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