Sr Examen
¿Cómo vas a descomponer esta sin(x)/4-1/(sin(x)*4)-cos(x)/4 expresión en fracciones?
Solución
Ha introducido
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sin(x) 1 cos(x) ------ - -------- - ------ 4 sin(x)*4 4
$$\left(\frac{\sin{\left(x \right)}}{4} - \frac{1}{4 \sin{\left(x \right)}}\right) - \frac{\cos{\left(x \right)}}{4}$$
sin(x)/4 - 1/(sin(x)*4) - cos(x)/4
Simplificación general
[src]
___ / pi\
\/ 2 *cos|x + --|
1 \ 4 /
- -------- - -----------------
4*sin(x) 4
$$- \frac{\sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)}}{4} - \frac{1}{4 \sin{\left(x \right)}}$$
-1/(4*sin(x)) - sqrt(2)*cos(x + pi/4)/4
Respuesta numérica
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0.25*sin(x) - 0.25/sin(x) - 0.25*cos(x)
0.25*sin(x) - 0.25/sin(x) - 0.25*cos(x)
Combinatoria
[src]
/ 2 \
-\1 - sin (x) + cos(x)*sin(x)/
-------------------------------
4*sin(x)
$$- \frac{- \sin^{2}{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} + 1}{4 \sin{\left(x \right)}}$$
-(1 - sin(x)^2 + cos(x)*sin(x))/(4*sin(x))
Potencias
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I*x -I*x / -I*x I*x\
e e I I*\- e + e /
- ---- - ----- - ------------------ - ------------------
8 8 / -I*x I*x\ 8
2*\- e + e /
$$- \frac{i \left(e^{i x} - e^{- i x}\right)}{8} - \frac{e^{i x}}{8} - \frac{e^{- i x}}{8} - \frac{i}{2 \left(e^{i x} - e^{- i x}\right)}$$
1 cos(x) sin(x) - -------- - ------ + ------ 4*sin(x) 4 4
$$\frac{\sin{\left(x \right)}}{4} - \frac{\cos{\left(x \right)}}{4} - \frac{1}{4 \sin{\left(x \right)}}$$
-1/(4*sin(x)) - cos(x)/4 + sin(x)/4
Denominador común
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1 cos(x) sin(x) - -------- - ------ + ------ 4*sin(x) 4 4
$$\frac{\sin{\left(x \right)}}{4} - \frac{\cos{\left(x \right)}}{4} - \frac{1}{4 \sin{\left(x \right)}}$$
-1/(4*sin(x)) - cos(x)/4 + sin(x)/4
Denominador racional
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2
-16 + 16*sin (x) - 16*cos(x)*sin(x)
-----------------------------------
64*sin(x)
$$\frac{16 \sin^{2}{\left(x \right)} - 16 \sin{\left(x \right)} \cos{\left(x \right)} - 16}{64 \sin{\left(x \right)}}$$
(-16 + 16*sin(x)^2 - 16*cos(x)*sin(x))/(64*sin(x))
Unión de expresiones racionales
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2
-1 + sin (x) - cos(x)*sin(x)
----------------------------
4*sin(x)
$$\frac{\sin^{2}{\left(x \right)} - \sin{\left(x \right)} \cos{\left(x \right)} - 1}{4 \sin{\left(x \right)}}$$
(-1 + sin(x)^2 - cos(x)*sin(x))/(4*sin(x))
Compilar la expresión
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1 cos(x) sin(x) - -------- - ------ + ------ 4*sin(x) 4 4
$$\frac{\sin{\left(x \right)}}{4} - \frac{\cos{\left(x \right)}}{4} - \frac{1}{4 \sin{\left(x \right)}}$$
-1/(4*sin(x)) - cos(x)/4 + sin(x)/4
Parte trigonométrica
[src]
___ / pi\
\/ 2 *cos|x + --|
1 \ 4 /
- -------- - -----------------
4*sin(x) 4
$$- \frac{\sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)}}{4} - \frac{1}{4 \sin{\left(x \right)}}$$
___ / 2/x pi\\
\/ 2 *|1 - tan |- + --||
1 \ \2 8 //
- -------- - ------------------------
4*sin(x) / 2/x pi\\
4*|1 + tan |- + --||
\ \2 8 //
$$- \frac{\sqrt{2} \left(1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}\right)}{4 \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right)} - \frac{1}{4 \sin{\left(x \right)}}$$
/ pi\
sin|x + --|
1 \ 2 / sin(x)
- -------- - ----------- + ------
4*sin(x) 4 4
$$\frac{\sin{\left(x \right)}}{4} - \frac{\sin{\left(x + \frac{\pi}{2} \right)}}{4} - \frac{1}{4 \sin{\left(x \right)}}$$
/ pi\
sec|x - --|
1 \ 2 / 1
- -------- - ----------- + -------------
4*sec(x) 4 / pi\
4*sec|x - --|
\ 2 /
$$- \frac{\sec{\left(x - \frac{\pi}{2} \right)}}{4} + \frac{1}{4 \sec{\left(x - \frac{\pi}{2} \right)}} - \frac{1}{4 \sec{\left(x \right)}}$$
___
csc(x) \/ 2
- ------ - --------------
4 / pi\
4*csc|-x + --|
\ 4 /
$$- \frac{\csc{\left(x \right)}}{4} - \frac{\sqrt{2}}{4 \csc{\left(- x + \frac{\pi}{4} \right)}}$$
/ cos(x)\
