Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
cos(6*a)/sin(2*a)+sin(6*a)/((cos(2)*a))
¿Cómo vas a descomponer esta cos(6*a)/sin(2*a)+sin(6*a)/((cos(2)*a)) expresión en fracciones?
Solución
Ha introducido
[src]
cos(6*a) sin(6*a) -------- + -------- sin(2*a) cos(2)*a
$$\frac{\cos{\left(6 a \right)}}{\sin{\left(2 a \right)}} + \frac{\sin{\left(6 a \right)}}{a \cos{\left(2 \right)}}$$
cos(6*a)/sin(2*a) + sin(6*a)/((cos(2)*a))
Respuesta numérica
[src]
cos(6*a)/sin(2*a) - 2.40299796172238*sin(6*a)/a
cos(6*a)/sin(2*a) - 2.40299796172238*sin(6*a)/a
Combinatoria
[src]
sin(2*a)*sin(6*a) + a*cos(2)*cos(6*a)
-------------------------------------
a*cos(2)*sin(2*a)
$$\frac{a \cos{\left(2 \right)} \cos{\left(6 a \right)} + \sin{\left(2 a \right)} \sin{\left(6 a \right)}}{a \sin{\left(2 a \right)} \cos{\left(2 \right)}}$$
(sin(2*a)*sin(6*a) + a*cos(2)*cos(6*a))/(a*cos(2)*sin(2*a))
Denominador racional
[src]
sin(2*a)*sin(6*a) + a*cos(2)*cos(6*a)
-------------------------------------
a*cos(2)*sin(2*a)
$$\frac{a \cos{\left(2 \right)} \cos{\left(6 a \right)} + \sin{\left(2 a \right)} \sin{\left(6 a \right)}}{a \sin{\left(2 a \right)} \cos{\left(2 \right)}}$$
(sin(2*a)*sin(6*a) + a*cos(2)*cos(6*a))/(a*cos(2)*sin(2*a))
Unión de expresiones racionales
[src]
sin(2*a)*sin(6*a) + a*cos(2)*cos(6*a)
-------------------------------------
a*cos(2)*sin(2*a)
$$\frac{a \cos{\left(2 \right)} \cos{\left(6 a \right)} + \sin{\left(2 a \right)} \sin{\left(6 a \right)}}{a \sin{\left(2 a \right)} \cos{\left(2 \right)}}$$
(sin(2*a)*sin(6*a) + a*cos(2)*cos(6*a))/(a*cos(2)*sin(2*a))
Denominador común
[src]
sin(2*a)*sin(6*a) + a*cos(2)*cos(6*a)
-------------------------------------
a*cos(2)*sin(2*a)
$$\frac{a \cos{\left(2 \right)} \cos{\left(6 a \right)} + \sin{\left(2 a \right)} \sin{\left(6 a \right)}}{a \sin{\left(2 a \right)} \cos{\left(2 \right)}}$$
(sin(2*a)*sin(6*a) + a*cos(2)*cos(6*a))/(a*cos(2)*sin(2*a))
Potencias
[src]
/ -6*I*a 6*I*a\
|e e |
2*I*|------- + ------| / -6*I*a 6*I*a\
\ 2 2 / I*\- e + e /
---------------------- - ----------------------
-2*I*a 2*I*a / -2*I 2*I\
- e + e |e e |
2*a*|----- + ----|
\ 2 2 /
$$\frac{2 i \left(\frac{e^{6 i a}}{2} + \frac{e^{- 6 i a}}{2}\right)}{e^{2 i a} - e^{- 2 i a}} - \frac{i \left(e^{6 i a} - e^{- 6 i a}\right)}{2 a \left(\frac{e^{- 2 i}}{2} + \frac{e^{2 i}}{2}\right)}$$
2*i*(exp(-6*i*a)/2 + exp(6*i*a)/2)/(-exp(-2*i*a) + exp(2*i*a)) - i*(-exp(-6*i*a) + exp(6*i*a))/(2*a*(exp(-2*i)/2 + exp(2*i)/2))
Parte trigonométrica
[src]
sec(2)*sin(6*a)
cos(6*a)*csc(2*a) + ---------------
a
$$\cos{\left(6 a \right)} \csc{\left(2 a \right)} + \frac{\sin{\left(6 a \right)} \sec{\left(2 \right)}}{a}$$
1 1
----------------- + ------------------
2 2/ pi\
csc (1) csc |-1 + --|
1 - ------------- \ 2 /
2/ pi\ -1 + -------------
