Sr Examen
¿Cómo vas a descomponer esta cos(3*pi/(2+a))/sin(pi-a) expresión en fracciones?
Solución
Ha introducido
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/ 3*pi\
cos|-----|
\2 + a/
-----------
sin(pi - a)
$$\frac{\cos{\left(\frac{3 \pi}{a + 2} \right)}}{\sin{\left(\pi - a \right)}}$$
cos((3*pi)/(2 + a))/sin(pi - a)
Simplificación general
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/ 3*pi\ cos|-----| \2 + a/ ---------- sin(a)
$$\frac{\cos{\left(\frac{3 \pi}{a + 2} \right)}}{\sin{\left(a \right)}}$$
cos(3*pi/(2 + a))/sin(a)
Denominador racional
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/ 3*pi\ cos|-----| \2 + a/ ---------- sin(a)
$$\frac{\cos{\left(\frac{3 \pi}{a + 2} \right)}}{\sin{\left(a \right)}}$$
cos(3*pi/(2 + a))/sin(a)
Combinatoria
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/ 3*pi\ cos|-----| \2 + a/ ---------- sin(a)
$$\frac{\cos{\left(\frac{3 \pi}{a + 2} \right)}}{\sin{\left(a \right)}}$$
cos(3*pi/(2 + a))/sin(a)
Potencias
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/ 3*pi\ cos|-----| \2 + a/ ---------- sin(a)
$$\frac{\cos{\left(\frac{3 \pi}{a + 2} \right)}}{\sin{\left(a \right)}}$$
/ -3*pi*I 3*pi*I\
| ------- ------|
| 2 + a 2 + a |
|e e |
2*I*|-------- + -------|
\ 2 2 /
---------------------------
I*(a - pi) I*(pi - a)
- e + e
$$\frac{2 i \left(\frac{e^{\frac{3 i \pi}{a + 2}}}{2} + \frac{e^{- \frac{3 i \pi}{a + 2}}}{2}\right)}{e^{i \left(\pi - a\right)} - e^{i \left(a - \pi\right)}}$$
2*i*(exp(-3*pi*i/(2 + a))/2 + exp(3*pi*i/(2 + a))/2)/(-exp(i*(a - pi)) + exp(i*(pi - a)))
Parte trigonométrica
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/ 1 \ / 2/ 3*pi \\ /a\
|1 + -------|*|1 - tan |---------||*tan|-|
| 2/a\| \ \2*(2 + a)// \2/
| tan |-||
\ \2//
------------------------------------------
/ 2/ 3*pi \\
2*|1 + tan |---------||
\ \2*(2 + a)//
$$\frac{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 - \tan^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)}\right) \tan{\left(\frac{a}{2} \right)}}{2 \left(\tan^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)} + 1\right)}$$
/ 3*pi\ cos|-----| \2 + a/ ---------- sin(a)
$$\frac{\cos{\left(\frac{3 \pi}{a + 2} \right)}}{\sin{\left(a \right)}}$$
/ 1 \ / 2/ 3*pi \\ /a\
|1 + -------|*|-1 + cot |---------||*cot|-|
| 2/a\| \ \2*(2 + a)// \2/
| cot |-||
\ \2//
-------------------------------------------
/ 2/ 3*pi \\
2*|1 + cot |---------||
\ \2*(2 + a)//
$$\frac{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \left(\cot^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)} - 1\right) \cot{\left(\frac{a}{2} \right)}}{2 \left(\cot^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)} + 1\right)}$$
/ pi\
sec|a - --|
\ 2 /
-----------
/ 3*pi\
sec|-----|
\2 + a/
$$\frac{\sec{\left(a - \frac{\pi}{2} \right)}}{\sec{\left(\frac{3 \pi}{a + 2} \right)}}$$
