Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sin(pi-a)/(sin(a)-cos(a)*tan(a/2))-cos(pi-a)
¿Cómo vas a descomponer esta sin(pi-a)/(sin(a)-cos(a)*tan(a/2))-cos(pi-a) expresión en fracciones?
Solución
Ha introducido
[src]
sin(pi - a)
---------------------- - cos(pi - a)
/a\
sin(a) - cos(a)*tan|-|
\2/
$$- \cos{\left(\pi - a \right)} + \frac{\sin{\left(\pi - a \right)}}{\sin{\left(a \right)} - \cos{\left(a \right)} \tan{\left(\frac{a}{2} \right)}}$$
sin(pi - a)/(sin(a) - cos(a)*tan(a/2)) - cos(pi - a)
Simplificación general
[src]
/ /a\ \
|- cos(a)*tan|-| + sin(a)|*cos(a) + sin(a)
\ \2/ /
------------------------------------------
/a\
- cos(a)*tan|-| + sin(a)
\2/
$$\frac{\left(\sin{\left(a \right)} - \cos{\left(a \right)} \tan{\left(\frac{a}{2} \right)}\right) \cos{\left(a \right)} + \sin{\left(a \right)}}{\sin{\left(a \right)} - \cos{\left(a \right)} \tan{\left(\frac{a}{2} \right)}}$$
((-cos(a)*tan(a/2) + sin(a))*cos(a) + sin(a))/(-cos(a)*tan(a/2) + sin(a))
Combinatoria
[src]
2 /a\
-sin(a) + cos (a)*tan|-| - cos(a)*sin(a)
\2/
----------------------------------------
/a\
-sin(a) + cos(a)*tan|-|
\2/
$$\frac{- \sin{\left(a \right)} \cos{\left(a \right)} - \sin{\left(a \right)} + \cos^{2}{\left(a \right)} \tan{\left(\frac{a}{2} \right)}}{- \sin{\left(a \right)} + \cos{\left(a \right)} \tan{\left(\frac{a}{2} \right)}}$$
(-sin(a) + cos(a)^2*tan(a/2) - cos(a)*sin(a))/(-sin(a) + cos(a)*tan(a/2))
Denominador común
[src]
sin(pi - a)
-cos(pi - a) - -----------------------
/a\
-sin(a) + cos(a)*tan|-|
\2/
$$- \cos{\left(\pi - a \right)} - \frac{\sin{\left(\pi - a \right)}}{- \sin{\left(a \right)} + \cos{\left(a \right)} \tan{\left(\frac{a}{2} \right)}}$$
-cos(pi - a) - sin(pi - a)/(-sin(a) + cos(a)*tan(a/2))
Potencias
[src]
sin(a)
------------------------ + cos(a)
/a\
- cos(a)*tan|-| + sin(a)
\2/
$$\cos{\left(a \right)} + \frac{\sin{\left(a \right)}}{\sin{\left(a \right)} - \cos{\left(a \right)} \tan{\left(\frac{a}{2} \right)}}$$
I*(pi - a) I*(a - pi) / I*(a - pi) I*(pi - a)\
e e I*\- e + e /
- ----------- - ----------- - -------------------------------------------------------------
2 2 / / I*a -I*a \\
| / I*a -I*a\ | --- -----||
| |e e | | 2 2 ||
| / -I*a I*a\ I*|---- + -----|*\- e + e /|
| I*\- e + e / \ 2 2 / |
2*|- ------------------ - ----------------------------------|
| 2 I*a -I*a |
| --- ----- |
| 2 2 |
\ e + e /
$$- \frac{e^{i \left(\pi - a\right)}}{2} - \frac{e^{i \left(a - \pi\right)}}{2} - \frac{i \left(e^{i \left(\pi - a\right)} - e^{i \left(a - \pi\right)}\right)}{2 \left(- \frac{i \left(- e^{\frac{i a}{2}} + e^{- \frac{i a}{2}}\right) \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right)}{e^{\frac{i a}{2}} + e^{- \frac{i a}{2}}} - \frac{i \left(e^{i a} - e^{- i a}\right)}{2}\right)}$$
