Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
sin(a)^2/cos(2*a)+tan(2*a)/2+1/(tan(a)+1)
¿Cómo vas a descomponer esta sin(a)^2/cos(2*a)+tan(2*a)/2+1/(tan(a)+1) expresión en fracciones?
Solución
Ha introducido
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2 sin (a) tan(2*a) 1 -------- + -------- + ---------- cos(2*a) 2 tan(a) + 1
$$\left(\frac{\sin^{2}{\left(a \right)}}{\cos{\left(2 a \right)}} + \frac{\tan{\left(2 a \right)}}{2}\right) + \frac{1}{\tan{\left(a \right)} + 1}$$
sin(a)^2/cos(2*a) + tan(2*a)/2 + 1/(tan(a) + 1)
Simplificación general
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4
1 2 sin (a)
------- - sin (a) - ------- + tan(a)
2 2
cos (a) cos (a)
------------------------------------
(1 + tan(a))*cos(2*a)
$$\frac{- \frac{\sin^{4}{\left(a \right)}}{\cos^{2}{\left(a \right)}} - \sin^{2}{\left(a \right)} + \tan{\left(a \right)} + \frac{1}{\cos^{2}{\left(a \right)}}}{\left(\tan{\left(a \right)} + 1\right) \cos{\left(2 a \right)}}$$
(cos(a)^(-2) - sin(a)^2 - sin(a)^4/cos(a)^2 + tan(a))/((1 + tan(a))*cos(2*a))
Denominador racional
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/ 2 \
2*cos(2*a) + (1 + tan(a))*\2*sin (a) + cos(2*a)*tan(2*a)/
---------------------------------------------------------
2*(1 + tan(a))*cos(2*a)
$$\frac{\left(2 \sin^{2}{\left(a \right)} + \cos{\left(2 a \right)} \tan{\left(2 a \right)}\right) \left(\tan{\left(a \right)} + 1\right) + 2 \cos{\left(2 a \right)}}{2 \left(\tan{\left(a \right)} + 1\right) \cos{\left(2 a \right)}}$$
(2*cos(2*a) + (1 + tan(a))*(2*sin(a)^2 + cos(2*a)*tan(2*a)))/(2*(1 + tan(a))*cos(2*a))
Unión de expresiones racionales
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/ 2 \
2*cos(2*a) + (1 + tan(a))*\2*sin (a) + cos(2*a)*tan(2*a)/
---------------------------------------------------------
2*(1 + tan(a))*cos(2*a)
$$\frac{\left(2 \sin^{2}{\left(a \right)} + \cos{\left(2 a \right)} \tan{\left(2 a \right)}\right) \left(\tan{\left(a \right)} + 1\right) + 2 \cos{\left(2 a \right)}}{2 \left(\tan{\left(a \right)} + 1\right) \cos{\left(2 a \right)}}$$
(2*cos(2*a) + (1 + tan(a))*(2*sin(a)^2 + cos(2*a)*tan(2*a)))/(2*(1 + tan(a))*cos(2*a))
Potencias
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2
1 tan(2*a) sin (a)
---------- + -------- + --------
1 + tan(a) 2 cos(2*a)
$$\frac{\sin^{2}{\left(a \right)}}{\cos{\left(2 a \right)}} + \frac{\tan{\left(2 a \right)}}{2} + \frac{1}{\tan{\left(a \right)} + 1}$$
2
/ -I*a I*a\ / 2*I*a -2*I*a\
1 \- e + e / I*\- e + e /
---------------------- - -------------------- + ----------------------
/ I*a -I*a\ / -2*I*a 2*I*a\ / -2*I*a 2*I*a\
I*\- e + e / |e e | 2*\e + e /
1 + ------------------ 4*|------- + ------|
I*a -I*a \ 2 2 /
e + e
