Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
(2*sin(a-3*pi)-cos(((-pi)/2)+a))/(5*sin(a-pi))
¿Cómo vas a descomponer esta (2*sin(a-3*pi)-cos(((-pi)/2)+a))/(5*sin(a-pi)) expresión en fracciones?
Solución
Ha introducido
[src]
/-pi \
2*sin(a - 3*pi) - cos|---- + a|
\ 2 /
-------------------------------
5*sin(a - pi)
$$\frac{2 \sin{\left(a - 3 \pi \right)} - \cos{\left(a + \frac{\left(-1\right) \pi}{2} \right)}}{5 \sin{\left(a - \pi \right)}}$$
(2*sin(a - 3*pi) - cos((-pi)/2 + a))/((5*sin(a - pi)))
Denominador común
[src]
-3*sin(a) ------------- 5*sin(a - pi)
$$- \frac{3 \sin{\left(a \right)}}{5 \sin{\left(a - \pi \right)}}$$
-3*sin(a)/(5*sin(a - pi))
Potencias
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3/5
$$\frac{3}{5}$$
/ / pi\ /pi \ \
| I*|a - --| I*|-- - a| |
| \ 2 / \2 / |
| e e / I*(-a + 3*pi) I*(a - 3*pi)\|
2*I*|- ----------- - ----------- - I*\- e + e /|
\ 2 2 /
------------------------------------------------------------------------
/ I*(pi - a) I*(a - pi)\
5*\- e + e /
$$\frac{2 i \left(- i \left(- e^{i \left(- a + 3 \pi\right)} + e^{i \left(a - 3 \pi\right)}\right) - \frac{e^{i \left(- a + \frac{\pi}{2}\right)}}{2} - \frac{e^{i \left(a - \frac{\pi}{2}\right)}}{2}\right)}{5 \left(- e^{i \left(\pi - a\right)} + e^{i \left(a - \pi\right)}\right)}$$
2*i*(-exp(i*(a - pi/2))/2 - exp(i*(pi/2 - a))/2 - i*(-exp(i*(-a + 3*pi)) + exp(i*(a - 3*pi))))/(5*(-exp(i*(pi - a)) + exp(i*(a - pi))))
Respuesta numérica
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0.2*(-cos((-pi)/2 + a) + 2.0*sin(a - 3*pi))/sin(a - pi)
0.2*(-cos((-pi)/2 + a) + 2.0*sin(a - 3*pi))/sin(a - pi)
Parte trigonométrica
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2/a pi\
cot |- + --|
1 \2 4 / 1 1
- - ------------ + --------- - ----------------------
2 10 10*sin(a) 2/a pi\
10*sin(a)*tan |- + --|
\2 4 /
$$- \frac{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{10} + \frac{1}{2} + \frac{1}{10 \sin{\left(a \right)}} - \frac{1}{10 \sin{\left(a \right)} \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
2 2
1 1 cos (a) cos (a)
- + --------- - --------------- - ----------------------
2 10*sin(a) 4/a pi\ 4/a pi\
40*sin |- + --| 40*sin(a)*sin |- + --|
\2 4 / \2 4 /
$$\frac{1}{2} - \frac{\cos^{2}{\left(a \right)}}{40 \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} + \frac{1}{10 \sin{\left(a \right)}} - \frac{\cos^{2}{\left(a \right)}}{40 \sin{\left(a \right)} \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
2/a pi\ 2/a pi\
cos |- + --| cos |- + --|
1 1 \2 4 / \2 4 /
- + -------------- - --------------- - ---------------------------
2 / pi\ 2/a pi\ / pi\ 2/a pi\
10*cos|a - --| 10*cos |- - --| 10*cos|a - --|*cos |- - --|
\ 2 / \2 4 / \ 2 / \2 4 /
$$\frac{1}{2} + \frac{1}{10 \cos{\left(a - \frac{\pi}{2} \right)}} - \frac{\cos^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{10 \cos^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)}} - \frac{\cos^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{10 \cos^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)} \cos{\left(a - \frac{\pi}{2} \right)}}$$
2/a pi\ 2/a\ 2/a pi\ / 2/a\\
cot |- + --| 1 + cot |-| cot |- + --|*|1 + cot |-||
1 \2 4 / \2/ \2 4 / \ \2//
- - ------------ + ----------- - --------------------------
