Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
cos(pi/4-a)/(1-sin(2a))^(-1)-(sin(pi/4-a))/(1+sin(2a))^(-1)
¿Cómo vas a descomponer esta cos(pi/4-a)/(1-sin(2a))^(-1)-(sin(pi/4-a))/(1+sin(2a))^(-1) expresión en fracciones?
Solución
Ha introducido
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/pi \ /pi \
cos|-- - a| sin|-- - a|
\4 / \4 /
-------------- - --------------
/ 1 \ / 1 \
|------------| |------------|
\1 - sin(2*a)/ \1 + sin(2*a)/
$$- \frac{\sin{\left(- a + \frac{\pi}{4} \right)}}{\frac{1}{\sin{\left(2 a \right)} + 1}} + \frac{\cos{\left(- a + \frac{\pi}{4} \right)}}{\frac{1}{1 - \sin{\left(2 a \right)}}}$$
cos(pi/4 - a)/1/(1 - sin(2*a)) - sin(pi/4 - a)/1/(1 + sin(2*a))
Simplificación general
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___
\/ 2 *(-sin(3*a) + sin(a))
--------------------------
2
$$\frac{\sqrt{2} \left(\sin{\left(a \right)} - \sin{\left(3 a \right)}\right)}{2}$$
sqrt(2)*(-sin(3*a) + sin(a))/2
Respuesta numérica
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(1.0 - sin(2*a))*cos(pi/4 - a) - (1.0 + sin(2*a))*sin(pi/4 - a)
(1.0 - sin(2*a))*cos(pi/4 - a) - (1.0 + sin(2*a))*sin(pi/4 - a)
Denominador común
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/ pi\ / pi\ / pi\ / pi\
- cos|a + --| - cos|a + --|*sin(2*a) - sin(2*a)*sin|a + --| + sin|a + --|
\ 4 / \ 4 / \ 4 / \ 4 /
$$- \sin{\left(2 a \right)} \sin{\left(a + \frac{\pi}{4} \right)} - \sin{\left(2 a \right)} \cos{\left(a + \frac{\pi}{4} \right)} + \sin{\left(a + \frac{\pi}{4} \right)} - \cos{\left(a + \frac{\pi}{4} \right)}$$
-cos(a + pi/4) - cos(a + pi/4)*sin(2*a) - sin(2*a)*sin(a + pi/4) + sin(a + pi/4)
Combinatoria
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/ pi\ / pi\ / pi\ / pi\
- cos|a + --| - cos|a + --|*sin(2*a) - sin(2*a)*sin|a + --| + sin|a + --|
\ 4 / \ 4 / \ 4 / \ 4 /
$$- \sin{\left(2 a \right)} \sin{\left(a + \frac{\pi}{4} \right)} - \sin{\left(2 a \right)} \cos{\left(a + \frac{\pi}{4} \right)} + \sin{\left(a + \frac{\pi}{4} \right)} - \cos{\left(a + \frac{\pi}{4} \right)}$$
-cos(a + pi/4) - cos(a + pi/4)*sin(2*a) - sin(2*a)*sin(a + pi/4) + sin(a + pi/4)
Denominador racional
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/ pi\ / pi\ / pi\ / pi\
sin(2*a)*sin|a - --| - cos|a - --|*sin(2*a) + cos|a - --| + sin|a - --|
\ 4 / \ 4 / \ 4 / \ 4 /
$$\sin{\left(2 a \right)} \sin{\left(a - \frac{\pi}{4} \right)} - \sin{\left(2 a \right)} \cos{\left(a - \frac{\pi}{4} \right)} + \sin{\left(a - \frac{\pi}{4} \right)} + \cos{\left(a - \frac{\pi}{4} \right)}$$
sin(2*a)*sin(a - pi/4) - cos(a - pi/4)*sin(2*a) + cos(a - pi/4) + sin(a - pi/4)
Unión de expresiones racionales
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/pi - 4*a\ /pi - 4*a\
(1 - sin(2*a))*cos|--------| - (1 + sin(2*a))*sin|--------|
\ 4 / \ 4 /
$$\left(1 - \sin{\left(2 a \right)}\right) \cos{\left(\frac{\pi - 4 a}{4} \right)} - \left(\sin{\left(2 a \right)} + 1\right) \sin{\left(\frac{\pi - 4 a}{4} \right)}$$
(1 - sin(2*a))*cos((pi - 4*a)/4) - (1 + sin(2*a))*sin((pi - 4*a)/4)
Potencias
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/ / pi\ / pi\\
/ / pi\ / pi\\ / / -2*I*a 2*I*a\\ | I*|a - --| I*|-a + --||
| I*|a - --| I*|-a + --|| | I*\- e + e /| | \ 4 / \ 4 /|
/ / -2*I*a 2*I*a\\ | \ 4 / \ 4 /| I*|1 - ----------------------|*\- e + e /
| I*\- e + e /| |e e | \ 2 /
|1 + ----------------------|*|----------- + ------------| + -------------------------------------------------------------
\ 2 / \ 2 2 / 2
$$\frac{i \left(- \frac{i \left(e^{2 i a} - e^{- 2 i a}\right)}{2} + 1\right) \left(e^{i \left(- a + \frac{\pi}{4}\right)} - e^{i \left(a - \frac{\pi}{4}\right)}\right)}{2} + \left(\frac{i \left(e^{2 i a} - e^{- 2 i a}\right)}{2} + 1\right) \left(\frac{e^{i \left(- a + \frac{\pi}{4}\right)}}{2} + \frac{e^{i \left(a - \frac{\pi}{4}\right)}}{2}\right)$$
/ pi\ / pi\
(1 - sin(2*a))*sin|a + --| - (1 + sin(2*a))*cos|a + --|
\ 4 / \ 4 /
$$\left(1 - \sin{\left(2 a \right)}\right) \sin{\left(a + \frac{\pi}{4} \right)} - \left(\sin{\left(2 a \right)} + 1\right) \cos{\left(a + \frac{\pi}{4} \right)}$$
(1 - sin(2*a))*sin(a + pi/4) - (1 + sin(2*a))*cos(a + pi/4)
Compilar la expresión
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/pi \ /pi \
(1 - sin(2*a))*cos|-- - a| - (1 + sin(2*a))*sin|-- - a|
\4 / \4 /
$$\left(1 - \sin{\left(2 a \right)}\right) \cos{\left(- a + \frac{\pi}{4} \right)} - \left(\sin{\left(2 a \right)} + 1\right) \sin{\left(- a + \frac{\pi}{4} \right)}$$
(1 - sin(2*a))*cos(pi/4 - a) - (1 + sin(2*a))*sin(pi/4 - a)