Sr Examen
¿Cómo vas a descomponer esta ctgx*(3tgy+ctgx)/(tgx+3ctgy) expresión en fracciones?
Solución
Ha introducido
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cot(x)*(3*tan(y) + cot(x))
--------------------------
tan(x) + 3*cot(y)
$$\frac{\left(3 \tan{\left(y \right)} + \cot{\left(x \right)}\right) \cot{\left(x \right)}}{\tan{\left(x \right)} + 3 \cot{\left(y \right)}}$$
(cot(x)*(3*tan(y) + cot(x)))/(tan(x) + 3*cot(y))
Respuesta numérica
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(3.0*tan(y) + cot(x))*cot(x)/(3.0*cot(y) + tan(x))
(3.0*tan(y) + cot(x))*cot(x)/(3.0*cot(y) + tan(x))
Denominador común
[src]
2
cot (x) + 3*cot(x)*tan(y)
-------------------------
3*cot(y) + tan(x)
$$\frac{3 \tan{\left(y \right)} \cot{\left(x \right)} + \cot^{2}{\left(x \right)}}{\tan{\left(x \right)} + 3 \cot{\left(y \right)}}$$
(cot(x)^2 + 3*cot(x)*tan(y))/(3*cot(y) + tan(x))
Potencias
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/ / I*y -I*y\ \
|3*I*\- e + e / |
|-------------------- + cot(x)|*cot(x)
| I*y -I*y |
\ e + e /
--------------------------------------
/ I*x -I*x\
I*\- e + e /
3*cot(y) + ------------------
I*x -I*x
e + e
$$\frac{\left(\frac{3 i \left(- e^{i y} + e^{- i y}\right)}{e^{i y} + e^{- i y}} + \cot{\left(x \right)}\right) \cot{\left(x \right)}}{\frac{i \left(- e^{i x} + e^{- i x}\right)}{e^{i x} + e^{- i x}} + 3 \cot{\left(y \right)}}$$
(3*i*(-exp(i*y) + exp(-i*y))/(exp(i*y) + exp(-i*y)) + cot(x))*cot(x)/(3*cot(y) + i*(-exp(i*x) + exp(-i*x))/(exp(i*x) + exp(-i*x)))
Parte trigonométrica
[src]
/ 3 \
|------ + cot(x)|*cot(x)
\cot(y) /
------------------------
1
------ + 3*cot(y)
cot(x)
$$\frac{\left(\cot{\left(x \right)} + \frac{3}{\cot{\left(y \right)}}\right) \cot{\left(x \right)}}{3 \cot{\left(y \right)} + \frac{1}{\cot{\left(x \right)}}}$$
1 ------ + 3*tan(y) tan(x) ------------------------ / 3 \ |------ + tan(x)|*tan(x) \tan(y) /
$$\frac{3 \tan{\left(y \right)} + \frac{1}{\tan{\left(x \right)}}}{\left(\tan{\left(x \right)} + \frac{3}{\tan{\left(y \right)}}\right) \tan{\left(x \right)}}$$
/ 1 \ |------ + 3*tan(y)|*tan(y) \tan(x) / -------------------------- (3 + tan(x)*tan(y))*tan(x)
$$\frac{\left(3 \tan{\left(y \right)} + \frac{1}{\tan{\left(x \right)}}\right) \tan{\left(y \right)}}{\left(\tan{\left(x \right)} \tan{\left(y \right)} + 3\right) \tan{\left(x \right)}}$$
/ / pi\ \ |sec|x - --| | | \ 2 / 3*sec(y) | / pi\ |----------- + -----------|*sec|x - --| | sec(x) / pi\| \ 2 / | sec|y - --|| \ \ 2 // --------------------------------------- / / pi\\ | 3*sec|y - --|| | sec(x) \ 2 /| |----------- + -------------|*sec(x) | / pi\ sec(y) | |sec|x - --| | \ \ 2 / /
$$\frac{\left(\frac{3 \sec{\left(y \right)}}{\sec{\left(y - \frac{\pi}{2} \right)}} + \frac{\sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)}}\right) \sec{\left(x - \frac{\pi}{2} \right)}}{\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{3 \sec{\left(y - \frac{\pi}{2} \right)}}{\sec{\left(y \right)}}\right) \sec{\left(x \right)}}$$
/ 2/x\\ / 3 \
|-1 + cot |-||*|------ + cot(x)|