2 10 /a\ /a\
20*cot|-| 20*cot|-|
\2/ \2/
$$- \frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right) \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{20 \cot{\left(\frac{a}{2} \right)}} + \frac{\cot^{2}{\left(\frac{a}{2} \right)} + 1}{20 \cot{\left(\frac{a}{2} \right)}} - \frac{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{10} + \frac{1}{2}$$
2/a\ 2/a\
1 + tan |-| 1 + tan |-|
1 1 \2/ \2/
- - --------------- + ----------- - ----------------------
2 2/a pi\ /a\ /a\ 2/a pi\
10*tan |- + --| 20*tan|-| 20*tan|-|*tan |- + --|
\2 4 / \2/ \2/ \2 4 /
$$\frac{\tan^{2}{\left(\frac{a}{2} \right)} + 1}{20 \tan{\left(\frac{a}{2} \right)}} - \frac{\tan^{2}{\left(\frac{a}{2} \right)} + 1}{20 \tan{\left(\frac{a}{2} \right)} \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} + \frac{1}{2} - \frac{1}{10 \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
/ /a\\
| tan|-||
| 1 /a\ \2/|
2*|------- + 3*tan|-| + ------|
| 2/a\ \2/ sin(a)|
|cos |-| |
\ \2/ /
-------------------------------
2
/ /a\\
5*|1 + tan|-||
\ \2//
$$\frac{2 \left(3 \tan{\left(\frac{a}{2} \right)} + \frac{1}{\cos^{2}{\left(\frac{a}{2} \right)}} + \frac{\tan{\left(\frac{a}{2} \right)}}{\sin{\left(a \right)}}\right)}{5 \left(\tan{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
/ / pi\ \
-|- cos|a - --| - 2*sin(a)|
\ \ 2 / /
----------------------------
5*sin(a)
$$- \frac{- 2 \sin{\left(a \right)} - \cos{\left(a - \frac{\pi}{2} \right)}}{5 \sin{\left(a \right)}}$$
/ 1 2 \
-|- ----------- - ------|*csc(a)
| /pi \ csc(a)|
| sec|-- - a| |
\ \2 / /
---------------------------------
5
$$- \frac{\left(- \frac{1}{\sec{\left(- a + \frac{\pi}{2} \right)}} - \frac{2}{\csc{\left(a \right)}}\right) \csc{\left(a \right)}}{5}$$
/ 1 \
| 1 - ------------ |
| 2/a pi\ |
| tan |- + --| |
/ 1 \ | \2 4 / 4 | /a\
-|1 + -------|*|- ---------------- - --------------------|*tan|-|
| 2/a\| | 1 / 1 \ /a\| \2/
| tan |-|| | 1 + ------------ |1 + -------|*tan|-||
\ \2// | 2/a pi\ | 2/a\| \2/|
| tan |- + --| | tan |-|| |
\ \2 4 / \ \2// /
------------------------------------------------------------------
10
$$- \frac{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \left(- \frac{1 - \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}} - \frac{4}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \tan{\left(\frac{a}{2} \right)}}\right) \tan{\left(\frac{a}{2} \right)}}{10}$$
/ 2/a pi\ /a\ \
| 1 - cot |- + --| 4*cot|-| |
/ 2/a\\ | \2 4 / \2/ |
-|1 + cot |-||*|- ---------------- - -----------|
\ \2// | 2/a pi\ 2/a\|
| 1 + cot |- + --| 1 + cot |-||
\ \2 4 / \2//
--------------------------------------------------
/a\
10*cot|-|
\2/
$$- \frac{\left(- \frac{1 - \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} - \frac{4 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)}{10 \cot{\left(\frac{a}{2} \right)}}$$
/ 7*pi\
-sin(a) + 2*cos|a - ----|
\ 2 /
-------------------------
/ 3*pi\
5*cos|a - ----|
\ 2 /
$$\frac{- \sin{\left(a \right)} + 2 \cos{\left(a - \frac{7 \pi}{2} \right)}}{5 \cos{\left(a - \frac{3 \pi}{2} \right)}}$$
/ 2/a pi\ /a\ \
| -1 + tan |- + --| 4*tan|-| |
/ 2/a\\ | \2 4 / \2/ |
-|1 + tan |-||*|- ----------------- - -----------|
\ \2// | 2/a pi\ 2/a\|