-|1 + ------|*cos(x)
\ sin(x)/
---------------------
4
$$- \frac{\left(1 + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \cos{\left(x \right)}}{4}$$
___ / 3*pi\
\/ 2 *sin|x + ----|
1 \ 4 /
- -------- - -------------------
4*sin(x) 4
$$- \frac{\sqrt{2} \sin{\left(x + \frac{3 \pi}{4} \right)}}{4} - \frac{1}{4 \sin{\left(x \right)}}$$
1 csc(x) 1 - -------- - ------ + -------- 4*sec(x) 4 4*csc(x)
$$- \frac{\csc{\left(x \right)}}{4} - \frac{1}{4 \sec{\left(x \right)}} + \frac{1}{4 \csc{\left(x \right)}}$$
1 cos(x) sin(x) - -------- - ------ + ------ 4*sin(x) 4 4
$$\frac{\sin{\left(x \right)}}{4} - \frac{\cos{\left(x \right)}}{4} - \frac{1}{4 \sin{\left(x \right)}}$$
/ pi\
cos|x - --|
1 cos(x) \ 2 /
- ------------- - ------ + -----------
/ pi\ 4 4
4*cos|x - --|
\ 2 /
$$- \frac{\cos{\left(x \right)}}{4} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{4} - \frac{1}{4 \cos{\left(x - \frac{\pi}{2} \right)}}$$
/ pi\
sec|x - --| ___
\ 2 / \/ 2
- ----------- - -------------
4 / pi\
4*sec|x + --|
\ 4 /
$$- \frac{\sec{\left(x - \frac{\pi}{2} \right)}}{4} - \frac{\sqrt{2}}{4 \sec{\left(x + \frac{\pi}{4} \right)}}$$
/x\ 2/x\ 2/x\
tan|-| 1 - tan |-| 1 + tan |-|
\2/ \2/ \2/
--------------- - --------------- - -----------
/ 2/x\\ / 2/x\\ /x\
2*|1 + tan |-|| 4*|1 + tan |-|| 8*tan|-|
\ \2// \ \2// \2/
$$- \frac{1 - \tan^{2}{\left(\frac{x}{2} \right)}}{4 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} - \frac{\tan^{2}{\left(\frac{x}{2} \right)} + 1}{8 \tan{\left(\frac{x}{2} \right)}} + \frac{\tan{\left(\frac{x}{2} \right)}}{2 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
___ 2/x pi\ / 2/x pi\\
\/ 2 *sin |- + --|*|-1 + cot |- + --||
1 \2 8 / \ \2 8 //
- -------- - --------------------------------------
4*sin(x) 4
$$- \frac{\sqrt{2} \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} - 1\right) \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{4} - \frac{1}{4 \sin{\left(x \right)}}$$
2/x\ ___ / 2/x pi\\
1 + tan |-| \/ 2 *|1 - tan |- + --||
\2/ \ \2 8 //
- ----------- - ------------------------
/x\ / 2/x pi\\
8*tan|-| 4*|1 + tan |- + --||
\2/ \ \2 8 //
$$- \frac{\sqrt{2} \left(1 - \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)}\right)}{4 \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right)} - \frac{\tan^{2}{\left(\frac{x}{2} \right)} + 1}{8 \tan{\left(\frac{x}{2} \right)}}$$
___ / pi\
\/ 2 *cos|x + --|
csc(x) \ 4 /
- ------ - -----------------
4 4
$$- \frac{\sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)}}{4} - \frac{\csc{\left(x \right)}}{4}$$
___ / pi\
\/ 2 *cos|x + --|
1 \ 4 /
- ------------- - -----------------
/ pi\ 4
4*cos|x - --|
\ 2 /
$$- \frac{\sqrt{2} \cos{\left(x + \frac{\pi}{4} \right)}}{4} - \frac{1}{4 \cos{\left(x - \frac{\pi}{2} \right)}}$$
/x\ 2/x\ 2/x\
cot|-| -1 + cot |-| 1 + cot |-|
\2/ \2/ \2/
--------------- - --------------- - -----------
/ 2/x\\ / 2/x\\ /x\
2*|1 + cot |-|| 4*|1 + cot |-|| 8*cot|-|
\ \2// \ \2// \2/
$$- \frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{4 \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} - \frac{\cot^{2}{\left(\frac{x}{2} \right)} + 1}{8 \cot{\left(\frac{x}{2} \right)}} + \frac{\cot{\left(\frac{x}{2} \right)}}{2 \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
1 csc(x) 1
- ------------- - ------ + --------
/pi \ 4 4*csc(x)
4*csc|-- - x|
\2 /
$$- \frac{\csc{\left(x \right)}}{4} - \frac{1}{4 \csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{1}{4 \csc{\left(x \right)}}$$
2/x\ ___ / 2/x pi\\
1 + cot |-| \/ 2 *|-1 + cot |- + --||
\2/ \ \2 8 //
- ----------- - -------------------------
/x\ / 2/x pi\\
8*cot|-| 4*|1 + cot |- + --||
\2/ \ \2 8 //
$$- \frac{\cot^{2}{\left(\frac{x}{2} \right)} + 1}{8 \cot{\left(\frac{x}{2} \right)}} - \frac{\sqrt{2} \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} - 1\right)}{4 \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1\right)}$$
-(1 + cot(x/2)^2)/(8*cot(x/2)) - sqrt(2)*(-1 + cot(x/2 + pi/8)^2)/(4*(1 + cot(x/2 + pi/8)^2))