-exp(i*(pi - a))/2 - exp(i*(a - pi))/2 - i*(-exp(i*(a - pi)) + exp(i*(pi - a)))/(2*(-i*(-exp(-i*a) + exp(i*a))/2 - i*(exp(i*a)/2 + exp(-i*a)/2)*(-exp(i*a/2) + exp(-i*a/2))/(exp(i*a/2) + exp(-i*a/2))))
Unión de expresiones racionales
[src]
/ /a\ \
|- cos(a)*tan|-| + sin(a)|*cos(a) + sin(a)
\ \2/ /
------------------------------------------
/a\
- cos(a)*tan|-| + sin(a)
\2/
$$\frac{\left(\sin{\left(a \right)} - \cos{\left(a \right)} \tan{\left(\frac{a}{2} \right)}\right) \cos{\left(a \right)} + \sin{\left(a \right)}}{\sin{\left(a \right)} - \cos{\left(a \right)} \tan{\left(\frac{a}{2} \right)}}$$
((-cos(a)*tan(a/2) + sin(a))*cos(a) + sin(a))/(-cos(a)*tan(a/2) + sin(a))
Respuesta numérica
[src]
-cos(pi - a) + sin(pi - a)/(-cos(a)*tan(a/2) + sin(a))
-cos(pi - a) + sin(pi - a)/(-cos(a)*tan(a/2) + sin(a))
Parte trigonométrica
[src]
1 1
------ + -------------------------------
sec(a) / /a\ \
| sec|-| |
| 1 \2/ |
|------ - -------------|*csc(a)
|csc(a) /a\ |
| csc|-|*sec(a)|
\ \2/ /
$$\frac{1}{\sec{\left(a \right)}} + \frac{1}{\left(\frac{1}{\csc{\left(a \right)}} - \frac{\sec{\left(\frac{a}{2} \right)}}{\csc{\left(\frac{a}{2} \right)} \sec{\left(a \right)}}\right) \csc{\left(a \right)}}$$
sin(a) / pi\
-------------------------------- + sin|a + --|
2/a\ / pi\ \ 2 /
2*sin |-|*sin|a + --|
\2/ \ 2 /
- --------------------- + sin(a)
sin(a)
$$\sin{\left(a + \frac{\pi}{2} \right)} + \frac{\sin{\left(a \right)}}{- \frac{2 \sin^{2}{\left(\frac{a}{2} \right)} \sin{\left(a + \frac{\pi}{2} \right)}}{\sin{\left(a \right)}} + \sin{\left(a \right)}}$$
2/ pi\
cos |a - --|
\ 2 /
- ------------ + cos(a)
-1 + cos(a)
$$\cos{\left(a \right)} - \frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos{\left(a \right)} - 1}$$
2/a\ /a\
-1 + tan |-| 2*tan|-|
\2/ \2/
- ------------ + --------------------------------------------------
2/a\ / /a\ 2/a\ \
1 + tan |-| | 2*cot|-| -1 + cot |-| |
\2/ / 2/a\\ | \2/ \2/ |
|1 + tan |-||*|----------- - --------------------|
\ \2// | 2/a\ / 2/a\\ /a\|
|1 + cot |-| |1 + cot |-||*cot|-||
\ \2/ \ \2// \2//
$$- \frac{\tan^{2}{\left(\frac{a}{2} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\left(- \frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right) \cot{\left(\frac{a}{2} \right)}} + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
/ 3*pi\ sin(a)
- sin|-a + ----| + --------------------------------
\ 2 / 2/a\ / pi\
2*sin |-|*sin|a + --|
\2/ \ 2 /
- --------------------- + sin(a)
sin(a)
$$- \sin{\left(- a + \frac{3 \pi}{2} \right)} + \frac{\sin{\left(a \right)}}{- \frac{2 \sin^{2}{\left(\frac{a}{2} \right)} \sin{\left(a + \frac{\pi}{2} \right)}}{\sin{\left(a \right)}} + \sin{\left(a \right)}}$$
2/a\ /a\
1 - cot |-| 2*cot|-|
\2/ \2/
- ----------- + --------------------------------------------------
2/a\ / /a\ / 2/a\\ /a\\
1 + cot |-| | 2*tan|-| |1 - tan |-||*tan|-||
\2/ / 2/a\\ | \2/ \ \2// \2/|
|1 + cot |-||*|----------- - --------------------|
