(- \frac{i \left(e^{i a} - e^{- i a}\right)}{2} - 1\right) \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2} + \frac{i \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right) \left(e^{i a} - e^{- i a}\right)}{2 \left(- \frac{i \left(e^{i a} - e^{- i a}\right)}{2} - 1\right)}\right)}$$
sin(a)
----------------------------------------
/ cos(a)*sin(a) \
(-1 + sin(a))*|- ------------- + cos(a)|
\ -1 + sin(a) /
$$\frac{\sin{\left(a \right)}}{\left(\sin{\left(a \right)} - 1\right) \left(\cos{\left(a \right)} - \frac{\sin{\left(a \right)} \cos{\left(a \right)}}{\sin{\left(a \right)} - 1}\right)}$$
sin(a)/((-1 + sin(a))*(-cos(a)*sin(a)/(-1 + sin(a)) + cos(a)))
Parte trigonométrica
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-sec(a) ----------- / pi\ sec|a - --| \ 2 /
$$- \frac{\sec{\left(a \right)}}{\sec{\left(a - \frac{\pi}{2} \right)}}$$
1
-------------------------------------------------------------------------------
/ 1 \ / 1 1 \ / pi\
|-1 + -----------|*|------ - -------------------------------------|*sec|a - --|
| / pi\| |sec(a) / 1 \ / pi\| \ 2 /
| sec|a - --|| | |-1 + -----------|*sec(a)*sec|a - --||
\ \ 2 // | | / pi\| \ 2 /|
| | sec|a - --|| |
\ \ \ 2 // /
$$\frac{1}{\left(-1 + \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}\right) \left(\frac{1}{\sec{\left(a \right)}} - \frac{1}{\left(-1 + \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}\right) \sec{\left(a \right)} \sec{\left(a - \frac{\pi}{2} \right)}}\right) \sec{\left(a - \frac{\pi}{2} \right)}}$$
sin(a)
----------------------------------------
/ cos(a)*sin(a) \
(-1 + sin(a))*|- ------------- + cos(a)|
\ -1 + sin(a) /
$$\frac{\sin{\left(a \right)}}{\left(\sin{\left(a \right)} - 1\right) \left(\cos{\left(a \right)} - \frac{\sin{\left(a \right)} \cos{\left(a \right)}}{\sin{\left(a \right)} - 1}\right)}$$
/a\
2*tan|-|
\2/
----------------------------------------------------------------------------------
/ /a\ \ / 2/a\ / 2/a\\ /a\ \
| 2*tan|-| | |1 - tan |-| 2*|1 - tan |-||*tan|-| |
/ 2/a\\ | \2/ | | \2/ \ \2// \2/ |
|1 + tan |-||*|-1 + -----------|*|----------- - ---------------------------------|
\ \2// | 2/a\| | 2/a\ / /a\ \|
| 1 + tan |-|| |1 + tan |-| 2 | 2*tan|-| ||
\ \2// | \2/ / 2/a\\ | \2/ ||
| |1 + tan |-|| *|-1 + -----------||
| \ \2// | 2/a\||
| | 1 + tan |-|||
\ \ \2///
$$\frac{2 \tan{\left(\frac{a}{2} \right)}}{\left(-1 + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\frac{1 - \tan^{2}{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} - \frac{2 \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right) \tan{\left(\frac{a}{2} \right)}}{\left(-1 + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
sin(a)
--------------------------------------------------
/ / pi\ \
| sin(a)*sin|a + --| |
| \ 2 / / pi\|
(-1 + sin(a))*|- ------------------ + sin|a + --||
\ -1 + sin(a) \ 2 //
$$\frac{\sin{\left(a \right)}}{\left(\sin{\left(a \right)} - 1\right) \left(\sin{\left(a + \frac{\pi}{2} \right)} - \frac{\sin{\left(a \right)} \sin{\left(a + \frac{\pi}{2} \right)}}{\sin{\left(a \right)} - 1}\right)}$$
-tan(a)
$$- \tan{\left(a \right)}$$
/a\
2*cot|-|
\2/