csc |-1 + --| 2
csc(2*a) \ 2 / csc (1)
------------- - --------------------------------------
/pi \ a*csc(6*a)
csc|-- - 6*a|
\2 /
$$\frac{\csc{\left(2 a \right)}}{\csc{\left(- 6 a + \frac{\pi}{2} \right)}} - \frac{\frac{1}{-1 + \frac{\csc^{2}{\left(-1 + \frac{\pi}{2} \right)}}{\csc^{2}{\left(1 \right)}}} + \frac{1}{- \frac{\csc^{2}{\left(1 \right)}}{\csc^{2}{\left(-1 + \frac{\pi}{2} \right)}} + 1}}{a \csc{\left(6 a \right)}}$$
/ 1 1 \
2*|----------- + ------------|*cot(3*a)
| 2 1 |
|1 - cot (1) -1 + -------|
/ 2 \ / 2 \ | 2 |
\1 + cot (a)/*\-1 + cot (3*a)/ \ cot (1)/
------------------------------ - ---------------------------------------
/ 2 \ / 2 \
2*\1 + cot (3*a)/*cot(a) a*\1 + cot (3*a)/
$$\frac{\left(\cot^{2}{\left(a \right)} + 1\right) \left(\cot^{2}{\left(3 a \right)} - 1\right)}{2 \left(\cot^{2}{\left(3 a \right)} + 1\right) \cot{\left(a \right)}} - \frac{2 \left(\frac{1}{-1 + \frac{1}{\cot^{2}{\left(1 \right)}}} + \frac{1}{1 - \cot^{2}{\left(1 \right)}}\right) \cot{\left(3 a \right)}}{a \left(\cot^{2}{\left(3 a \right)} + 1\right)}$$
csc(2*a) sec(2) -------- + ---------- sec(6*a) a*csc(6*a)
$$\frac{\csc{\left(2 a \right)}}{\sec{\left(6 a \right)}} + \frac{\sec{\left(2 \right)}}{a \csc{\left(6 a \right)}}$$
/ 1 1 \
2*|----------- + ------------|*tan(3*a)
| 1 2 |
|1 - ------- -1 + tan (1)|
/ 2 \ / 2 \ | 2 |
\1 + tan (a)/*\1 - tan (3*a)/ \ tan (1) /
----------------------------- - ---------------------------------------
/ 2 \ / 2 \
2*\1 + tan (3*a)/*tan(a) a*\1 + tan (3*a)/
$$\frac{\left(1 - \tan^{2}{\left(3 a \right)}\right) \left(\tan^{2}{\left(a \right)} + 1\right)}{2 \left(\tan^{2}{\left(3 a \right)} + 1\right) \tan{\left(a \right)}} - \frac{2 \left(\frac{1}{-1 + \tan^{2}{\left(1 \right)}} + \frac{1}{1 - \frac{1}{\tan^{2}{\left(1 \right)}}}\right) \tan{\left(3 a \right)}}{a \left(\tan^{2}{\left(3 a \right)} + 1\right)}$$
/ pi\
csc|-2 + --|
csc(2*a) \ 2 /
------------- + ------------
/pi \ a*csc(6*a)
csc|-- - 6*a|
\2 /
$$\frac{\csc{\left(2 a \right)}}{\csc{\left(- 6 a + \frac{\pi}{2} \right)}} + \frac{\csc{\left(-2 + \frac{\pi}{2} \right)}}{a \csc{\left(6 a \right)}}$$
cos(6*a) sec(2)*sin(6*a) -------- + --------------- sin(2*a) a
$$\frac{\cos{\left(6 a \right)}}{\sin{\left(2 a \right)}} + \frac{\sin{\left(6 a \right)} \sec{\left(2 \right)}}{a}$$
/ 1 \
| 1 - ------|
| 1 cos(2)|
|------------ + ----------|*sin(6*a)
| 2 2 |
\-1 + tan (1) /
cos(6*a)*csc(2*a) - ------------------------------------
a
$$\cos{\left(6 a \right)} \csc{\left(2 a \right)} - \frac{\left(\frac{1}{-1 + \tan^{2}{\left(1 \right)}} + \frac{1 - \frac{1}{\cos{\left(2 \right)}}}{2}\right) \sin{\left(6 a \right)}}{a}$$
/ 1 1 \
|------------- + --------------|*sin(6*a)
| 2 4 |
| sin (2) 4*sin (1)|
/pi \ |1 - --------- -1 + ---------|
sin|-- + 6*a| | 4 2 |
