Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sin(3*pi/2+a)/(3*cos(pi-a))еслиa=2*pi/3
¿Cómo vas a descomponer esta sin(3*pi/2+a)/(3*cos(pi-a))еслиa=2*pi/3 expresión en fracciones?
Solución
Ha introducido
[src]
/3*pi \ sin|---- + a| \ 2 / ------------- 3*cos(pi - a)
$$\frac{\sin{\left(a + \frac{3 \pi}{2} \right)}}{3 \cos{\left(\pi - a \right)}}$$
sin((3*pi)/2 + a)/((3*cos(pi - a)))
Sustitución de una condición
[src]
sin((3*pi)/2 + a)/((3*cos(pi - a))) con a = (2*pi)/3
sin((3*pi)/2 + a)/((3*cos(pi - a)))
$$\frac{\sin{\left(a + \frac{3 \pi}{2} \right)}}{3 \cos{\left(\pi - a \right)}}$$
sin((3*pi)/2 + ((2*pi)/3^1))/((3*cos(pi - ((2*pi)/3^1))))
$$\frac{\sin{\left(((2*pi)/3^{1)} + \frac{3 \pi}{2} \right)}}{3 \cos{\left(\pi - ((2*pi)/3^{1)} \right)}}$$
sin((3*pi)/2 + (2*pi)/3)/((3*cos(pi - 2*pi/3)))
$$\frac{\sin{\left(\frac{2 \pi}{3} + \frac{3 \pi}{2} \right)}}{3 \cos{\left(\pi - \frac{2 \pi}{3} \right)}}$$
1/3
$$\frac{1}{3}$$
1/3
Respuesta numérica
[src]
0.333333333333333*sin((3*pi)/2 + a)/cos(pi - a)
0.333333333333333*sin((3*pi)/2 + a)/cos(pi - a)
Potencias
[src]
1/3
$$\frac{1}{3}$$
/ / 3*pi\ / 3*pi\\
| I*|-a - ----| I*|a + ----||
| \ 2 / \ 2 /|
-I*\- e + e /
--------------------------------------
/ I*(pi - a) I*(a - pi)\
|3*e 3*e |
2*|------------- + -------------|
\ 2 2 /
$$- \frac{i \left(- e^{i \left(- a - \frac{3 \pi}{2}\right)} + e^{i \left(a + \frac{3 \pi}{2}\right)}\right)}{2 \left(\frac{3 e^{i \left(\pi - a\right)}}{2} + \frac{3 e^{i \left(a - \pi\right)}}{2}\right)}$$
-i*(-exp(i*(-a - 3*pi/2)) + exp(i*(a + 3*pi/2)))/(2*(3*exp(i*(pi - a))/2 + 3*exp(i*(a - pi))/2))
Denominador común
[src]
-cos(a) ------------- 3*cos(pi - a)
$$- \frac{\cos{\left(a \right)}}{3 \cos{\left(\pi - a \right)}}$$
-cos(a)/(3*cos(pi - a))
Unión de expresiones racionales
[src]
/2*a + 3*pi\
-sin|----------|
\ 2 /
-----------------
3*cos(a)
$$- \frac{\sin{\left(\frac{2 a + 3 \pi}{2} \right)}}{3 \cos{\left(a \right)}}$$
-sin((2*a + 3*pi)/2)/(3*cos(a))
Parte trigonométrica
[src]
-sec(a) ------------- 3*sec(pi + a)
$$- \frac{\sec{\left(a \right)}}{3 \sec{\left(a + \pi \right)}}$$
/ /a\ \
2 | 2*cot|-| |
/ 2/a\\ | \2/ | /a pi\
-|1 + cot |-|| *|1 + -----------|*cot|- + --|
\ \4// | 2/a\| \2 4 /
| 1 + cot |-||
\ \2//
----------------------------------------------
/ 2/a\\ 2/a\
12*|1 - cot |-||*cot |-|
\ \2// \4/
$$- \frac{\left(1 + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2} \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{12 \left(1 - \cot^{2}{\left(\frac{a}{2} \right)}\right) \cot^{2}{\left(\frac{a}{4} \right)}}$$
2/a\ / 1 \ /a pi\
-csc |-|*|1 + ------|*csc|- + --|
\2/ \ csc(a)/ \2 4 /
----------------------------------
/ 2/a\ \
| csc |-| |
| \2/ | / a pi\
3*|1 - ------------|*csc|- - + --|
| 2/pi a\| \ 2 4 /
| csc |-- - -||
\ \2 2//
$$- \frac{\left(1 + \frac{1}{\csc{\left(a \right)}}\right) \csc^{2}{\left(\frac{a}{2} \right)} \csc{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{3 \left(- \frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{a}{2} + \frac{\pi}{4} \right)}}$$
/pi \ csc|-- - a| \2 / ----------- 3*sec(a)
$$\frac{\csc{\left(- a + \frac{\pi}{2} \right)}}{3 \sec{\left(a \right)}}$$
sec(a)
-------------
/pi \
3*csc|-- - a|
\2 /
