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