\ \2// \cot(y) /
--------------------------------
/ 1 \ /x\
2*|------ + 3*cot(y)|*cot|-|
\cot(x) / \2/
$$\frac{\left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \left(\cot{\left(x \right)} + \frac{3}{\cot{\left(y \right)}}\right)}{2 \left(3 \cot{\left(y \right)} + \frac{1}{\cot{\left(x \right)}}\right) \cot{\left(\frac{x}{2} \right)}}$$
/ 2 \
2 | sin(2*x) 6*sin (y)|
sin (y)*|--------- + ---------|*sin(2*x)
| 2 sin(2*y)|
\2*sin (x) /
----------------------------------------
/ 2 2 \
| 4*sin (x)*sin (y)| 2
|3 + -----------------|*sin (x)*sin(2*y)
\ sin(2*x)*sin(2*y)/
$$\frac{\left(\frac{6 \sin^{2}{\left(y \right)}}{\sin{\left(2 y \right)}} + \frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(x \right)}}\right) \sin{\left(2 x \right)} \sin^{2}{\left(y \right)}}{\left(\frac{4 \sin^{2}{\left(x \right)} \sin^{2}{\left(y \right)}}{\sin{\left(2 x \right)} \sin{\left(2 y \right)}} + 3\right) \sin^{2}{\left(x \right)} \sin{\left(2 y \right)}}$$
/ 2 \ | sin(2*x) 6*sin (y)| |--------- + ---------|*sin(2*x) | 2 sin(2*y)| \2*sin (x) / ---------------------------------- / 2 \ |2*sin (x) 3*sin(2*y)| 2 2*|--------- + ----------|*sin (x) | sin(2*x) 2 | \ 2*sin (y) /
$$\frac{\left(\frac{6 \sin^{2}{\left(y \right)}}{\sin{\left(2 y \right)}} + \frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(x \right)}}\right) \sin{\left(2 x \right)}}{2 \left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + \frac{3 \sin{\left(2 y \right)}}{2 \sin^{2}{\left(y \right)}}\right) \sin^{2}{\left(x \right)}}$$
(3*tan(y) + cot(x))*cot(x)*tan(y)
---------------------------------
3 + tan(x)*tan(y)
$$\frac{\left(3 \tan{\left(y \right)} + \cot{\left(x \right)}\right) \tan{\left(y \right)} \cot{\left(x \right)}}{\tan{\left(x \right)} \tan{\left(y \right)} + 3}$$
/ 3 \ |------ + cot(x)|*cot(x) \cot(y) / -------------------------- / 1 \ |3 + -------------|*cot(y) \ cot(x)*cot(y)/
$$\frac{\left(\cot{\left(x \right)} + \frac{3}{\cot{\left(y \right)}}\right) \cot{\left(x \right)}}{\left(3 + \frac{1}{\cot{\left(x \right)} \cot{\left(y \right)}}\right) \cot{\left(y \right)}}$$
/ / pi\ \ |sec|x - --| | | \ 2 / 3*sec(y) | /pi \ |----------- + -----------|*sec|-- - x| | sec(x) / pi\| \2 / | sec|y - --|| \ \ 2 // --------------------------------------- / / pi\\ | 3*sec|y - --|| | sec(x) \ 2 /| |----------- + -------------|*sec(x) | / pi\ sec(y) | |sec|x - --| | \ \ 2 / /
$$\frac{\left(\frac{3 \sec{\left(y \right)}}{\sec{\left(y - \frac{\pi}{2} \right)}} + \frac{\sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)}}\right) \sec{\left(- x + \frac{\pi}{2} \right)}}{\left(\frac{\sec{\left(x \right)}}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{3 \sec{\left(y - \frac{\pi}{2} \right)}}{\sec{\left(y \right)}}\right) \sec{\left(x \right)}}$$
/ /pi \\ | 3*csc|-- - y|| | csc(x) \2 /| |----------- + -------------|*csc(x) | /pi \ csc(y) | |csc|-- - x| | \ \2 / / --------------------------------------- / /pi \ \ |csc|-- - x| | | \2 / 3*csc(y) | /pi \ |----------- + -----------|*csc|-- - x| | csc(x) /pi \| \2 / | csc|-- - y|| \ \2 //