$$- \frac{\left(e^{i a} - e^{- i a}\right)^{2}}{4 \left(\frac{e^{2 i a}}{2} + \frac{e^{- 2 i a}}{2}\right)} + \frac{i \left(- e^{2 i a} + e^{- 2 i a}\right)}{2 \left(e^{2 i a} + e^{- 2 i a}\right)} + \frac{1}{\frac{i \left(- e^{i a} + e^{- i a}\right)}{e^{i a} + e^{- i a}} + 1}$$
1/(1 + i*(-exp(i*a) + exp(-i*a))/(exp(i*a) + exp(-i*a))) - (-exp(-i*a) + exp(i*a))^2/(4*(exp(-2*i*a)/2 + exp(2*i*a)/2)) + i*(-exp(2*i*a) + exp(-2*i*a))/(2*(exp(-2*i*a) + exp(2*i*a)))
Respuesta numérica
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1/(1.0 + tan(a)) + 0.5*tan(2*a) + sin(a)^2/cos(2*a)
1/(1.0 + tan(a)) + 0.5*tan(2*a) + sin(a)^2/cos(2*a)
Compilar la expresión
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2
1 tan(2*a) sin (a)
---------- + -------- + --------
1 + tan(a) 2 cos(2*a)
$$\frac{\sin^{2}{\left(a \right)}}{\cos{\left(2 a \right)}} + \frac{\tan{\left(2 a \right)}}{2} + \frac{1}{\tan{\left(a \right)} + 1}$$
1/(1 + tan(a)) + tan(2*a)/2 + sin(a)^2/cos(2*a)
Denominador común
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2 2 tan(2*a) sin (a) + sin (a)*tan(a) + cos(2*a) -------- + ----------------------------------- 2 cos(2*a)*tan(a) + cos(2*a)
$$\frac{\tan{\left(2 a \right)}}{2} + \frac{\sin^{2}{\left(a \right)} \tan{\left(a \right)} + \sin^{2}{\left(a \right)} + \cos{\left(2 a \right)}}{\cos{\left(2 a \right)} \tan{\left(a \right)} + \cos{\left(2 a \right)}}$$
tan(2*a)/2 + (sin(a)^2 + sin(a)^2*tan(a) + cos(2*a))/(cos(2*a)*tan(a) + cos(2*a))
Parte trigonométrica
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/pi \ /pi \
csc|-- - 2*a| csc|-- - 2*a|
1 1 \2 / \2 /
- - + --------------- + ------------- + -------------
2 /pi \ 2 2*csc(2*a)
csc|-- - a|
\2 /
1 + -----------
csc(a)
$$\frac{\csc{\left(- 2 a + \frac{\pi}{2} \right)}}{2} - \frac{1}{2} + \frac{\csc{\left(- 2 a + \frac{\pi}{2} \right)}}{2 \csc{\left(2 a \right)}} + \frac{1}{1 + \frac{\csc{\left(- a + \frac{\pi}{2} \right)}}{\csc{\left(a \right)}}}$$
1
-1 + --------
1 cos(2*a) tan(2*a)
---------- + ------------- + --------
tan(a) + 1 2 2
$$\frac{-1 + \frac{1}{\cos{\left(2 a \right)}}}{2} + \frac{\tan{\left(2 a \right)}}{2} + \frac{1}{\tan{\left(a \right)} + 1}$$
1
-1 + --------
1 cos(2*a) tan(2*a)
---------- + ------------- + --------
1 + tan(a) 2 2
$$\frac{-1 + \frac{1}{\cos{\left(2 a \right)}}}{2} + \frac{\tan{\left(2 a \right)}}{2} + \frac{1}{\tan{\left(a \right)} + 1}$$
2
1 sin (a) sin(2*a)
---------- + -------- + ----------
sin(a) cos(2*a) 2*cos(2*a)
1 + ------
cos(a)
$$\frac{\sin^{2}{\left(a \right)}}{\cos{\left(2 a \right)}} + \frac{\sin{\left(2 a \right)}}{2 \cos{\left(2 a \right)}} + \frac{1}{\frac{\sin{\left(a \right)}}{\cos{\left(a \right)}} + 1}$$
/pi \ /pi \
csc|-- - 2*a| csc|-- - 2*a|
1 \2 / \2 /
--------------- + ------------- + -------------
/pi \ 2 2*csc(2*a)
csc|-- - a| csc (a)
\2 /
1 + -----------
csc(a)