\2 / \ 4*sin (1) sin (2) /
------------- - -----------------------------------------
sin(2*a) a
$$\frac{\sin{\left(6 a + \frac{\pi}{2} \right)}}{\sin{\left(2 a \right)}} - \frac{\left(\frac{1}{-1 + \frac{4 \sin^{4}{\left(1 \right)}}{\sin^{2}{\left(2 \right)}}} + \frac{1}{- \frac{\sin^{2}{\left(2 \right)}}{4 \sin^{4}{\left(1 \right)}} + 1}\right) \sin{\left(6 a \right)}}{a}$$
/1 2 cos(4)\
|- + (1 - cos(2)) - ------|*sin(6*a)
cos(6*a) \2 2 /
-------- - -------------------------------------
sin(2*a) a*(1 - 2*cos(2) + cos(4))
$$\frac{\cos{\left(6 a \right)}}{\sin{\left(2 a \right)}} - \frac{\left(- \frac{\cos{\left(4 \right)}}{2} + \frac{1}{2} + \left(1 - \cos{\left(2 \right)}\right)^{2}\right) \sin{\left(6 a \right)}}{a \left(\cos{\left(4 \right)} - 2 \cos{\left(2 \right)} + 1\right)}$$
/ 1 \
| 1 - ------|
2 | 1 cos(2)|
2*cos (3*a)*|------------ + ----------|*tan(3*a)
| 2 2 |
cos(6*a) \-1 + tan (1) /
-------- - ------------------------------------------------
sin(2*a) a
$$\frac{\cos{\left(6 a \right)}}{\sin{\left(2 a \right)}} - \frac{2 \left(\frac{1}{-1 + \tan^{2}{\left(1 \right)}} + \frac{1 - \frac{1}{\cos{\left(2 \right)}}}{2}\right) \cos^{2}{\left(3 a \right)} \tan{\left(3 a \right)}}{a}$$
/pi \
sec|-- - 2*a|
\2 / sec(2)
------------- + ---------------
sec(6*a) / pi\
a*sec|6*a - --|
\ 2 /
$$\frac{\sec{\left(- 2 a + \frac{\pi}{2} \right)}}{\sec{\left(6 a \right)}} + \frac{\sec{\left(2 \right)}}{a \sec{\left(6 a - \frac{\pi}{2} \right)}}$$
/pi \
sin|-- + 6*a|
\2 / sin(6*a)
------------- + -------------
sin(2*a) / pi\
a*sin|2 + --|
\ 2 /
$$\frac{\sin{\left(6 a + \frac{\pi}{2} \right)}}{\sin{\left(2 a \right)}} + \frac{\sin{\left(6 a \right)}}{a \sin{\left(\frac{\pi}{2} + 2 \right)}}$$
/ pi\
cos|6*a - --|
cos(6*a) \ 2 /
------------- + -------------
/ pi\ a*cos(2)
cos|2*a - --|
\ 2 /
$$\frac{\cos{\left(6 a \right)}}{\cos{\left(2 a - \frac{\pi}{2} \right)}} + \frac{\cos{\left(6 a - \frac{\pi}{2} \right)}}{a \cos{\left(2 \right)}}$$
/ 2 \ / 2 \ / 2 \
\1 + cot (a)/*\-1 + cot (3*a)/ 2*\1 + cot (1)/*cot(3*a)
------------------------------ + --------------------------------
/ 2 \ / 2 \ / 2 \
2*\1 + cot (3*a)/*cot(a) a*\1 + cot (3*a)/*\-1 + cot (1)/
$$\frac{\left(\cot^{2}{\left(a \right)} + 1\right) \left(\cot^{2}{\left(3 a \right)} - 1\right)}{2 \left(\cot^{2}{\left(3 a \right)} + 1\right) \cot{\left(a \right)}} + \frac{2 \left(\cot^{2}{\left(1 \right)} + 1\right) \cot{\left(3 a \right)}}{a \left(-1 + \cot^{2}{\left(1 \right)}\right) \left(\cot^{2}{\left(3 a \right)} + 1\right)}$$
/ 1 1 \ / pi\
|---------------- + -----------------|*cos|6*a - --|
| 2 2/ pi\| \ 2 /
| cos (1) cos |1 - --||
|1 - ------------ \ 2 /|
| 2/ pi\ -1 + ------------|
| cos |1 - --| 2 |
cos(6*a) \ \ 2 / cos (1) /
------------- - ----------------------------------------------------
