Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sin(x)/(2*sqrt(1-cos(x)^2)*sqrt(acos(cos(x))))
¿Cómo vas a descomponer esta sin(x)/(2*sqrt(1-cos(x)^2)*sqrt(acos(cos(x)))) expresión en fracciones?
Solución
Ha introducido
[src]
sin(x)
-----------------------------------
_____________
/ 2 ______________
2*\/ 1 - cos (x) *\/ acos(cos(x))
$$\frac{\sin{\left(x \right)}}{2 \sqrt{1 - \cos^{2}{\left(x \right)}} \sqrt{\operatorname{acos}{\left(\cos{\left(x \right)} \right)}}}$$
sin(x)/(((2*sqrt(1 - cos(x)^2))*sqrt(acos(cos(x)))))
Simplificación general
[src]
sin(x)
-------------------------------
_________
/ 2 ______________
2*\/ sin (x) *\/ acos(cos(x))
$$\frac{\sin{\left(x \right)}}{2 \sqrt{\sin^{2}{\left(x \right)}} \sqrt{\operatorname{acos}{\left(\cos{\left(x \right)} \right)}}}$$
sin(x)/(2*sqrt(sin(x)^2)*sqrt(acos(cos(x))))
Respuesta numérica
[src]
0.5*(1.0 - cos(x)^2)^(-0.5)*acos(cos(x))^(-0.5)*sin(x)
0.5*(1.0 - cos(x)^2)^(-0.5)*acos(cos(x))^(-0.5)*sin(x)
Potencias
[src]
/ -I*x I*x\
-I*\- e + e /
-------------------------------------------------------
_____________________
/ 2 ____________________
/ / I*x -I*x\ / / I*x -I*x\
/ |e e | / |e e |
4* / 1 - |---- + -----| * / acos|---- + -----|
\/ \ 2 2 / \/ \ 2 2 /
$$- \frac{i \left(e^{i x} - e^{- i x}\right)}{4 \sqrt{1 - \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right)^{2}} \sqrt{\operatorname{acos}{\left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2} \right)}}}$$
sin(x)
---------------------------------
____________________________
/ / 2 \
2*\/ \1 - cos (x)/*acos(cos(x))
$$\frac{\sin{\left(x \right)}}{2 \sqrt{\left(1 - \cos^{2}{\left(x \right)}\right) \operatorname{acos}{\left(\cos{\left(x \right)} \right)}}}$$
sin(x)/(2*sqrt((1 - cos(x)^2)*acos(cos(x))))
Denominador racional
[src]
_____________
/ 2
-\/ 1 - cos (x) *sin(x)
---------------------------------
/ 2 \ ______________
2*\-1 + cos (x)/*\/ acos(cos(x))
$$- \frac{\sqrt{1 - \cos^{2}{\left(x \right)}} \sin{\left(x \right)}}{2 \left(\cos^{2}{\left(x \right)} - 1\right) \sqrt{\operatorname{acos}{\left(\cos{\left(x \right)} \right)}}}$$
-sqrt(1 - cos(x)^2)*sin(x)/(2*(-1 + cos(x)^2)*sqrt(acos(cos(x))))
Combinatoria
[src]
sin(x)
--------------------------------------------------
_____________________________ ______________
2*\/ -(1 + cos(x))*(-1 + cos(x)) *\/ acos(cos(x))
$$\frac{\sin{\left(x \right)}}{2 \sqrt{- \left(\cos{\left(x \right)} - 1\right) \left(\cos{\left(x \right)} + 1\right)} \sqrt{\operatorname{acos}{\left(\cos{\left(x \right)} \right)}}}$$
sin(x)/(2*sqrt(-(1 + cos(x))*(-1 + cos(x)))*sqrt(acos(cos(x))))
Parte trigonométrica
[src]
/x\
cot|-|
\2/
----------------------------------------------------------------------
________________ ____________________
/ 2/x\ / / 2/x\\
/ cot |-| / |-1 + cot |-||
/ \2/ / 2/x\\ / | \2/|
2* / -------------- *|1 + cot |-||* / acos|------------|
/ 2 \ \2// / | 2/x\ |
/ / 2/x\\ / |1 + cot |-| |
/ |1 + cot |-|| \/ \ \2/ /
\/ \ \2//
$$\frac{\cot{\left(\frac{x}{2} \right)}}{2 \sqrt{\frac{\cot^{2}{\left(\frac{x}{2} \right)}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \sqrt{\operatorname{acos}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}}$$
/x\
tan|-|
\2/
---------------------------------------------------------------------
________________ ___________________
/ 2/x\ / / 2/x\\
/ tan |-| / |1 - tan |-||
/ \2/ / 2/x\\ / | \2/|
2* / -------------- *|1 + tan |-||* / acos|-----------|
/ 2 \ \2// / | 2/x\|
/ / 2/x\\ / |1 + tan |-||
/ |1 + tan |-|| \/ \ \2//
\/ \ \2//
$$\frac{\tan{\left(\frac{x}{2} \right)}}{2 \sqrt{\frac{\tan^{2}{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \sqrt{\operatorname{acos}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}}$$
___ /x\
\/ 2 *cot|-|
\2/
-------------------------------------------------------------------