\ \2// | 2/a\ 2/a\ |
|1 + tan |-| 1 + tan |-| |
\ \2/ \2/ /
$$- \frac{1 - \cot^{2}{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\left(- \frac{\left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right) \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
sin(a)
------------------------ + cos(a)
/a\
- cos(a)*tan|-| + sin(a)
\2/
$$\cos{\left(a \right)} + \frac{\sin{\left(a \right)}}{\sin{\left(a \right)} - \cos{\left(a \right)} \tan{\left(\frac{a}{2} \right)}}$$
2/a\ /a\
1 - tan |-| 2*tan|-|
\2/ \2/
----------- + --------------------------------------------------
2/a\ / /a\ / 2/a\\ /a\\
1 + tan |-| | 2*tan|-| |1 - tan |-||*tan|-||
\2/ / 2/a\\ | \2/ \ \2// \2/|
|1 + tan |-||*|----------- - --------------------|
\ \2// | 2/a\ 2/a\ |
|1 + tan |-| 1 + tan |-| |
\ \2/ \2/ /
$$\frac{1 - \tan^{2}{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\left(- \frac{\left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right) \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
/ 2/a\\
2*|-1 + cot |-||
\ \2//
1 + ----------------
2/a\
1 + cot |-|
\2/
$$\frac{2 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} + 1$$
2/a\ /a\
-1 + cot |-| 2*cot|-|
\2/ \2/
------------ + --------------------------------------------------
2/a\ / /a\ 2/a\ \
1 + cot |-| | 2*cot|-| -1 + cot |-| |
\2/ / 2/a\\ | \2/ \2/ |
|1 + cot |-||*|----------- - --------------------|
\ \2// | 2/a\ / 2/a\\ /a\|
|1 + cot |-| |1 + cot |-||*cot|-||
\ \2/ \ \2// \2//
$$\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\left(- \frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right) \cot{\left(\frac{a}{2} \right)}} + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
/ pi\
1 + 2*sin|a + --|
\ 2 /
$$2 \sin{\left(a + \frac{\pi}{2} \right)} + 1$$
1 1
------ + ----------------------------------------------
sec(a) / /a\ \
| sec|-| |
| 1 \2/ | / pi\
|----------- - ------------------|*sec|a - --|
| / pi\ /a pi\| \ 2 /
|sec|a - --| sec(a)*sec|- - --||
\ \ 2 / \2 2 //
$$\frac{1}{\sec{\left(a \right)}} + \frac{1}{\left(- \frac{\sec{\left(\frac{a}{2} \right)}}{\sec{\left(a \right)} \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}\right) \sec{\left(a - \frac{\pi}{2} \right)}}$$
1 1
------ + ----------------------------------------------
sec(a) / /a\ \
| sec|-| |
| 1 \2/ | /pi \
|----------- - ------------------|*sec|-- - a|
| / pi\ /a pi\| \2 /
|sec|a - --| sec(a)*sec|- - --||
\ \ 2 / \2 2 //
$$\frac{1}{\sec{\left(a \right)}} + \frac{1}{\left(- \frac{\sec{\left(\frac{a}{2} \right)}}{\sec{\left(a \right)} \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}\right) \sec{\left(- a + \frac{\pi}{2} \right)}}$$
/ pi\
cos|a - --|
\ 2 /
---------------------------------- + cos(a)
/a pi\
cos(a)*cos|- - --|
\2 2 / / pi\
- ------------------ + cos|a - --|
/a\ \ 2 /
cos|-|
\2/
