Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sin(t)/(1+cos(t))+sin(t)/(1-cos(t))
¿Cómo vas a descomponer esta sin(t)/(1+cos(t))+sin(t)/(1-cos(t)) expresión en fracciones?
Solución
Ha introducido
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sin(t) sin(t) ---------- + ---------- 1 + cos(t) 1 - cos(t)
$$\frac{\sin{\left(t \right)}}{\cos{\left(t \right)} + 1} + \frac{\sin{\left(t \right)}}{1 - \cos{\left(t \right)}}$$
sin(t)/(1 + cos(t)) + sin(t)/(1 - cos(t))
Respuesta numérica
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sin(t)/(1.0 - cos(t)) + sin(t)/(1.0 + cos(t))
sin(t)/(1.0 - cos(t)) + sin(t)/(1.0 + cos(t))
Denominador racional
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(1 - cos(t))*sin(t) + (1 + cos(t))*sin(t)
-----------------------------------------
(1 - cos(t))*(1 + cos(t))
$$\frac{\left(1 - \cos{\left(t \right)}\right) \sin{\left(t \right)} + \left(\cos{\left(t \right)} + 1\right) \sin{\left(t \right)}}{\left(1 - \cos{\left(t \right)}\right) \left(\cos{\left(t \right)} + 1\right)}$$
((1 - cos(t))*sin(t) + (1 + cos(t))*sin(t))/((1 - cos(t))*(1 + cos(t)))
Denominador común
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-2*sin(t)
------------
2
-1 + cos (t)
$$- \frac{2 \sin{\left(t \right)}}{\cos^{2}{\left(t \right)} - 1}$$
-2*sin(t)/(-1 + cos(t)^2)
Potencias
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/ -I*t I*t\ / -I*t I*t\
I*\- e + e / I*\- e + e /
- -------------------- - --------------------
/ I*t -I*t\ / I*t -I*t\
| e e | | e e |
2*|1 + ---- + -----| 2*|1 - ---- - -----|
\ 2 2 / \ 2 2 /
$$- \frac{i \left(e^{i t} - e^{- i t}\right)}{2 \left(\frac{e^{i t}}{2} + 1 + \frac{e^{- i t}}{2}\right)} - \frac{i \left(e^{i t} - e^{- i t}\right)}{2 \left(- \frac{e^{i t}}{2} + 1 - \frac{e^{- i t}}{2}\right)}$$
-i*(-exp(-i*t) + exp(i*t))/(2*(1 + exp(i*t)/2 + exp(-i*t)/2)) - i*(-exp(-i*t) + exp(i*t))/(2*(1 - exp(i*t)/2 - exp(-i*t)/2))
Unión de expresiones racionales
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2*sin(t) ------------------------- (1 - cos(t))*(1 + cos(t))
$$\frac{2 \sin{\left(t \right)}}{\left(1 - \cos{\left(t \right)}\right) \left(\cos{\left(t \right)} + 1\right)}$$
2*sin(t)/((1 - cos(t))*(1 + cos(t)))
Combinatoria
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-2*sin(t) -------------------------- (1 + cos(t))*(-1 + cos(t))
$$- \frac{2 \sin{\left(t \right)}}{\left(\cos{\left(t \right)} - 1\right) \left(\cos{\left(t \right)} + 1\right)}$$
-2*sin(t)/((1 + cos(t))*(-1 + cos(t)))
Parte trigonométrica
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2 ------ sin(t)
$$\frac{2}{\sin{\left(t \right)}}$$
1 1 ------------------- + ------------------- / 1 \ / 1 \ |1 + ------|*csc(t) |1 - ------|*csc(t) \ sec(t)/ \ sec(t)/
$$\frac{1}{\left(1 + \frac{1}{\sec{\left(t \right)}}\right) \csc{\left(t \right)}} + \frac{1}{\left(1 - \frac{1}{\sec{\left(t \right)}}\right) \csc{\left(t \right)}}$$
2*csc(t)
$$2 \csc{\left(t \right)}$$
2/t\
1 + tan |-|
\2/
-----------
/t\
tan|-|
\2/