| 1 + tan |- + --| 1 + tan |-||
\ \2 4 / \2//
---------------------------------------------------
/a\
10*tan|-|
\2/
$$- \frac{\left(- \frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} - \frac{4 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}{10 \tan{\left(\frac{a}{2} \right)}}$$
3/5
$$\frac{3}{5}$$
/ 1 2 \ / 3*pi\
|- ------ + -------------|*sec|a - ----|
| csc(a) / 7*pi\| \ 2 /
| sec|a - ----||
\ \ 2 //
----------------------------------------
5
$$\frac{\left(\frac{2}{\sec{\left(a - \frac{7 \pi}{2} \right)}} - \frac{1}{\csc{\left(a \right)}}\right) \sec{\left(a - \frac{3 \pi}{2} \right)}}{5}$$
/ pi\ 2/a pi\ 2/a pi\ / pi\
sec|a - --| sec |- - --| sec |- - --|*sec|a - --|
1 \ 2 / \2 4 / \2 4 / \ 2 /
- + ----------- - --------------- - ------------------------
2 10 2/a pi\ 2/a pi\
10*sec |- + --| 10*sec |- + --|
\2 4 / \2 4 /
$$- \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)} \sec{\left(a - \frac{\pi}{2} \right)}}{10 \sec^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} - \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}{10 \sec^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} + \frac{\sec{\left(a - \frac{\pi}{2} \right)}}{10} + \frac{1}{2}$$
2/a pi\ 2/a pi\
csc |- + --| csc |- + --|*csc(a)
1 csc(a) \2 4 / \2 4 /
- + ------ - ----------------- - -------------------
2 10 2/ a pi\ 2/ a pi\
10*csc |- - + --| 10*csc |- - + --|
\ 2 4 / \ 2 4 /
$$\frac{\csc{\left(a \right)}}{10} - \frac{\csc{\left(a \right)} \csc^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{10 \csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{4} \right)}} + \frac{1}{2} - \frac{\csc^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{10 \csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{4} \right)}}$$
1 1 1 1
- - --------------- + --------- - ----------------------
2 2/a pi\ 10*sin(a) 2/a pi\
10*tan |- + --| 10*sin(a)*tan |- + --|
\2 4 / \2 4 /
$$\frac{1}{2} - \frac{1}{10 \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} + \frac{1}{10 \sin{\left(a \right)}} - \frac{1}{10 \sin{\left(a \right)} \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
/ 1 \
| -1 + ------------ |
| 2/a pi\ |
| cot |- + --| |
/ 1 \ | \2 4 / 4 | /a\
-|1 + -------|*|- ----------------- - --------------------|*cot|-|
| 2/a\| | 1 / 1 \ /a\| \2/
| cot |-|| | 1 + ------------ |1 + -------|*cot|-||
\ \2// | 2/a pi\ | 2/a\| \2/|
| cot |- + --| | cot |-|| |
\ \2 4 / \ \2// /
-------------------------------------------------------------------
10
$$- \frac{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \left(- \frac{-1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}}{1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}} - \frac{4}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \cot{\left(\frac{a}{2} \right)}}\right) \cot{\left(\frac{a}{2} \right)}}{10}$$
-(1 + cot(a/2)^(-2))*(-(-1 + cot(a/2 + pi/4)^(-2))/(1 + cot(a/2 + pi/4)^(-2)) - 4/((1 + cot(a/2)^(-2))*cot(a/2)))*cot(a/2)/10
Unión de expresiones racionales
[src]
/ /-pi + 2*a\ \
-|- cos|---------| - 2*sin(a)|
\ \ 2 / /
-------------------------------
5*sin(a)
$$- \frac{- 2 \sin{\left(a \right)} - \cos{\left(\frac{2 a - \pi}{2} \right)}}{5 \sin{\left(a \right)}}$$
-(-cos((-pi + 2*a)/2) - 2*sin(a))/(5*sin(a))