$$\cos{\left(a \right)} + \frac{\cos{\left(a - \frac{\pi}{2} \right)}}{\cos{\left(a - \frac{\pi}{2} \right)} - \frac{\cos{\left(a \right)} \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{a}{2} \right)}}}$$
2/a\ 2/a\
-1 + cot |-| 4*cot |-|
\2/ \2/
------------ - ----------------------------------
2/a\ / 2/a\\
1 + cot |-| 2 | -1 + cot |-||
\2/ / 2/a\\ | \2/|
|1 + cot |-|| *|-1 + ------------|
\ \2// | 2/a\ |
| 1 + cot |-| |
\ \2/ /
$$\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} - \frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} - 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
/pi \
cos|-- - a|
\2 /
---------------------------------- + cos(a)
/a pi\
cos(a)*cos|- - --|
\2 2 / / pi\
- ------------------ + cos|a - --|
/a\ \ 2 /
cos|-|
\2/
$$\cos{\left(a \right)} + \frac{\cos{\left(- a + \frac{\pi}{2} \right)}}{\cos{\left(a - \frac{\pi}{2} \right)} - \frac{\cos{\left(a \right)} \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos{\left(\frac{a}{2} \right)}}}$$
-cos(a) + cos(2*a) ------------------ -1 + cos(a)
$$\frac{- \cos{\left(a \right)} + \cos{\left(2 a \right)}}{\cos{\left(a \right)} - 1}$$
sin(a)
--------------------------- + cos(a)
2/a\
2*sin |-|*cos(a)
\2/
- ---------------- + sin(a)
sin(a)
$$\cos{\left(a \right)} + \frac{\sin{\left(a \right)}}{- \frac{2 \sin^{2}{\left(\frac{a}{2} \right)} \cos{\left(a \right)}}{\sin{\left(a \right)}} + \sin{\left(a \right)}}$$
2
sin (a) / pi\
- ---------------- + sin|a + --|
/ pi\ \ 2 /
-1 + sin|a + --|
\ 2 /
$$\sin{\left(a + \frac{\pi}{2} \right)} - \frac{\sin^{2}{\left(a \right)}}{\sin{\left(a + \frac{\pi}{2} \right)} - 1}$$
1
1 - -------
2/a\
tan |-|
\2/ 2
- ----------- + ---------------------------------------------------------
1 / /a\ / 2/a\\ /a\\
1 + ------- | 2*tan|-| |1 - tan |-||*tan|-||
2/a\ / 1 \ | \2/ \ \2// \2/| /a\
tan |-| |1 + -------|*|----------- - --------------------|*tan|-|
\2/ | 2/a\| | 2/a\ 2/a\ | \2/
| tan |-|| |1 + tan |-| 1 + tan |-| |
\ \2// \ \2/ \2/ /
$$- \frac{1 - \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}} + \frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \left(- \frac{\left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right) \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right) \tan{\left(\frac{a}{2} \right)}}$$
1 1
----------- - --------------------------
/pi \ / 1 \ 2
csc|-- - a| |-1 + -----------|*csc (a)
\2 / | /pi \|
| csc|-- - a||
\ \2 //
$$\frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}} - \frac{1}{\left(-1 + \frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}}\right) \csc^{2}{\left(a \right)}}$$
1 1
----------- + ------------------------------------
/pi \ / /pi a\ \
csc|-- - a| | csc|-- - -| |
\2 / | 1 \2 2/ |
|------ - ------------------|*csc(a)
|csc(a) /a\ /pi \|
| csc|-|*csc|-- - a||
\ \2/ \2 //
