Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sin(a)/(1-sin(a))-cos(a)/(sin(a))
¿Cómo vas a descomponer esta sin(a)/(1-sin(a))-cos(a)/(sin(a)) expresión en fracciones?
Solución
Ha introducido
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sin(a) cos(a) ---------- - ------ 1 - sin(a) sin(a)
$$- \frac{\cos{\left(a \right)}}{\sin{\left(a \right)}} + \frac{\sin{\left(a \right)}}{1 - \sin{\left(a \right)}}$$
sin(a)/(1 - sin(a)) - cos(a)/sin(a)
Simplificación general
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sin(a) cos(a) - ----------- - ------ -1 + sin(a) sin(a)
$$- \frac{\cos{\left(a \right)}}{\sin{\left(a \right)}} - \frac{\sin{\left(a \right)}}{\sin{\left(a \right)} - 1}$$
-sin(a)/(-1 + sin(a)) - cos(a)/sin(a)
Respuesta numérica
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sin(a)/(1.0 - sin(a)) - cos(a)/sin(a)
sin(a)/(1.0 - sin(a)) - cos(a)/sin(a)
Denominador racional
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2
sin (a) - (1 - sin(a))*cos(a)
-----------------------------
(1 - sin(a))*sin(a)
$$\frac{- \left(1 - \sin{\left(a \right)}\right) \cos{\left(a \right)} + \sin^{2}{\left(a \right)}}{\left(1 - \sin{\left(a \right)}\right) \sin{\left(a \right)}}$$
(sin(a)^2 - (1 - sin(a))*cos(a))/((1 - sin(a))*sin(a))
Potencias
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/ I*a -I*a\
|e e |
2*I*|---- + -----| / -I*a I*a\
\ 2 2 / I*\- e + e /
- ------------------ - --------------------------
-I*a I*a / / -I*a I*a\\
- e + e | I*\- e + e /|
2*|1 + ------------------|
\ 2 /
$$- \frac{2 i \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right)}{e^{i a} - e^{- i a}} - \frac{i \left(e^{i a} - e^{- i a}\right)}{2 \left(\frac{i \left(e^{i a} - e^{- i a}\right)}{2} + 1\right)}$$
-2*i*(exp(i*a)/2 + exp(-i*a)/2)/(-exp(-i*a) + exp(i*a)) - i*(-exp(-i*a) + exp(i*a))/(2*(1 + i*(-exp(-i*a) + exp(i*a))/2))
Combinatoria
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/ 2 \
-\sin (a) - cos(a) + cos(a)*sin(a)/
------------------------------------
(-1 + sin(a))*sin(a)
$$- \frac{\sin^{2}{\left(a \right)} + \sin{\left(a \right)} \cos{\left(a \right)} - \cos{\left(a \right)}}{\left(\sin{\left(a \right)} - 1\right) \sin{\left(a \right)}}$$
-(sin(a)^2 - cos(a) + cos(a)*sin(a))/((-1 + sin(a))*sin(a))
Unión de expresiones racionales
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2
sin (a) - (1 - sin(a))*cos(a)
-----------------------------
(1 - sin(a))*sin(a)
$$\frac{- \left(1 - \sin{\left(a \right)}\right) \cos{\left(a \right)} + \sin^{2}{\left(a \right)}}{\left(1 - \sin{\left(a \right)}\right) \sin{\left(a \right)}}$$
(sin(a)^2 - (1 - sin(a))*cos(a))/((1 - sin(a))*sin(a))
Denominador común
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-cos(a) + cos(a)*sin(a) + sin(a)
-1 - --------------------------------
2
sin (a) - sin(a)
$$-1 - \frac{\sin{\left(a \right)} \cos{\left(a \right)} + \sin{\left(a \right)} - \cos{\left(a \right)}}{\sin^{2}{\left(a \right)} - \sin{\left(a \right)}}$$
-1 - (-cos(a) + cos(a)*sin(a) + sin(a))/(sin(a)^2 - sin(a))
Parte trigonométrica
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/a\
tan|-|
\2/ 1 sin(a)
------ - -------- - -----------
2 /a\ -1 + sin(a)
2*tan|-|
\2/
$$\frac{\tan{\left(\frac{a}{2} \right)}}{2} - \frac{1}{2 \tan{\left(\frac{a}{2} \right)}} - \frac{\sin{\left(a \right)}}{\sin{\left(a \right)} - 1}$$