$$\frac{\left(\frac{\csc{\left(x \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{3 \csc{\left(- y + \frac{\pi}{2} \right)}}{\csc{\left(y \right)}}\right) \csc{\left(x \right)}}{\left(\frac{3 \csc{\left(y \right)}}{\csc{\left(- y + \frac{\pi}{2} \right)}} + \frac{\csc{\left(- x + \frac{\pi}{2} \right)}}{\csc{\left(x \right)}}\right) \csc{\left(- x + \frac{\pi}{2} \right)}}$$
/ 1 \
|------ + 3*tan(y)|*cos(x)*csc(x)
\tan(x) /
---------------------------------
3
------ + tan(x)
tan(y)
$$\frac{\left(3 \tan{\left(y \right)} + \frac{1}{\tan{\left(x \right)}}\right) \cos{\left(x \right)} \csc{\left(x \right)}}{\tan{\left(x \right)} + \frac{3}{\tan{\left(y \right)}}}$$
/ / pi\\ | 3*cos|y - --|| | cos(x) \ 2 /| |----------- + -------------|*cos(x) | / pi\ cos(y) | |cos|x - --| | \ \ 2 / / --------------------------------------- / / pi\ \ |cos|x - --| | | \ 2 / 3*cos(y) | / pi\ |----------- + -----------|*cos|x - --| | cos(x) / pi\| \ 2 / | cos|y - --|| \ \ 2 //
$$\frac{\left(\frac{\cos{\left(x \right)}}{\cos{\left(x - \frac{\pi}{2} \right)}} + \frac{3 \cos{\left(y - \frac{\pi}{2} \right)}}{\cos{\left(y \right)}}\right) \cos{\left(x \right)}}{\left(\frac{3 \cos{\left(y \right)}}{\cos{\left(y - \frac{\pi}{2} \right)}} + \frac{\cos{\left(x - \frac{\pi}{2} \right)}}{\cos{\left(x \right)}}\right) \cos{\left(x - \frac{\pi}{2} \right)}}$$
/ / pi\ \ |sec|x - --| | | \ 2 / 3*sec(y) | / pi\ |----------- + -----------|*sec(y)*sec|x - --| | sec(x) / pi\| \ 2 / | sec|y - --|| \ \ 2 // ------------------------------------------------ / sec(x)*sec(y) \ / pi\ |3 + -----------------------|*sec(x)*sec|y - --| | / pi\ / pi\| \ 2 / | sec|x - --|*sec|y - --|| \ \ 2 / \ 2 //
$$\frac{\left(\frac{3 \sec{\left(y \right)}}{\sec{\left(y - \frac{\pi}{2} \right)}} + \frac{\sec{\left(x - \frac{\pi}{2} \right)}}{\sec{\left(x \right)}}\right) \sec{\left(y \right)} \sec{\left(x - \frac{\pi}{2} \right)}}{\left(\frac{\sec{\left(x \right)} \sec{\left(y \right)}}{\sec{\left(x - \frac{\pi}{2} \right)} \sec{\left(y - \frac{\pi}{2} \right)}} + 3\right) \sec{\left(x \right)} \sec{\left(y - \frac{\pi}{2} \right)}}$$
/ 2/x\\ /x\
|-1 + cot |-||*(3*tan(y) + cot(x))*tan|-|
\ \2// \2/
-----------------------------------------
2*(3*cot(y) + tan(x))
$$\frac{\left(3 \tan{\left(y \right)} + \cot{\left(x \right)}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right) \tan{\left(\frac{x}{2} \right)}}{2 \left(\tan{\left(x \right)} + 3 \cot{\left(y \right)}\right)}$$
/csc(x) 3*sec(y)\ |------ + --------|*csc(x) \sec(x) csc(y) / -------------------------- /sec(x) 3*csc(y)\ |------ + --------|*sec(x) \csc(x) sec(y) /
$$\frac{\left(\frac{\csc{\left(x \right)}}{\sec{\left(x \right)}} + \frac{3 \sec{\left(y \right)}}{\csc{\left(y \right)}}\right) \csc{\left(x \right)}}{\left(\frac{3 \csc{\left(y \right)}}{\sec{\left(y \right)}} + \frac{\sec{\left(x \right)}}{\csc{\left(x \right)}}\right) \sec{\left(x \right)}}$$