$$\frac{\csc{\left(- 2 a + \frac{\pi}{2} \right)}}{2 \csc{\left(2 a \right)}} + \frac{\csc{\left(- 2 a + \frac{\pi}{2} \right)}}{\csc^{2}{\left(a \right)}} + \frac{1}{1 + \frac{\csc{\left(- a + \frac{\pi}{2} \right)}}{\csc{\left(a \right)}}}$$
2
1 1 tan(2*a) 1 + tan (a)
- - + ---------- + -------- + ---------------
2 1 + tan(a) 2 / 2 \
2*\1 - tan (a)/
$$\frac{\tan{\left(2 a \right)}}{2} - \frac{1}{2} + \frac{1}{\tan{\left(a \right)} + 1} + \frac{\tan^{2}{\left(a \right)} + 1}{2 \left(1 - \tan^{2}{\left(a \right)}\right)}$$
1 sec(2*a) sec(2*a)
---------- + -------- + ----------
sec(a) 2 2*csc(2*a)
1 + ------ csc (a)
csc(a)
$$\frac{\sec{\left(2 a \right)}}{2 \csc{\left(2 a \right)}} + \frac{\sec{\left(2 a \right)}}{\csc^{2}{\left(a \right)}} + \frac{1}{1 + \frac{\sec{\left(a \right)}}{\csc{\left(a \right)}}}$$
2
1 1 1 1 + cot (a)
- - + ---------- + ---------- + ----------------
2 1 2*cot(2*a) / 2 \
1 + ------ 2*\-1 + cot (a)/
cot(a)
$$- \frac{1}{2} + \frac{1}{2 \cot{\left(2 a \right)}} + \frac{\cot^{2}{\left(a \right)} + 1}{2 \left(\cot^{2}{\left(a \right)} - 1\right)} + \frac{1}{1 + \frac{1}{\cot{\left(a \right)}}}$$
1 1 1 tan(2*a) - - + ---------- + ---------- + -------- 2 1 + tan(a) 2*cos(2*a) 2
$$\frac{\tan{\left(2 a \right)}}{2} - \frac{1}{2} + \frac{1}{2 \cos{\left(2 a \right)}} + \frac{1}{\tan{\left(a \right)} + 1}$$
/ pi\
cos|2*a - --|
1 1 1 \ 2 /
- - + --------------- + ---------- + -------------
2 / pi\ 2*cos(2*a) 2*cos(2*a)
cos|a - --|
\ 2 /
1 + -----------
cos(a)
$$- \frac{1}{2} + \frac{\cos{\left(2 a - \frac{\pi}{2} \right)}}{2 \cos{\left(2 a \right)}} + \frac{1}{2 \cos{\left(2 a \right)}} + \frac{1}{1 + \frac{\cos{\left(a - \frac{\pi}{2} \right)}}{\cos{\left(a \right)}}}$$
2/ pi\ / pi\
cos |a - --| cos|2*a - --|
1 \ 2 / \ 2 /
--------------- + ------------ + -------------
/ pi\ cos(2*a) 2*cos(2*a)
cos|a - --|
\ 2 /
1 + -----------
cos(a)
$$\frac{\cos^{2}{\left(a - \frac{\pi}{2} \right)}}{\cos{\left(2 a \right)}} + \frac{\cos{\left(2 a - \frac{\pi}{2} \right)}}{2 \cos{\left(2 a \right)}} + \frac{1}{1 + \frac{\cos{\left(a - \frac{\pi}{2} \right)}}{\cos{\left(a \right)}}}$$
1 sec(2*a) sec(2*a)
--------------- + ------------ + ---------------
sec(a) 2/ pi\ / pi\
1 + ----------- sec |a - --| 2*sec|2*a - --|
/ pi\ \ 2 / \ 2 /
sec|a - --|
\ 2 /
$$\frac{\sec{\left(2 a \right)}}{2 \sec{\left(2 a - \frac{\pi}{2} \right)}} + \frac{\sec{\left(2 a \right)}}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} + \frac{1}{\frac{\sec{\left(a \right)}}{\sec{\left(a - \frac{\pi}{2} \right)}} + 1}$$
1 1 sec(2*a) tan(2*a) - - + ---------- + -------- + -------- 2 1 + tan(a) 2 2
$$\frac{\tan{\left(2 a \right)}}{2} + \frac{\sec{\left(2 a \right)}}{2} - \frac{1}{2} + \frac{1}{\tan{\left(a \right)} + 1}$$
2 2
1 sin (a) sin (2*a)
------------- + ------------- + ---------
2 /pi \ sin(4*a)
2*sin (a) sin|-- + 2*a|
1 + --------- \2 /
sin(2*a)