$$\frac{\sec{\left(a \right)}}{3 \csc{\left(- a + \frac{\pi}{2} \right)}}$$
/ 2/a\\ /a pi\
-2*|1 + tan |-||*tan|- + --|
\ \2// \2 4 /
-----------------------------------
/ 2/a pi\\ / 2/a\\
3*|1 + tan |- + --||*|-1 + tan |-||
\ \2 4 // \ \2//
$$- \frac{2 \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{3 \left(\tan^{2}{\left(\frac{a}{2} \right)} - 1\right) \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)}$$
-cos(pi + a) ------------- 3*cos(a)
$$- \frac{\cos{\left(a + \pi \right)}}{3 \cos{\left(a \right)}}$$
2/a\ /a pi\
-tan |-|*(1 + sin(a))*cot|- + --|
\4/ \2 4 /
----------------------------------
/ 2/a\\ 4/a\
12*|1 - cot |-||*sin |-|
\ \2// \4/
$$- \frac{\left(\sin{\left(a \right)} + 1\right) \tan^{2}{\left(\frac{a}{4} \right)} \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{12 \left(1 - \cot^{2}{\left(\frac{a}{2} \right)}\right) \sin^{4}{\left(\frac{a}{4} \right)}}$$
2/a\ / 1 \ /a pi\ /a pi\
-csc |-|*|1 - ------|*(1 + sin(a))*cos|- + --|*csc|- + --|
\2/ \ cos(a)/ \2 4 / \2 4 /
-----------------------------------------------------------
6
$$- \frac{\left(1 - \frac{1}{\cos{\left(a \right)}}\right) \left(\sin{\left(a \right)} + 1\right) \cos{\left(\frac{a}{2} + \frac{\pi}{4} \right)} \csc^{2}{\left(\frac{a}{2} \right)} \csc{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{6}$$
/ 1 \
-2*|1 + -------|
| 2/a\|
| tan |-||
\ \2//
----------------------------------------------
/ 1 \ / 1 \ /a pi\
3*|1 + ------------|*|1 - -------|*tan|- + --|
| 2/a pi\| | 2/a\| \2 4 /
| tan |- + --|| | tan |-||
\ \2 4 // \ \2//
$$- \frac{2 \left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)}{3 \left(1 - \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}\right) \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
/ / pi\\ /a pi\
-|1 + cos|a - --||*cos|- + --|
\ \ 2 // \2 4 /
---------------------------------------------
/ 2/a\ \
| cos |-| |
| \2/ | 2/a pi\ /a pi\
3*|1 - ------------|*cos |- - --|*cos|- - --|
| 2/a pi\| \2 2 / \2 4 /
| cos |- - --||
\ \2 2 //
$$- \frac{\left(\cos{\left(a - \frac{\pi}{2} \right)} + 1\right) \cos{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{3 \left(- \frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)} \cos{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}$$
-(1 + sin(a))*cos(a) -------------------------------------- / 2 \ | sin (a) | 2/a\ 2/a pi\ 6*|1 - ---------|*sin |-|*sin |- + --| | 4/a\| \2/ \2 4 / | 4*sin |-|| \ \2//
$$- \frac{\left(\sin{\left(a \right)} + 1\right) \cos{\left(a \right)}}{6 \left(1 - \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)} \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
2/a pi\ / 1 \ /a pi\
-sec |- - --|*|1 + -----------|*sec|- - --|
\2 2 / | / pi\| \2 4 /
| sec|a - --||
\ \ 2 //
--------------------------------------------
/ 2/a pi\\
| sec |- - --||
| \2 2 /| /a pi\
3*|1 - ------------|*sec|- + --|
| 2/a\ | \2 4 /
| sec |-| |
\ \2/ /
$$- \frac{\left(1 + \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}\right) \sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)} \sec{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}{3 \left(1 - \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right) \sec{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
/ 1 \
-2*|1 + -------|
