Sr Examen
¿Cómo vas a descomponer esta cos(a-b)*b/((sin(a)*sin(a)))-1 expresión en fracciones?
Solución
Ha introducido
[src]
cos(a - b)*b ------------- - 1 sin(a)*sin(a)
$$\frac{b \cos{\left(a - b \right)}}{\sin{\left(a \right)} \sin{\left(a \right)}} - 1$$
(cos(a - b)*b)/((sin(a)*sin(a))) - 1
Simplificación general
[src]
b*cos(a - b)
-1 + ------------
2
sin (a)
$$\frac{b \cos{\left(a - b \right)}}{\sin^{2}{\left(a \right)}} - 1$$
-1 + b*cos(a - b)/sin(a)^2
Unión de expresiones racionales
[src]
2
- sin (a) + b*cos(a - b)
------------------------
2
sin (a)
$$\frac{b \cos{\left(a - b \right)} - \sin^{2}{\left(a \right)}}{\sin^{2}{\left(a \right)}}$$
(-sin(a)^2 + b*cos(a - b))/sin(a)^2
Combinatoria
[src]
2
- sin (a) + b*cos(a - b)
------------------------
2
sin (a)
$$\frac{b \cos{\left(a - b \right)} - \sin^{2}{\left(a \right)}}{\sin^{2}{\left(a \right)}}$$
(-sin(a)^2 + b*cos(a - b))/sin(a)^2
Denominador común
[src]
b*cos(a - b)
-1 + ------------
2
sin (a)
$$\frac{b \cos{\left(a - b \right)}}{\sin^{2}{\left(a \right)}} - 1$$
-1 + b*cos(a - b)/sin(a)^2
Potencias
[src]
/ I*(a - b) I*(b - a)\
|e e |
4*b*|---------- + ----------|
\ 2 2 /
-1 - -----------------------------
2
/ -I*a I*a\
\- e + e /
$$- \frac{4 b \left(\frac{e^{i \left(- a + b\right)}}{2} + \frac{e^{i \left(a - b\right)}}{2}\right)}{\left(e^{i a} - e^{- i a}\right)^{2}} - 1$$
b*cos(a - b)
-1 + ------------
2
sin (a)
$$\frac{b \cos{\left(a - b \right)}}{\sin^{2}{\left(a \right)}} - 1$$
-1 + b*cos(a - b)/sin(a)^2
Compilar la expresión
[src]
b*cos(a - b)
-1 + ------------
2
sin (a)
$$\frac{b \cos{\left(a - b \right)}}{\sin^{2}{\left(a \right)}} - 1$$
-1 + b*cos(a - b)/sin(a)^2
Denominador racional
[src]
2
- sin (a) + b*cos(a - b)
------------------------
2
sin (a)
$$\frac{b \cos{\left(a - b \right)} - \sin^{2}{\left(a \right)}}{\sin^{2}{\left(a \right)}}$$
(-sin(a)^2 + b*cos(a - b))/sin(a)^2
Parte trigonométrica
[src]
/ 4/a - b\\
| 4*sin |-----||
2 | \ 2 /|
b*sin (a)*|1 - -------------|
| 2 |
\ sin (a - b) /
-1 + --------------------------------------
/ 4/a - b\\
| 4*sin |-----||
| \ 2 /| 4/a\ 4/a\
16*|1 + -------------|*cos |-|*sin |-|
| 2 | \2/ \2/
\ sin (a - b) /
$$\frac{b \left(- \frac{4 \sin^{4}{\left(\frac{a - b}{2} \right)}}{\sin^{2}{\left(a - b \right)}} + 1\right) \sin^{2}{\left(a \right)}}{16 \left(\frac{4 \sin^{4}{\left(\frac{a - b}{2} \right)}}{\sin^{2}{\left(a - b \right)}} + 1\right) \sin^{4}{\left(\frac{a}{2} \right)} \cos^{4}{\left(\frac{a}{2} \right)}} - 1$$
4
/ 2/a\\ / 2/a b\\
b*|1 + tan |-|| *|1 - tan |- - -||
\ \4// \ \2 2//
-1 + ------------------------------------------
4
/ 2/a b\\ / 2/a\\ 2/a\
4*|1 + tan |- - -||*|1 - tan |-|| *tan |-|
\ \2 2// \ \4// \2/
$$\frac{b \left(1 - \tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}\right) \left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{4}}{4 \left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{4} \left(\tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1\right) \tan^{2}{\left(\frac{a}{2} \right)}} - 1$$
/ 2/a b\\
b*|1 - tan |- - -||
\ \2 2//
-1 + -----------------------------------
/ 2/a b\\ 4/a\ 2/a\
4*|1 + tan |- - -||*cos |-|*tan |-|
\ \2 2// \2/ \2/
$$\frac{b \left(1 - \tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}\right)}{4 \left(\tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1\right) \cos^{4}{\left(\frac{a}{2} \right)} \tan^{2}{\left(\frac{a}{2} \right)}} - 1$$
/ 2/pi b a\\
| csc |-- + - - -||
2/a\ 2/pi a\ | \2 2 2/|
b*csc |-|*csc |-- - -|*|1 - ----------------|
\2/ \2 2/ | 2/a b\ |
| csc |- - -| |
\ \2 2/ /
-1 + ---------------------------------------------
/ 2/pi b a\\
| csc |-- + - - -||