$$\frac{\sin^{2}{\left(a \right)}}{\sin{\left(2 a + \frac{\pi}{2} \right)}} + \frac{\sin^{2}{\left(2 a \right)}}{\sin{\left(4 a \right)}} + \frac{1}{\frac{2 \sin^{2}{\left(a \right)}}{\sin{\left(2 a \right)}} + 1}$$
1 1 sec(2*a) sec(2*a)
- - + --------------- + -------- + ---------------
2 sec(a) 2 / pi\
1 + ----------- 2*sec|2*a - --|
/ pi\ \ 2 /
sec|a - --|
\ 2 /
$$\frac{\sec{\left(2 a \right)}}{2} + \frac{\sec{\left(2 a \right)}}{2 \sec{\left(2 a - \frac{\pi}{2} \right)}} - \frac{1}{2} + \frac{1}{\frac{\sec{\left(a \right)}}{\sec{\left(a - \frac{\pi}{2} \right)}} + 1}$$
2/a\ / 2 \
4*cot |-|*\1 + cot (a)/
1 1 \2/
---------- + ---------- + -----------------------------
1 2*cot(2*a) 2
1 + ------ / 2/a\\ / 2 \
cot(a) |1 + cot |-|| *\-1 + cot (a)/
\ \2//
$$\frac{1}{2 \cot{\left(2 a \right)}} + \frac{4 \left(\cot^{2}{\left(a \right)} + 1\right) \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2} \left(\cot^{2}{\left(a \right)} - 1\right)} + \frac{1}{1 + \frac{1}{\cot{\left(a \right)}}}$$
2 2
1 sin (a) sin (2*a)
------------- + -------- + ---------
2 cos(2*a) sin(4*a)
2*sin (a)
1 + ---------
sin(2*a)
$$\frac{\sin^{2}{\left(a \right)}}{\cos{\left(2 a \right)}} + \frac{\sin^{2}{\left(2 a \right)}}{\sin{\left(4 a \right)}} + \frac{1}{\frac{2 \sin^{2}{\left(a \right)}}{\sin{\left(2 a \right)}} + 1}$$
2/a\ / 2 \
4*tan |-|*\1 + tan (a)/
1 tan(2*a) \2/
---------- + -------- + ----------------------------
1 + tan(a) 2 2
/ 2/a\\ / 2 \
|1 + tan |-|| *\1 - tan (a)/
\ \2//
$$\frac{\tan{\left(2 a \right)}}{2} + \frac{1}{\tan{\left(a \right)} + 1} + \frac{4 \left(\tan^{2}{\left(a \right)} + 1\right) \tan^{2}{\left(\frac{a}{2} \right)}}{\left(1 - \tan^{2}{\left(a \right)}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
2
1 1 1 sin (2*a)
- - + ------------- + --------------- + ---------
2 2 /pi \ sin(4*a)
2*sin (a) 2*sin|-- + 2*a|
1 + --------- \2 /
sin(2*a)
$$\frac{\sin^{2}{\left(2 a \right)}}{\sin{\left(4 a \right)}} - \frac{1}{2} + \frac{1}{2 \sin{\left(2 a + \frac{\pi}{2} \right)}} + \frac{1}{\frac{2 \sin^{2}{\left(a \right)}}{\sin{\left(2 a \right)}} + 1}$$
-1/2 + 1/(1 + 2*sin(a)^2/sin(2*a)) + 1/(2*sin(pi/2 + 2*a)) + sin(2*a)^2/sin(4*a)
Combinatoria
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2 2
2*sin (a) + 2*cos(2*a) + cos(2*a)*tan(2*a) + 2*sin (a)*tan(a) + cos(2*a)*tan(a)*tan(2*a)
----------------------------------------------------------------------------------------
2*(1 + tan(a))*cos(2*a)
$$\frac{2 \sin^{2}{\left(a \right)} \tan{\left(a \right)} + 2 \sin^{2}{\left(a \right)} + \cos{\left(2 a \right)} \tan{\left(a \right)} \tan{\left(2 a \right)} + \cos{\left(2 a \right)} \tan{\left(2 a \right)} + 2 \cos{\left(2 a \right)}}{2 \left(\tan{\left(a \right)} + 1\right) \cos{\left(2 a \right)}}$$
(2*sin(a)^2 + 2*cos(2*a) + cos(2*a)*tan(2*a) + 2*sin(a)^2*tan(a) + cos(2*a)*tan(a)*tan(2*a))/(2*(1 + tan(a))*cos(2*a))