csc(a) ---------- / 3*pi\ sec|-----| \2 + a/
$$\frac{\csc{\left(a \right)}}{\sec{\left(\frac{3 \pi}{a + 2} \right)}}$$
/ 3*pi\
cos|-----|
\2 + a/
-----------
/ pi\
cos|a - --|
\ 2 /
$$\frac{\cos{\left(\frac{3 \pi}{a + 2} \right)}}{\cos{\left(a - \frac{\pi}{2} \right)}}$$
/ 2/a\\ / 2/ 3*pi \\
|1 + cot |-||*|1 - tan |---------||
\ \2// \ \2*(2 + a)//
-----------------------------------
/ 2/ 3*pi \\ /a\
2*|1 + tan |---------||*cot|-|
\ \2*(2 + a)// \2/
$$\frac{\left(1 - \tan^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)}\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)}{2 \left(\tan^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)} + 1\right) \cot{\left(\frac{a}{2} \right)}}$$
/ 3*pi\
cos|-----|
\2 + a/
-----------
/pi \
cos|-- - a|
\2 /
$$\frac{\cos{\left(\frac{3 \pi}{a + 2} \right)}}{\cos{\left(- a + \frac{\pi}{2} \right)}}$$
/ 2/a\\ / 2/ 3*pi \\
|1 + cot |-||*|-1 + cot |---------||
\ \2// \ \2*(2 + a)//
------------------------------------
/ 2/ 3*pi \\ /a\
2*|1 + cot |---------||*cot|-|
\ \2*(2 + a)// \2/
$$\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)} - 1\right)}{2 \left(\cot^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)} + 1\right) \cot{\left(\frac{a}{2} \right)}}$$
/pi \
sec|-- - a|
\2 /
-----------
/ 3*pi\
sec|-----|
\2 + a/
$$\frac{\sec{\left(- a + \frac{\pi}{2} \right)}}{\sec{\left(\frac{3 \pi}{a + 2} \right)}}$$
/ 2/a\\ / 2/ 3*pi \\
|1 + tan |-||*|-1 + cot |---------||
\ \2// \ \2*(2 + a)//
------------------------------------
/ 2/ 3*pi \\ /a\
2*|1 + cot |---------||*tan|-|
\ \2*(2 + a)// \2/
$$\frac{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)} - 1\right)}{2 \left(\cot^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)} + 1\right) \tan{\left(\frac{a}{2} \right)}}$$
csc(a) --------------- /pi 3*pi\ csc|-- - -----| \2 2 + a/
$$\frac{\csc{\left(a \right)}}{\csc{\left(\frac{\pi}{2} - \frac{3 \pi}{a + 2} \right)}}$$
/pi 3*pi\
sin|-- + -----|
\2 2 + a/
---------------
sin(a)
$$\frac{\sin{\left(\frac{\pi}{2} + \frac{3 \pi}{a + 2} \right)}}{\sin{\left(a \right)}}$$
/ 2/a\\ / 2/ 3*pi \\
|1 + tan |-||*|1 - tan |---------||
\ \2// \ \2*(2 + a)//
-----------------------------------
/ 2/ 3*pi \\ /a\
2*|1 + tan |---------||*tan|-|
\ \2*(2 + a)// \2/
$$\frac{\left(1 - \tan^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}{2 \left(\tan^{2}{\left(\frac{3 \pi}{2 \left(a + 2\right)} \right)} + 1\right) \tan{\left(\frac{a}{2} \right)}}$$
/ 3*pi\ cos|-----|*csc(a) \2 + a/
$$\cos{\left(\frac{3 \pi}{a + 2} \right)} \csc{\left(a \right)}$$
cos(3*pi/(2 + a))*csc(a)
Unión de expresiones racionales
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/ 3*pi\ cos|-----| \2 + a/ ---------- sin(a)
$$\frac{\cos{\left(\frac{3 \pi}{a + 2} \right)}}{\sin{\left(a \right)}}$$
cos(3*pi/(2 + a))/sin(a)