____________________
__________________ / / 2/x\\
/ 2 / |-1 + cot |-||
/ 2/x\\ / -1 + cot (x) / | \2/|
|1 + cot |-||* / 1 - ------------ * / acos|------------|
\ \2// / 2 / | 2/x\ |
\/ 1 + cot (x) / |1 + cot |-| |
\/ \ \2/ /
$$\frac{\sqrt{2} \cot{\left(\frac{x}{2} \right)}}{\sqrt{- \frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} + 1} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \sqrt{\operatorname{acos}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}}$$
1
----------------------------------------------
_____________ ______________
/ 1 / / 1 \
2* / 1 - ------- * / acos|------| *csc(x)
/ 2 \/ \sec(x)/
\/ sec (x)
$$\frac{1}{2 \sqrt{1 - \frac{1}{\sec^{2}{\left(x \right)}}} \sqrt{\operatorname{acos}{\left(\frac{1}{\sec{\left(x \right)}} \right)}} \csc{\left(x \right)}}$$
sin(x)
--------------------------------------
_________ ___________________
/ 2 / / / pi\\
2*\/ sin (x) * / acos|sin|x + --||
\/ \ \ 2 //
$$\frac{\sin{\left(x \right)}}{2 \sqrt{\sin^{2}{\left(x \right)}} \sqrt{\operatorname{acos}{\left(\sin{\left(x + \frac{\pi}{2} \right)} \right)}}}$$
___
\/ 2
---------------------------------------------------
______________ ______________
/ 1 / / 1 \ / pi\
2* / 1 - -------- * / acos|------| *sec|x - --|
\/ sec(2*x) \/ \sec(x)/ \ 2 /
$$\frac{\sqrt{2}}{2 \sqrt{1 - \frac{1}{\sec{\left(2 x \right)}}} \sqrt{\operatorname{acos}{\left(\frac{1}{\sec{\left(x \right)}} \right)}} \sec{\left(x - \frac{\pi}{2} \right)}}$$
___ /x\
\/ 2 *tan|-|
\2/
-----------------------------------------------------------------
___________________
_________________ / / 2/x\\
/ 2 / |1 - tan |-||
/ 2/x\\ / 1 - tan (x) / | \2/|
|1 + tan |-||* / 1 - ----------- * / acos|-----------|
\ \2// / 2 / | 2/x\|
\/ 1 + tan (x) / |1 + tan |-||
\/ \ \2//
$$\frac{\sqrt{2} \tan{\left(\frac{x}{2} \right)}}{\sqrt{- \frac{1 - \tan^{2}{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} + 1} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \sqrt{\operatorname{acos}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}}$$
___
\/ 2 *sin(x)
-------------------------------------------------
___________________ ___________________
/ /pi \ / / / pi\\
2* / 1 - sin|-- + 2*x| * / acos|sin|x + --||
\/ \2 / \/ \ \ 2 //
$$\frac{\sqrt{2} \sin{\left(x \right)}}{2 \sqrt{1 - \sin{\left(2 x + \frac{\pi}{2} \right)}} \sqrt{\operatorname{acos}{\left(\sin{\left(x + \frac{\pi}{2} \right)} \right)}}}$$
/x\
cot|-|
\2/
--------------------------------------------------------------------------
_____________________
/ 2 ____________________
/ / 2/x\\ / / 2/x\\
/ |-1 + cot |-|| / |-1 + cot |-||
/ 2/x\\ / \ \2// / | \2/|
|1 + cot |-||* / 1 - --------------- * / acos|------------|
\ \2// / 2 / | 2/x\ |
/ / 2/x\\ / |1 + cot |-| |
/ |1 + cot |-|| \/ \ \2/ /
\/ \ \2//
$$\frac{\cot{\left(\frac{x}{2} \right)}}{\sqrt{- \frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 1} \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right) \sqrt{\operatorname{acos}{\left(\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}}$$
sin(x)
------------------------------------------------
__________________ ___________________
/ 2/ pi\ / / / pi\\
2* / 1 - sin |x + --| * / acos|sin|x + --||
\/ \ 2 / \/ \ \ 2 //
$$\frac{\sin{\left(x \right)}}{2 \sqrt{1 - \sin^{2}{\left(x + \frac{\pi}{2} \right)}} \sqrt{\operatorname{acos}{\left(\sin{\left(x + \frac{\pi}{2} \right)} \right)}}}$$
/ pi\
cos|x - --|
\ 2 /
-------------------------------------
______________
/ 2/ pi\ ______________
2* / cos |x - --| *\/ acos(cos(x))
\/ \ 2 /
$$\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{2 \sqrt{\cos^{2}{\left(x - \frac{\pi}{2} \right)}} \sqrt{\operatorname{acos}{\left(\cos{\left(x \right)} \right)}}}$$
/ pi\
cos|x - --|
\ 2 /