/ pi\ a
cos|2*a - --|
\ 2 /
$$\frac{\cos{\left(6 a \right)}}{\cos{\left(2 a - \frac{\pi}{2} \right)}} - \frac{\left(\frac{1}{-1 + \frac{\cos^{2}{\left(1 - \frac{\pi}{2} \right)}}{\cos^{2}{\left(1 \right)}}} + \frac{1}{- \frac{\cos^{2}{\left(1 \right)}}{\cos^{2}{\left(1 - \frac{\pi}{2} \right)}} + 1}\right) \cos{\left(6 a - \frac{\pi}{2} \right)}}{a}$$
/ 1 1 \
|----------- + ------------|*sin(6*a)
| 1 2 |
|1 - ------- -1 + tan (1)|
| 2 |
cos(6*a) \ tan (1) /
-------- - -------------------------------------
sin(2*a) a
$$\frac{\cos{\left(6 a \right)}}{\sin{\left(2 a \right)}} - \frac{\left(\frac{1}{-1 + \tan^{2}{\left(1 \right)}} + \frac{1}{1 - \frac{1}{\tan^{2}{\left(1 \right)}}}\right) \sin{\left(6 a \right)}}{a}$$
1 1
---------------- + -----------------
2/ pi\ 2
sec |1 - --| sec (1)
\ 2 / -1 + ------------
/ pi\ 1 - ------------ 2/ pi\
sec|2*a - --| 2 sec |1 - --|
\ 2 / sec (1) \ 2 /
------------- - ------------------------------------
sec(6*a) / pi\
a*sec|6*a - --|
\ 2 /
$$\frac{\sec{\left(2 a - \frac{\pi}{2} \right)}}{\sec{\left(6 a \right)}} - \frac{\frac{1}{-1 + \frac{\sec^{2}{\left(1 \right)}}{\sec^{2}{\left(1 - \frac{\pi}{2} \right)}}} + \frac{1}{- \frac{\sec^{2}{\left(1 - \frac{\pi}{2} \right)}}{\sec^{2}{\left(1 \right)}} + 1}}{a \sec{\left(6 a - \frac{\pi}{2} \right)}}$$
/ 1 \
| 1 - ------|
| 1 cos(2)|
|------------ + ----------|*sin(6*a)
| 2 2 |
cos(6*a) \-1 + tan (1) /
-------- - ------------------------------------
sin(2*a) a
$$\frac{\cos{\left(6 a \right)}}{\sin{\left(2 a \right)}} - \frac{\left(\frac{1}{-1 + \tan^{2}{\left(1 \right)}} + \frac{1 - \frac{1}{\cos{\left(2 \right)}}}{2}\right) \sin{\left(6 a \right)}}{a}$$
/ pi\
sec|2*a - --|
\ 2 / sec(2)
------------- + ---------------
sec(6*a) / pi\
a*sec|6*a - --|
\ 2 /
$$\frac{\sec{\left(2 a - \frac{\pi}{2} \right)}}{\sec{\left(6 a \right)}} + \frac{\sec{\left(2 \right)}}{a \sec{\left(6 a - \frac{\pi}{2} \right)}}$$
sin(6*a)
cos(6*a)*csc(2*a) + --------
a*cos(2)
$$\cos{\left(6 a \right)} \csc{\left(2 a \right)} + \frac{\sin{\left(6 a \right)}}{a \cos{\left(2 \right)}}$$
/ 2 \ / 2 \ / 2 \
\1 + tan (a)/*\1 - tan (3*a)/ 2*\1 + tan (1)/*tan(3*a)
----------------------------- + -------------------------------
/ 2 \ / 2 \ / 2 \
2*\1 + tan (3*a)/*tan(a) a*\1 + tan (3*a)/*\1 - tan (1)/
$$\frac{\left(1 - \tan^{2}{\left(3 a \right)}\right) \left(\tan^{2}{\left(a \right)} + 1\right)}{2 \left(\tan^{2}{\left(3 a \right)} + 1\right) \tan{\left(a \right)}} + \frac{2 \left(1 + \tan^{2}{\left(1 \right)}\right) \tan{\left(3 a \right)}}{a \left(1 - \tan^{2}{\left(1 \right)}\right) \left(\tan^{2}{\left(3 a \right)} + 1\right)}$$
(1 + tan(a)^2)*(1 - tan(3*a)^2)/(2*(1 + tan(3*a)^2)*tan(a)) + 2*(1 + tan(1)^2)*tan(3*a)/(a*(1 + tan(3*a)^2)*(1 - tan(1)^2))