$$\frac{\tan^{2}{\left(\frac{t}{2} \right)} + 1}{\tan{\left(\frac{t}{2} \right)}}$$
/t\ /t\
2*cot|-| 2*cot|-|
\2/ \2/
-------------------------------- + --------------------------------
/ 2/t\\ / 2/t\\
| -1 + cot |-|| | -1 + cot |-||
/ 2/t\\ | \2/| / 2/t\\ | \2/|
|1 + cot |-||*|1 + ------------| |1 + cot |-||*|1 - ------------|
\ \2// | 2/t\ | \ \2// | 2/t\ |
| 1 + cot |-| | | 1 + cot |-| |
\ \2/ / \ \2/ /
$$\frac{2 \cot{\left(\frac{t}{2} \right)}}{\left(\frac{\cot^{2}{\left(\frac{t}{2} \right)} - 1}{\cot^{2}{\left(\frac{t}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)} + \frac{2 \cot{\left(\frac{t}{2} \right)}}{\left(- \frac{\cot^{2}{\left(\frac{t}{2} \right)} - 1}{\cot^{2}{\left(\frac{t}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)}$$
2 ----------- / pi\ cos|t - --| \ 2 /
$$\frac{2}{\cos{\left(t - \frac{\pi}{2} \right)}}$$
/ pi\
2*sec|t - --|
\ 2 /
$$2 \sec{\left(t - \frac{\pi}{2} \right)}$$
1 1 ------------------------ + ------------------------ / 1 \ / pi\ / 1 \ / pi\ |1 + ------|*sec|t - --| |1 - ------|*sec|t - --| \ sec(t)/ \ 2 / \ sec(t)/ \ 2 /
$$\frac{1}{\left(1 + \frac{1}{\sec{\left(t \right)}}\right) \sec{\left(t - \frac{\pi}{2} \right)}} + \frac{1}{\left(1 - \frac{1}{\sec{\left(t \right)}}\right) \sec{\left(t - \frac{\pi}{2} \right)}}$$
2/t\
1 + cot |-|
\2/
-----------
/t\
cot|-|
\2/
$$\frac{\cot^{2}{\left(\frac{t}{2} \right)} + 1}{\cot{\left(\frac{t}{2} \right)}}$$
/t\ /t\
2*tan|-| 2*tan|-|
\2/ \2/
------------------------------- + -------------------------------
/ 2/t\\ / 2/t\\
| 1 - tan |-|| | 1 - tan |-||
/ 2/t\\ | \2/| / 2/t\\ | \2/|
|1 + tan |-||*|1 + -----------| |1 + tan |-||*|1 - -----------|
\ \2// | 2/t\| \ \2// | 2/t\|
| 1 + tan |-|| | 1 + tan |-||
\ \2// \ \2//
$$\frac{2 \tan{\left(\frac{t}{2} \right)}}{\left(\frac{1 - \tan^{2}{\left(\frac{t}{2} \right)}}{\tan^{2}{\left(\frac{t}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)} + \frac{2 \tan{\left(\frac{t}{2} \right)}}{\left(- \frac{1 - \tan^{2}{\left(\frac{t}{2} \right)}}{\tan^{2}{\left(\frac{t}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)}$$
sin(t) sin(t)
--------------- + ---------------
/ pi\ / pi\
1 - sin|t + --| 1 + sin|t + --|
\ 2 / \ 2 /
$$\frac{\sin{\left(t \right)}}{\sin{\left(t + \frac{\pi}{2} \right)} + 1} + \frac{\sin{\left(t \right)}}{1 - \sin{\left(t + \frac{\pi}{2} \right)}}$$
/ pi\ / pi\ cos|t - --| cos|t - --| \ 2 / \ 2 / ----------- + ----------- 1 - cos(t) 1 + cos(t)
$$\frac{\cos{\left(t - \frac{\pi}{2} \right)}}{\cos{\left(t \right)} + 1} + \frac{\cos{\left(t - \frac{\pi}{2} \right)}}{1 - \cos{\left(t \right)}}$$
1 1 ------------------------ + ------------------------ / 1 \ / 1 \ |1 + -----------|*csc(t) |1 - -----------|*csc(t) | /pi \| | /pi \| | csc|-- - t|| | csc|-- - t|| \ \2 // \ \2 //
$$\frac{1}{\left(1 + \frac{1}{\csc{\left(- t + \frac{\pi}{2} \right)}}\right) \csc{\left(t \right)}} + \frac{1}{\left(1 - \frac{1}{\csc{\left(- t + \frac{\pi}{2} \right)}}\right) \csc{\left(t \right)}}$$
1/((1 + 1/csc(pi/2 - t))*csc(t)) + 1/((1 - 1/csc(pi/2 - t))*csc(t))