$$\frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}} + \frac{1}{\left(\frac{1}{\csc{\left(a \right)}} - \frac{\csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc{\left(\frac{a}{2} \right)} \csc{\left(- a + \frac{\pi}{2} \right)}}\right) \csc{\left(a \right)}}$$
1 1
------ - --------------------------
sec(a) / 1 \ 2/ pi\
|-1 + ------|*sec |a - --|
\ sec(a)/ \ 2 /
$$\frac{1}{\sec{\left(a \right)}} - \frac{1}{\left(-1 + \frac{1}{\sec{\left(a \right)}}\right) \sec^{2}{\left(a - \frac{\pi}{2} \right)}}$$
2/a\ 2/a\
1 - tan |-| 4*tan |-|
\2/ \2/
----------- - ---------------------------------
2/a\ / 2/a\\
1 + tan |-| 2 | 1 - tan |-||
\2/ / 2/a\\ | \2/|
|1 + tan |-|| *|-1 + -----------|
\ \2// | 2/a\|
| 1 + tan |-||
\ \2//
$$\frac{1 - \tan^{2}{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} - \frac{4 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} - 1\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
1 + 2*cos(a)
$$2 \cos{\left(a \right)} + 1$$
/ 2/a\\
2*|1 - tan |-||
\ \2//
1 + ---------------
2/a\
1 + tan |-|
\2/
$$\frac{2 \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + 1$$
2
1 + -----------
/pi \
csc|-- - a|
\2 /
$$1 + \frac{2}{\csc{\left(- a + \frac{\pi}{2} \right)}}$$
2
sin (a)
- ----------- + cos(a)
-1 + cos(a)
$$\cos{\left(a \right)} - \frac{\sin^{2}{\left(a \right)}}{\cos{\left(a \right)} - 1}$$
1
-1 + -------
2/a\
cot |-|
\2/ 2
- ------------ + ---------------------------------------------------------
1 / /a\ 2/a\ \
1 + ------- | 2*cot|-| -1 + cot |-| |
2/a\ / 1 \ | \2/ \2/ | /a\
cot |-| |1 + -------|*|----------- - --------------------|*cot|-|
\2/ | 2/a\| | 2/a\ / 2/a\\ /a\| \2/
| cot |-|| |1 + cot |-| |1 + cot |-||*cot|-||
\ \2// \ \2/ \ \2// \2//
$$- \frac{-1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}} + \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \left(- \frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right) \cot{\left(\frac{a}{2} \right)}} + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}\right) \cot{\left(\frac{a}{2} \right)}}$$
sin(a)
------------------------ + cos(a)
/a\
cos(a)*sin|-|
\2/
- ------------- + sin(a)
/a\
cos|-|
\2/
$$\cos{\left(a \right)} + \frac{\sin{\left(a \right)}}{- \frac{\sin{\left(\frac{a}{2} \right)} \cos{\left(a \right)}}{\cos{\left(\frac{a}{2} \right)}} + \sin{\left(a \right)}}$$
2
1 + ------
sec(a)
$$1 + \frac{2}{\sec{\left(a \right)}}$$
1 + 2/sec(a)
Denominador racional
[src]
2 /a\
cos(a)*sin(a) - cos (a)*tan|-| + sin(a)
\2/
---------------------------------------
/a\
- cos(a)*tan|-| + sin(a)
\2/
$$\frac{\sin{\left(a \right)} \cos{\left(a \right)} + \sin{\left(a \right)} - \cos^{2}{\left(a \right)} \tan{\left(\frac{a}{2} \right)}}{\sin{\left(a \right)} - \cos{\left(a \right)} \tan{\left(\frac{a}{2} \right)}}$$
(cos(a)*sin(a) - cos(a)^2*tan(a/2) + sin(a))/(-cos(a)*tan(a/2) + sin(a))