Sr Examen
¿Cómo vas a descomponer esta sin(п+a)*ctg(п-a)/(п/2-a) expresión en fracciones?
Solución
Ha introducido
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sin(pi + a)*cot(pi - a)
-----------------------
pi
-- - a
2
$$\frac{\sin{\left(a + \pi \right)} \cot{\left(\pi - a \right)}}{- a + \frac{\pi}{2}}$$
(sin(pi + a)*cot(pi - a))/(pi/2 - a)
Simplificación general
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-2*cos(a) --------- -pi + 2*a
$$- \frac{2 \cos{\left(a \right)}}{2 a - \pi}$$
-2*cos(a)/(-pi + 2*a)
Respuesta numérica
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cot(pi - a)*sin(pi + a)/(1.5707963267949 - a)
cot(pi - a)*sin(pi + a)/(1.5707963267949 - a)
Potencias
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cot(a)*sin(a)
-------------
pi
-- - a
2
$$\frac{\sin{\left(a \right)} \cot{\left(a \right)}}{- a + \frac{\pi}{2}}$$
/ I*(-pi - a) I*(pi + a)\
I*\- e + e /*cot(a)
---------------------------------------
/pi \
2*|-- - a|
\2 /
$$\frac{i \left(- e^{i \left(- a - \pi\right)} + e^{i \left(a + \pi\right)}\right) \cot{\left(a \right)}}{2 \left(- a + \frac{\pi}{2}\right)}$$
i*(-exp(i*(-pi - a)) + exp(i*(pi + a)))*cot(a)/(2*(pi/2 - a))
Denominador común
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-2*cot(pi - a)*sin(pi + a)
--------------------------
-pi + 2*a
$$- \frac{2 \sin{\left(a + \pi \right)} \cot{\left(\pi - a \right)}}{2 a - \pi}$$
-2*cot(pi - a)*sin(pi + a)/(-pi + 2*a)
Denominador racional
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-2*cot(a)*sin(a) ---------------- -pi + 2*a
$$- \frac{2 \sin{\left(a \right)} \cot{\left(a \right)}}{2 a - \pi}$$
-2*cot(a)*sin(a)/(-pi + 2*a)
Unión de expresiones racionales
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2*cot(a)*sin(a)
---------------
pi - 2*a
$$\frac{2 \sin{\left(a \right)} \cot{\left(a \right)}}{\pi - 2 a}$$
2*cot(a)*sin(a)/(pi - 2*a)
Parte trigonométrica
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2*cot(a) ----------------------------- / 1 \ /pi \ /a\ |1 + -------|*|-- - a|*cot|-| | 2/a\| \2 / \2/ | cot |-|| \ \2//
$$\frac{2 \cot{\left(a \right)}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \left(- a + \frac{\pi}{2}\right) \cot{\left(\frac{a}{2} \right)}}$$
1 --------------- /pi \ |-- - a|*sec(a) \2 /
$$\frac{1}{\left(- a + \frac{\pi}{2}\right) \sec{\left(a \right)}}$$
/a\
2*cot(a)*tan|-|
\2/
----------------------
/ 2/a\\ /pi \
|1 + tan |-||*|-- - a|
\ \2// \2 /
$$\frac{2 \tan{\left(\frac{a}{2} \right)} \cot{\left(a \right)}}{\left(- a + \frac{\pi}{2}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
/a\
2*cot|-|
\2/
-----------------------------
/ 2/a\\ /pi \
|1 + cot |-||*|-- - a|*tan(a)
\ \2// \2 /
$$\frac{2 \cot{\left(\frac{a}{2} \right)}}{\left(- a + \frac{\pi}{2}\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right) \tan{\left(a \right)}}$$
1 -------------------- /pi \ /pi \ |-- - a|*csc|-- - a| \2 / \2 /
$$\frac{1}{\left(- a + \frac{\pi}{2}\right) \csc{\left(- a + \frac{\pi}{2} \right)}}$$
2/a\
1 - tan |-|
\2/
----------------------
/ 2/a\\ /pi \
|1 + tan |-||*|-- - a|
\ \2// \2 /
$$\frac{1 - \tan^{2}{\left(\frac{a}{2} \right)}}{\left(- a + \frac{\pi}{2}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
/ pi\
-sec|a - --|
\ 2 /
---------------------------
/pi \ / pi\
|-- - a|*sec(a)*sec|a + --|
\2 / \ 2 /
$$- \frac{\sec{\left(a - \frac{\pi}{2} \right)}}{\left(- a + \frac{\pi}{2}\right) \sec{\left(a \right)} \sec{\left(a + \frac{\pi}{2} \right)}}$$
2 ------------------------------------ / 1 \ /pi \ /a\ |1 + -------|*|-- - a|*tan(a)*tan|-| | 2/a\| \2 / \2/ | tan |-|| \ \2//
$$\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \left(- a + \frac{\pi}{2}\right) \tan{\left(\frac{a}{2} \right)} \tan{\left(a \right)}}$$
cos(a) ------ pi -- - a 2
$$\frac{\cos{\left(a \right)}}{- a + \frac{\pi}{2}}$$
/ pi\ sin|a + --| \ 2 / ----------- pi -- - a 2
$$\frac{\sin{\left(a + \frac{\pi}{2} \right)}}{- a + \frac{\pi}{2}}$$
sin(2*a) ----------------- /pi \ 2*|-- - a|*sin(a) \2 /
$$\frac{\sin{\left(2 a \right)}}{2 \left(- a + \frac{\pi}{2}\right) \sin{\left(a \right)}}$$
/ pi\
-cos(a)*cos|a + --|
\ 2 /
--------------------
/pi \ /pi \
|-- - a|*cos|-- - a|
\2 / \2 /
$$- \frac{\cos{\left(a \right)} \cos{\left(a + \frac{\pi}{2} \right)}}{\left(- a + \frac{\pi}{2}\right) \cos{\left(- a + \frac{\pi}{2} \right)}}$$
2/a\
-1 + cot |-|
\2/
----------------------
/ 2/a\\ /pi \
|1 + cot |-||*|-- - a|
\ \2// \2 /
$$\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\left(- a + \frac{\pi}{2}\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
(-1 + cot(a/2)^2)/((1 + cot(a/2)^2)*(pi/2 - a))
Combinatoria
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-2*cot(a)*sin(a) ---------------- -pi + 2*a
$$- \frac{2 \sin{\left(a \right)} \cot{\left(a \right)}}{2 a - \pi}$$
-2*cot(a)*sin(a)/(-pi + 2*a)