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