/a\ /a\
tan|-| 2*tan|-|
\2/ 1 \2/
------ - -------- - --------------------------------
2 /a\ / /a\ \
2*tan|-| | 2*tan|-| |
\2/ / 2/a\\ | \2/ |
|1 + tan |-||*|-1 + -----------|
\ \2// | 2/a\|
| 1 + tan |-||
\ \2//
$$\frac{\tan{\left(\frac{a}{2} \right)}}{2} - \frac{1}{2 \tan{\left(\frac{a}{2} \right)}} - \frac{2 \tan{\left(\frac{a}{2} \right)}}{\left(-1 + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
sin(a)
-cot(a) - -----------
-1 + sin(a)
$$- \cot{\left(a \right)} - \frac{\sin{\left(a \right)}}{\sin{\left(a \right)} - 1}$$
/ pi\
cos|a - --|
\ 2 / cos(a)
--------------- - -----------
/ pi\ / pi\
1 - cos|a - --| cos|a - --|
\ 2 / \ 2 /
$$- \frac{\cos{\left(a \right)}}{\cos{\left(a - \frac{\pi}{2} \right)}} + \frac{\cos{\left(a - \frac{\pi}{2} \right)}}{1 - \cos{\left(a - \frac{\pi}{2} \right)}}$$
1 csc(a) ------------------- - ----------- / 1 \ /pi \ |1 - ------|*csc(a) csc|-- - a| \ csc(a)/ \2 /
$$- \frac{\csc{\left(a \right)}}{\csc{\left(- a + \frac{\pi}{2} \right)}} + \frac{1}{\left(1 - \frac{1}{\csc{\left(a \right)}}\right) \csc{\left(a \right)}}$$
/a\ /a\ tan|-| cot|-| \2/ \2/ sin(a) ------ - ------ - ----------- 2 2 -1 + sin(a)
$$\frac{\tan{\left(\frac{a}{2} \right)}}{2} - \frac{\cot{\left(\frac{a}{2} \right)}}{2} - \frac{\sin{\left(a \right)}}{\sin{\left(a \right)} - 1}$$
2/a\
sin |-|
\2/ sin(a) sin(a)
------- - ----------- - ---------
sin(a) -1 + sin(a) 2/a\
4*sin |-|
\2/
$$\frac{\sin^{2}{\left(\frac{a}{2} \right)}}{\sin{\left(a \right)}} - \frac{\sin{\left(a \right)}}{4 \sin^{2}{\left(\frac{a}{2} \right)}} - \frac{\sin{\left(a \right)}}{\sin{\left(a \right)} - 1}$$
sin(a) sin(2*a)
---------- - ---------
1 - sin(a) 2
2*sin (a)
$$- \frac{\sin{\left(2 a \right)}}{2 \sin^{2}{\left(a \right)}} + \frac{\sin{\left(a \right)}}{1 - \sin{\left(a \right)}}$$
/ pi\
sin|a + --|
sin(a) \ 2 /
---------- - -----------
1 - sin(a) sin(a)
$$- \frac{\sin{\left(a + \frac{\pi}{2} \right)}}{\sin{\left(a \right)}} + \frac{\sin{\left(a \right)}}{1 - \sin{\left(a \right)}}$$
/pi a\ /a\
csc|-- - -| csc|-|
\2 2/ 1 \2/
----------- - -------------------- - -------------
/a\ / 1 \ /pi a\
2*csc|-| |-1 + ------|*csc(a) 2*csc|-- - -|
\2/ \ csc(a)/ \2 2/
$$- \frac{\csc{\left(\frac{a}{2} \right)}}{2 \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + \frac{\csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{2 \csc{\left(\frac{a}{2} \right)}} - \frac{1}{\left(-1 + \frac{1}{\csc{\left(a \right)}}\right) \csc{\left(a \right)}}$$
/a\ /a\
cot|-| 2*cot|-|
1 \2/ \2/
-------- - ------ - --------------------------------
/a\ 2 / /a\ \
2*cot|-| | 2*cot|-| |
\2/ / 2/a\\ | \2/ |
|1 + cot |-||*|-1 + -----------|
\ \2// | 2/a\|
| 1 + cot |-||
\ \2//
$$- \frac{\cot{\left(\frac{a}{2} \right)}}{2} + \frac{1}{2 \cot{\left(\frac{a}{2} \right)}} - \frac{2 \cot{\left(\frac{a}{2} \right)}}{\left(-1 + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
/a pi\ / pi\ /a\
cos|- - --| cos|a - --| cos|-|
\2 2 / \ 2 / \2/
----------- - ---------------- - -------------
/a\ / pi\ /a pi\
2*cos|-| -1 + cos|a - --| 2*cos|- - --|
\2/ \ 2 / \2 2 /
$$- \frac{\cos{\left(\frac{a}{2} \right)}}{2 \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + \frac{\cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{2 \cos{\left(\frac{a}{2} \right)}} - \frac{\cos{\left(a - \frac{\pi}{2} \right)}}{\cos{\left(a - \frac{\pi}{2} \right)} - 1}$$
