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