-----------------------------------
_____________
/ 2 ______________
2*\/ 1 - cos (x) *\/ acos(cos(x))
$$\frac{\cos{\left(x - \frac{\pi}{2} \right)}}{2 \sqrt{1 - \cos^{2}{\left(x \right)}} \sqrt{\operatorname{acos}{\left(\cos{\left(x \right)} \right)}}}$$
1
---------------------------------------------------
_____________ ______________
/ 1 / / 1 \ / pi\
2* / 1 - ------- * / acos|------| *sec|x - --|
/ 2 \/ \sec(x)/ \ 2 /
\/ sec (x)
$$\frac{1}{2 \sqrt{1 - \frac{1}{\sec^{2}{\left(x \right)}}} \sqrt{\operatorname{acos}{\left(\frac{1}{\sec{\left(x \right)}} \right)}} \sec{\left(x - \frac{\pi}{2} \right)}}$$
sin(x)
-------------------------------
_________
/ 2 ______________
2*\/ sin (x) *\/ acos(cos(x))
$$\frac{\sin{\left(x \right)}}{2 \sqrt{\sin^{2}{\left(x \right)}} \sqrt{\operatorname{acos}{\left(\cos{\left(x \right)} \right)}}}$$
1
-----------------------------------------------------------
__________________ ___________________
/ 1 / / 1 \
2* / 1 - ------------ * / acos|-----------| *csc(x)
/ 2/pi \ / | /pi \|
/ csc |-- - x| / |csc|-- - x||
\/ \2 / \/ \ \2 //
$$\frac{1}{2 \sqrt{1 - \frac{1}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}} \sqrt{\operatorname{acos}{\left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} \right)}} \csc{\left(x \right)}}$$
/x\
tan|-|
\2/
------------------------------------------------------------------------
____________________
/ 2 ___________________
/ / 2/x\\ / / 2/x\\
/ |1 - tan |-|| / |1 - tan |-||
/ 2/x\\ / \ \2// / | \2/|
|1 + tan |-||* / 1 - -------------- * / acos|-----------|
\ \2// / 2 / | 2/x\|
/ / 2/x\\ / |1 + tan |-||
/ |1 + tan |-|| \/ \ \2//
\/ \ \2//
$$\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{- \frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}} + 1} \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \sqrt{\operatorname{acos}{\left(\frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} \right)}}}$$
1
-------------------------------------------------
_________ ___________________
/ 1 / / 1 \
2* / ------- * / acos|-----------| *csc(x)
/ 2 / | /pi \|
\/ csc (x) / |csc|-- - x||
\/ \ \2 //
$$\frac{1}{2 \sqrt{\frac{1}{\csc^{2}{\left(x \right)}}} \sqrt{\operatorname{acos}{\left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} \right)}} \csc{\left(x \right)}}$$
___
\/ 2 *sin(x)
-----------------------------------
______________ ______________
2*\/ 1 - cos(2*x) *\/ acos(cos(x))
$$\frac{\sqrt{2} \sin{\left(x \right)}}{2 \sqrt{1 - \cos{\left(2 x \right)}} \sqrt{\operatorname{acos}{\left(\cos{\left(x \right)} \right)}}}$$
1
-----------------------------------------------------
______________ ______________
/ 1 / / 1 \ / pi\
2* / ------------ * / acos|------| *sec|x - --|
/ 2/ pi\ \/ \sec(x)/ \ 2 /
/ sec |x - --|
\/ \ 2 /
$$\frac{1}{2 \sqrt{\frac{1}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}} \sqrt{\operatorname{acos}{\left(\frac{1}{\sec{\left(x \right)}} \right)}} \sec{\left(x - \frac{\pi}{2} \right)}}$$
___ / pi\
\/ 2 *cos|x - --|
\ 2 /
-----------------------------------
______________ ______________
2*\/ 1 - cos(2*x) *\/ acos(cos(x))
$$\frac{\sqrt{2} \cos{\left(x - \frac{\pi}{2} \right)}}{2 \sqrt{1 - \cos{\left(2 x \right)}} \sqrt{\operatorname{acos}{\left(\cos{\left(x \right)} \right)}}}$$
___
\/ 2
------------------------------------------------------------
___________________ ___________________
/ 1 / / 1 \
2* / 1 - ------------- * / acos|-----------| *csc(x)
/ /pi \ / | /pi \|
/ csc|-- - 2*x| / |csc|-- - x||
\/ \2 / \/ \ \2 //
$$\frac{\sqrt{2}}{2 \sqrt{1 - \frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}}} \sqrt{\operatorname{acos}{\left(\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} \right)}} \csc{\left(x \right)}}$$
sqrt(2)/(2*sqrt(1 - 1/csc(pi/2 - 2*x))*sqrt(acos(1/csc(pi/2 - x)))*csc(x))