Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sin(pi-a)/sin(3*pi/2+a)+cos(pi-a)/cos(pi/(2+a))-1
¿Cómo vas a descomponer esta sin(pi-a)/sin(3*pi/2+a)+cos(pi-a)/cos(pi/(2+a))-1 expresión en fracciones?
Solución
Ha introducido
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sin(pi - a) cos(pi - a) ------------- + ----------- - 1 /3*pi \ / pi \ sin|---- + a| cos|-----| \ 2 / \2 + a/
$$\left(\frac{\sin{\left(\pi - a \right)}}{\sin{\left(a + \frac{3 \pi}{2} \right)}} + \frac{\cos{\left(\pi - a \right)}}{\cos{\left(\frac{\pi}{a + 2} \right)}}\right) - 1$$
sin(pi - a)/sin((3*pi)/2 + a) + cos(pi - a)/cos(pi/(2 + a)) - 1
Simplificación general
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cos(a)
-1 - tan(a) - ----------
/ pi \
cos|-----|
\2 + a/
$$- \frac{\cos{\left(a \right)}}{\cos{\left(\frac{\pi}{a + 2} \right)}} - \tan{\left(a \right)} - 1$$
-1 - tan(a) - cos(a)/cos(pi/(2 + a))
Respuesta numérica
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-1.0 + cos(pi - a)/cos(pi/(2 + a)) + sin(pi - a)/sin((3*pi)/2 + a)
-1.0 + cos(pi - a)/cos(pi/(2 + a)) + sin(pi - a)/sin((3*pi)/2 + a)
Denominador común
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/ pi \
cos(a)*cos(pi - a) - cos|-----|*sin(pi - a)
\2 + a/
-1 + -------------------------------------------
/ pi \
cos(a)*cos|-----|
\2 + a/
$$\frac{- \sin{\left(\pi - a \right)} \cos{\left(\frac{\pi}{a + 2} \right)} + \cos{\left(a \right)} \cos{\left(\pi - a \right)}}{\cos{\left(a \right)} \cos{\left(\frac{\pi}{a + 2} \right)}} - 1$$
-1 + (cos(a)*cos(pi - a) - cos(pi/(2 + a))*sin(pi - a))/(cos(a)*cos(pi/(2 + a)))
Potencias
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I*(pi - a) I*(a - pi)
e e
----------- + ----------- I*(a - pi) I*(pi - a)
2 2 - e + e
-1 + ------------------------- + --------------------------------
pi*I -pi*I / 3*pi\ / 3*pi\
----- ------ I*|-a - ----| I*|a + ----|
2 + a 2 + a \ 2 / \ 2 /
e e - e + e
------ + -------
2 2
$$\frac{\frac{e^{i \left(\pi - a\right)}}{2} + \frac{e^{i \left(a - \pi\right)}}{2}}{\frac{e^{\frac{i \pi}{a + 2}}}{2} + \frac{e^{- \frac{i \pi}{a + 2}}}{2}} + \frac{e^{i \left(\pi - a\right)} - e^{i \left(a - \pi\right)}}{- e^{i \left(- a - \frac{3 \pi}{2}\right)} + e^{i \left(a + \frac{3 \pi}{2}\right)}} - 1$$
sin(a) cos(a)
-1 - ------ - ----------
cos(a) / pi \
cos|-----|
\2 + a/
$$- \frac{\sin{\left(a \right)}}{\cos{\left(a \right)}} - \frac{\cos{\left(a \right)}}{\cos{\left(\frac{\pi}{a + 2} \right)}} - 1$$
-1 - sin(a)/cos(a) - cos(a)/cos(pi/(2 + a))
Unión de expresiones racionales
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/ pi \ /2*a + 3*pi\ / pi \ /2*a + 3*pi\
cos|-----|*sin(a) - cos(a)*sin|----------| - cos|-----|*sin|----------|
\2 + a/ \ 2 / \2 + a/ \ 2 /
-----------------------------------------------------------------------
/ pi \ /2*a + 3*pi\
cos|-----|*sin|----------|
\2 + a/ \ 2 /
$$\frac{\sin{\left(a \right)} \cos{\left(\frac{\pi}{a + 2} \right)} - \sin{\left(\frac{2 a + 3 \pi}{2} \right)} \cos{\left(a \right)} - \sin{\left(\frac{2 a + 3 \pi}{2} \right)} \cos{\left(\frac{\pi}{a + 2} \right)}}{\sin{\left(\frac{2 a + 3 \pi}{2} \right)} \cos{\left(\frac{\pi}{a + 2} \right)}}$$
(cos(pi/(2 + a))*sin(a) - cos(a)*sin((2*a + 3*pi)/2) - cos(pi/(2 + a))*sin((2*a + 3*pi)/2))/(cos(pi/(2 + a))*sin((2*a + 3*pi)/2))
Compilar la expresión
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cos(pi - a) sin(pi - a)
-1 + ----------- + -------------
/ pi \ /3*pi \
cos|-----| sin|---- + a|
\2 + a/ \ 2 /
$$\frac{\sin{\left(\pi - a \right)}}{\sin{\left(a + \frac{3 \pi}{2} \right)}} - 1 + \frac{\cos{\left(\pi - a \right)}}{\cos{\left(\frac{\pi}{a + 2} \right)}}$$
-1 + cos(pi - a)/cos(pi/(2 + a)) + sin(pi - a)/sin((3*pi)/2 + a)
Denominador racional
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2 / pi \ / pi \
- cos (a) - cos(a)*cos|-----| - cos|-----|*sin(a)
\2 + a/ \2 + a/
-------------------------------------------------
/ pi \
cos(a)*cos|-----|
\2 + a/
$$\frac{- \sin{\left(a \right)} \cos{\left(\frac{\pi}{a + 2} \right)} - \cos^{2}{\left(a \right)} - \cos{\left(a \right)} \cos{\left(\frac{\pi}{a + 2} \right)}}{\cos{\left(a \right)} \cos{\left(\frac{\pi}{a + 2} \right)}}$$
(-cos(a)^2 - cos(a)*cos(pi/(2 + a)) - cos(pi/(2 + a))*sin(a))/(cos(a)*cos(pi/(2 + a)))
Combinatoria
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/ 2 / pi \ / pi \ \
-|cos (a) + cos(a)*cos|-----| + cos|-----|*sin(a)|
\ \2 + a/ \2 + a/ /
---------------------------------------------------
/ pi \
cos(a)*cos|-----|
\2 + a/
$$- \frac{\sin{\left(a \right)} \cos{\left(\frac{\pi}{a + 2} \right)} + \cos^{2}{\left(a \right)} + \cos{\left(a \right)} \cos{\left(\frac{\pi}{a + 2} \right)}}{\cos{\left(a \right)} \cos{\left(\frac{\pi}{a + 2} \right)}}$$
-(cos(a)^2 + cos(a)*cos(pi/(2 + a)) + cos(pi/(2 + a))*sin(a))/(cos(a)*cos(pi/(2 + a)))