/cos(x) 3*sin(y)\ |------ + --------|*cos(x) \sin(x) cos(y) / -------------------------- /sin(x) 3*cos(y)\ |------ + --------|*sin(x) \cos(x) sin(y) /
$$\frac{\left(\frac{3 \sin{\left(y \right)}}{\cos{\left(y \right)}} + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\right) \cos{\left(x \right)}}{\left(\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{3 \cos{\left(y \right)}}{\sin{\left(y \right)}}\right) \sin{\left(x \right)}}$$
/ 2 \ | sin(2*x) 6*sin (y)| / pi\ |--------- + ---------|*sin|x + --| | 2 sin(2*y)| \ 2 / \2*sin (x) / ----------------------------------- / 2 \ |2*sin (x) 3*sin(2*y)| |--------- + ----------|*sin(x) | sin(2*x) 2 | \ 2*sin (y) /
$$\frac{\left(\frac{6 \sin^{2}{\left(y \right)}}{\sin{\left(2 y \right)}} + \frac{\sin{\left(2 x \right)}}{2 \sin^{2}{\left(x \right)}}\right) \sin{\left(x + \frac{\pi}{2} \right)}}{\left(\frac{2 \sin^{2}{\left(x \right)}}{\sin{\left(2 x \right)}} + \frac{3 \sin{\left(2 y \right)}}{2 \sin^{2}{\left(y \right)}}\right) \sin{\left(x \right)}}$$
/ 2/x\\ / 1 \
|1 - tan |-||*|------ + 3*tan(y)|
\ \2// \tan(x) /
---------------------------------
/ 3 \ /x\
2*|------ + tan(x)|*tan|-|
\tan(y) / \2/
$$\frac{\left(1 - \tan^{2}{\left(\frac{x}{2} \right)}\right) \left(3 \tan{\left(y \right)} + \frac{1}{\tan{\left(x \right)}}\right)}{2 \left(\tan{\left(x \right)} + \frac{3}{\tan{\left(y \right)}}\right) \tan{\left(\frac{x}{2} \right)}}$$
/ /pi \\ | 3*csc|-- - y|| | csc(x) \2 /| /pi \ |----------- + -------------|*csc(x)*csc|-- - y| | /pi \ csc(y) | \2 / |csc|-- - x| | \ \2 / / ------------------------------------------------ / /pi \ /pi \\ | csc|-- - x|*csc|-- - y|| | \2 / \2 /| /pi \ |3 + -----------------------|*csc(y)*csc|-- - x| \ csc(x)*csc(y) / \2 /
$$\frac{\left(\frac{\csc{\left(x \right)}}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{3 \csc{\left(- y + \frac{\pi}{2} \right)}}{\csc{\left(y \right)}}\right) \csc{\left(x \right)} \csc{\left(- y + \frac{\pi}{2} \right)}}{\left(3 + \frac{\csc{\left(- x + \frac{\pi}{2} \right)} \csc{\left(- y + \frac{\pi}{2} \right)}}{\csc{\left(x \right)} \csc{\left(y \right)}}\right) \csc{\left(y \right)} \csc{\left(- x + \frac{\pi}{2} \right)}}$$
/ / pi\\ | 3*cos|y - --|| | cos(x) \ 2 /| / pi\ |----------- + -------------|*cos(x)*cos|y - --| | / pi\ cos(y) | \ 2 / |cos|x - --| | \ \ 2 / / ------------------------------------------------ / / pi\ / pi\\ | cos|x - --|*cos|y - --|| | \ 2 / \ 2 /| / pi\ |3 + -----------------------|*cos(y)*cos|x - --| \ cos(x)*cos(y) / \ 2 /
$$\frac{\left(\frac{\cos{\left(x \right)}}{\cos{\left(x - \frac{\pi}{2} \right)}} + \frac{3 \cos{\left(y - \frac{\pi}{2} \right)}}{\cos{\left(y \right)}}\right) \cos{\left(x \right)} \cos{\left(y - \frac{\pi}{2} \right)}}{\left(3 + \frac{\cos{\left(x - \frac{\pi}{2} \right)} \cos{\left(y - \frac{\pi}{2} \right)}}{\cos{\left(x \right)} \cos{\left(y \right)}}\right) \cos{\left(y \right)} \cos{\left(x - \frac{\pi}{2} \right)}}$$
(cos(x)/cos(x - pi/2) + 3*cos(y - pi/2)/cos(y))*cos(x)*cos(y - pi/2)/((3 + cos(x - pi/2)*cos(y - pi/2)/(cos(x)*cos(y)))*cos(y)*cos(x - pi/2))