| 2/a\|
| cot |-||
\ \2//
-----------------------------------------------
/ 1 \ / 1 \ /a pi\
3*|1 + ------------|*|-1 + -------|*cot|- + --|
| 2/a pi\| | 2/a\| \2 4 /
| cot |- + --|| | cot |-||
\ \2 4 // \ \2//
$$- \frac{2 \left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)}{3 \left(-1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}\right) \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
/ 2/a\\ /a pi\
-2*|1 + cot |-||*cot|- + --|
\ \2// \2 4 /
----------------------------------
/ 2/a pi\\ / 2/a\\
3*|1 + cot |- + --||*|1 - cot |-||
\ \2 4 // \ \2//
$$- \frac{2 \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right) \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{3 \left(1 - \cot^{2}{\left(\frac{a}{2} \right)}\right) \left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)}$$
-2*cos(a)
------------------------------
/ 2 \
| sin (a) |
3*(1 - cos(a))*|1 - ---------|
| 4/a\|
| 4*sin |-||
\ \2//
$$- \frac{2 \cos{\left(a \right)}}{3 \left(1 - \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \left(1 - \cos{\left(a \right)}\right)}$$
/ pi\ sin|a + --| \ 2 / ----------- 3*cos(a)
$$\frac{\sin{\left(a + \frac{\pi}{2} \right)}}{3 \cos{\left(a \right)}}$$
/ /a\ \
2 | 2*tan|-| |
/ 2/a\\ | \2/ |
-|1 + tan |-|| *|1 + -----------|
\ \4// | 2/a\|
| 1 + tan |-||
\ \2//
------------------------------------
/ 1 \ 2/a\ /a pi\
12*|1 - -------|*tan |-|*tan|- + --|
| 2/a\| \4/ \2 4 /
| tan |-||
\ \2//
$$- \frac{\left(1 + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}}{12 \left(1 - \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \tan^{2}{\left(\frac{a}{4} \right)} \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
1/3
$$\frac{1}{3}$$
/a pi\
-(1 + sin(a))*cot|- + --|
\2 4 /
--------------------------
/ 2/a\\ 2/a\
3*|1 - cot |-||*sin |-|
\ \2// \2/
$$- \frac{\left(\sin{\left(a \right)} + 1\right) \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{3 \left(1 - \cot^{2}{\left(\frac{a}{2} \right)}\right) \sin^{2}{\left(\frac{a}{2} \right)}}$$
/ 1 \
-|1 - ------|
\ cos(a)/
----------------------------------------
/ 1 \ 2/a\ /a pi\
3*|1 + ------------|*sin |-|*tan|- + --|
| 2/a pi\| \2/ \2 4 /
| tan |- + --||
\ \2 4 //
$$- \frac{1 - \frac{1}{\cos{\left(a \right)}}}{3 \left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)} \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
/ /a\\ / 1 \
|1 - tan|-||*|------ + tan(a)|
\ \2// \cos(a) /
------------------------------
/ /a\\
3*|1 + tan|-||
\ \2//
$$\frac{\left(1 - \tan{\left(\frac{a}{2} \right)}\right) \left(\tan{\left(a \right)} + \frac{1}{\cos{\left(a \right)}}\right)}{3 \left(\tan{\left(\frac{a}{2} \right)} + 1\right)}$$
-cos(a)
----------------
/ 3*pi\
3*sin|-a + ----|
\ 2 /
$$- \frac{\cos{\left(a \right)}}{3 \sin{\left(- a + \frac{3 \pi}{2} \right)}}$$
/ 1 \
-|1 - ------|*(1 + sin(a))
\ cos(a)/
------------------------------
4/a\ 2/a\ /a pi\
24*cos |-|*tan |-|*tan|- + --|
\4/ \4/ \2 4 /
$$- \frac{\left(1 - \frac{1}{\cos{\left(a \right)}}\right) \left(\sin{\left(a \right)} + 1\right)}{24 \cos^{4}{\left(\frac{a}{4} \right)} \tan^{2}{\left(\frac{a}{4} \right)} \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
-(1 - 1/cos(a))*(1 + sin(a))/(24*cos(a/4)^4*tan(a/4)^2*tan(a/2 + pi/4))