-----------------------------------------------------------------------------------
/ /a\ \ / 2/a\ / 2/a\\ /a\ \
| 2*cot|-| | |-1 + cot |-| 2*|-1 + cot |-||*cot|-| |
/ 2/a\\ | \2/ | | \2/ \ \2// \2/ |
|1 + cot |-||*|-1 + -----------|*|------------ - ---------------------------------|
\ \2// | 2/a\| | 2/a\ / /a\ \|
| 1 + cot |-|| |1 + cot |-| 2 | 2*cot|-| ||
\ \2// | \2/ / 2/a\\ | \2/ ||
| |1 + cot |-|| *|-1 + -----------||
| \ \2// | 2/a\||
| | 1 + cot |-|||
\ \ \2///
$$\frac{2 \cot{\left(\frac{a}{2} \right)}}{\left(-1 + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} - \frac{2 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right) \cot{\left(\frac{a}{2} \right)}}{\left(-1 + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
/ pi\
-cos|a - --|
\ 2 /
-------------
cos(a)
$$- \frac{\cos{\left(a - \frac{\pi}{2} \right)}}{\cos{\left(a \right)}}$$
/ pi\
cos|a - --|
\ 2 /
--------------------------------------------------
/ / pi\ \
| cos(a)*cos|a - --| |
/ / pi\\ | \ 2 / |
|-1 + cos|a - --||*|- ------------------ + cos(a)|
\ \ 2 // | / pi\ |
| -1 + cos|a - --| |
\ \ 2 / /
$$\frac{\cos{\left(a - \frac{\pi}{2} \right)}}{\left(\cos{\left(a \right)} - \frac{\cos{\left(a \right)} \cos{\left(a - \frac{\pi}{2} \right)}}{\cos{\left(a - \frac{\pi}{2} \right)} - 1}\right) \left(\cos{\left(a - \frac{\pi}{2} \right)} - 1\right)}$$
1
---------------------------------------------------------------------
/ 1 \ / 1 1 \
|-1 + ------|*|----------- - --------------------------------|*csc(a)
\ csc(a)/ | /pi \ / 1 \ /pi \|
|csc|-- - a| |-1 + ------|*csc(a)*csc|-- - a||
\ \2 / \ csc(a)/ \2 //
$$\frac{1}{\left(-1 + \frac{1}{\csc{\left(a \right)}}\right) \left(\frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}} - \frac{1}{\left(-1 + \frac{1}{\csc{\left(a \right)}}\right) \csc{\left(a \right)} \csc{\left(- a + \frac{\pi}{2} \right)}}\right) \csc{\left(a \right)}}$$
-1 ------ cot(a)
$$- \frac{1}{\cot{\left(a \right)}}$$
2 -2*sin (a) ---------- sin(2*a)
$$- \frac{2 \sin^{2}{\left(a \right)}}{\sin{\left(2 a \right)}}$$
1
-----------------------------------------------------------
/ 1 \ / 1 1 \
|-1 + ------|*|------ - ---------------------------|*csc(a)
\ csc(a)/ |sec(a) / 1 \ |
| |-1 + ------|*csc(a)*sec(a)|
\ \ csc(a)/ /
$$\frac{1}{\left(-1 + \frac{1}{\csc{\left(a \right)}}\right) \left(\frac{1}{\sec{\left(a \right)}} - \frac{1}{\left(-1 + \frac{1}{\csc{\left(a \right)}}\right) \csc{\left(a \right)} \sec{\left(a \right)}}\right) \csc{\left(a \right)}}$$
/pi \
-csc|-- - a|
\2 /
-------------
csc(a)
$$- \frac{\csc{\left(- a + \frac{\pi}{2} \right)}}{\csc{\left(a \right)}}$$
-csc(pi/2 - a)/csc(a)
Denominador común
[src]
-sin(a) -------- cos(a)
$$- \frac{\sin{\left(a \right)}}{\cos{\left(a \right)}}$$
-sin(a)/cos(a)
Compilar la expresión
[src]
sin(a)
----------------------------------------
/ cos(a)*sin(a) \
(-1 + sin(a))*|- ------------- + cos(a)|
\ -1 + sin(a) /
$$\frac{\sin{\left(a \right)}}{\left(\sin{\left(a \right)} - 1\right) \left(\cos{\left(a \right)} - \frac{\sin{\left(a \right)} \cos{\left(a \right)}}{\sin{\left(a \right)} - 1}\right)}$$
sin(a)/((-1 + sin(a))*(-cos(a)*sin(a)/(-1 + sin(a)) + cos(a)))