2/a\
sin |-|
\2/ sin(a) sin(a)
------- - ----------- - --------------
sin(a) -1 + sin(a) 2*(1 - cos(a))
$$\frac{\sin^{2}{\left(\frac{a}{2} \right)}}{\sin{\left(a \right)}} - \frac{\sin{\left(a \right)}}{\sin{\left(a \right)} - 1} - \frac{\sin{\left(a \right)}}{2 \left(1 - \cos{\left(a \right)}\right)}$$
/ pi\
sec|a - --|
1 \ 2 /
----------------------------- - -----------
/ 1 \ / pi\ sec(a)
|1 - -----------|*sec|a - --|
| / pi\| \ 2 /
| sec|a - --||
\ \ 2 //
$$- \frac{\sec{\left(a - \frac{\pi}{2} \right)}}{\sec{\left(a \right)}} + \frac{1}{\left(1 - \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}\right) \sec{\left(a - \frac{\pi}{2} \right)}}$$
1 csc(a) ------------------- - ------ / 1 \ sec(a) |1 - ------|*csc(a) \ csc(a)/
$$- \frac{\csc{\left(a \right)}}{\sec{\left(a \right)}} + \frac{1}{\left(1 - \frac{1}{\csc{\left(a \right)}}\right) \csc{\left(a \right)}}$$
1 sin(a) - ------ + ---------- tan(a) 1 - sin(a)
$$- \frac{1}{\tan{\left(a \right)}} + \frac{\sin{\left(a \right)}}{1 - \sin{\left(a \right)}}$$
1 sin(a) - ------ - ----------- tan(a) -1 + sin(a)
$$- \frac{1}{\tan{\left(a \right)}} - \frac{\sin{\left(a \right)}}{\sin{\left(a \right)} - 1}$$
2/a\ /a\
-1 + cot |-| 2*cot|-|
\2/ \2/
- ------------ + -------------------------------
/a\ / /a\ \
2*cot|-| | 2*cot|-| |
\2/ / 2/a\\ | \2/ |
|1 + cot |-||*|1 - -----------|
\ \2// | 2/a\|
| 1 + cot |-||
\ \2//
$$- \frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{2 \cot{\left(\frac{a}{2} \right)}} + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\left(1 - \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
/a\
2*tan|-|
1 \2/
- ------ + -------------------------------
tan(a) / /a\ \
| 2*tan|-| |
/ 2/a\\ | \2/ |
|1 + tan |-||*|1 - -----------|
\ \2// | 2/a\|
| 1 + tan |-||
\ \2//
$$- \frac{1}{\tan{\left(a \right)}} + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\left(1 - \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
/a\ /a pi\
sec|-| sec|- - --|
\2/ 1 \2 2 /
------------- - ------------------------------ - -----------
/a pi\ / 1 \ / pi\ /a\
2*sec|- - --| |-1 + -----------|*sec|a - --| 2*sec|-|
\2 2 / | / pi\| \ 2 / \2/
| sec|a - --||
\ \ 2 //
$$\frac{\sec{\left(\frac{a}{2} \right)}}{2 \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - \frac{\sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{2 \sec{\left(\frac{a}{2} \right)}} - \frac{1}{\left(-1 + \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}\right) \sec{\left(a - \frac{\pi}{2} \right)}}$$
/a\
2*cot|-|
\2/
-cot(a) + -------------------------------
/ /a\ \
| 2*cot|-| |
/ 2/a\\ | \2/ |
|1 + cot |-||*|1 - -----------|
\ \2// | 2/a\|
| 1 + cot |-||
\ \2//
$$- \cot{\left(a \right)} + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\left(1 - \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
2/a\ /a\
1 - tan |-| 2*tan|-|
\2/ \2/
- ----------- + -------------------------------
/a\ / /a\ \
2*tan|-| | 2*tan|-| |
\2/ / 2/a\\ | \2/ |
|1 + tan |-||*|1 - -----------|
\ \2// | 2/a\|
| 1 + tan |-||
\ \2//
$$- \frac{1 - \tan^{2}{\left(\frac{a}{2} \right)}}{2 \tan{\left(\frac{a}{2} \right)}} + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\left(1 - \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
-(1 - tan(a/2)^2)/(2*tan(a/2)) + 2*tan(a/2)/((1 + tan(a/2)^2)*(1 - 2*tan(a/2)/(1 + tan(a/2)^2)))