| \2 2 2/|
4*|1 + ----------------|
| 2/a b\ |
| csc |- - -| |
\ \2 2/ /
$$\frac{b \left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{b}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}\right) \csc^{2}{\left(\frac{a}{2} \right)} \csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{4 \left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{b}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}\right)} - 1$$
2
/ 2/a\\ / 2/a b\\
b*|1 + tan |-|| *|1 - tan |- - -||
\ \2// \ \2 2//
-1 + ----------------------------------
/ 2/a b\\ 2/a\
4*|1 + tan |- - -||*tan |-|
\ \2 2// \2/
$$\frac{b \left(1 - \tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}{4 \left(\tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1\right) \tan^{2}{\left(\frac{a}{2} \right)}} - 1$$
/ 2 \
| / / pi\\ / pi\ / pi \|
2*|-2 + |1 - sin|a + --|| + 2*sin|a + --| + b*sin|a + -- - b||
\ \ \ 2 // \ 2 / \ 2 //
---------------------------------------------------------------
/pi \
1 - sin|-- + 2*a|
\2 /
$$\frac{2 \left(b \sin{\left(a - b + \frac{\pi}{2} \right)} + \left(1 - \sin{\left(a + \frac{\pi}{2} \right)}\right)^{2} + 2 \sin{\left(a + \frac{\pi}{2} \right)} - 2\right)}{1 - \sin{\left(2 a + \frac{\pi}{2} \right)}}$$
/ 4/a b\\
| 4*sin |- - -||
2 | \2 2/|
b*sin (a)*|1 - -------------|
| 2 |
\ sin (a - b) /
-1 + -------------------------------------------
/ 4/a b\\
| 4*sin |- - -||
| \2 2/| 4/a\ 4/pi a\
16*|1 + -------------|*sin |-|*sin |-- + -|
| 2 | \2/ \2 2/
\ sin (a - b) /
$$\frac{b \left(- \frac{4 \sin^{4}{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\sin^{2}{\left(a - b \right)}} + 1\right) \sin^{2}{\left(a \right)}}{16 \left(\frac{4 \sin^{4}{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\sin^{2}{\left(a - b \right)}} + 1\right) \sin^{4}{\left(\frac{a}{2} \right)} \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{2} \right)}} - 1$$
2
b*csc (a)
-1 + ---------------
/ pi \
csc|b + -- - a|
\ 2 /
$$\frac{b \csc^{2}{\left(a \right)}}{\csc{\left(- a + b + \frac{\pi}{2} \right)}} - 1$$
2/ pi\
b*sec |a - --|
\ 2 /
-1 + --------------
sec(a - b)
$$\frac{b \sec^{2}{\left(a - \frac{\pi}{2} \right)}}{\sec{\left(a - b \right)}} - 1$$
/ 2 \
2*\-2 + (1 - cos(a)) + 2*cos(a) + b*cos(a - b)/
------------------------------------------------
1 - cos(2*a)
$$\frac{2 \left(b \cos{\left(a - b \right)} + \left(1 - \cos{\left(a \right)}\right)^{2} + 2 \cos{\left(a \right)} - 2\right)}{1 - \cos{\left(2 a \right)}}$$
/ 2 \
| / 2/a\\ / 2/a\\ / 2/a b\\|
| | -1 + cot |-|| 2*|-1 + cot |-|| b*|-1 + cot |- - -|||
| | \2/| \ \2// \ \2 2//|
2*|-2 + |1 - ------------| + ---------------- + --------------------|
| | 2/a\ | 2/a\ 2/a b\ |
| | 1 + cot |-| | 1 + cot |-| 1 + cot |- - -| |
\ \ \2/ / \2/ \2 2/ /
----------------------------------------------------------------------
2
-1 + cot (a)
1 - ------------
2
1 + cot (a)
$$\frac{2 \left(\frac{b \left(\cot^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1} + \left(- \frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} + 1\right)^{2} + \frac{2 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} - 2\right)}{- \frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} + 1}$$
/ 2 \
| / 1 \ 2 b |
2*|-2 + |1 - ------| + ------ + ----------|
\ \ sec(a)/ sec(a) sec(a - b)/
--------------------------------------------
1
1 - --------
sec(2*a)
$$\frac{2 \left(\frac{b}{\sec{\left(a - b \right)}} + \left(1 - \frac{1}{\sec{\left(a \right)}}\right)^{2} - 2 + \frac{2}{\sec{\left(a \right)}}\right)}{1 - \frac{1}{\sec{\left(2 a \right)}}}$$
2
b*csc (a)
-1 + ----------
sec(a - b)
$$\frac{b \csc^{2}{\left(a \right)}}{\sec{\left(a - b \right)}} - 1$$
b*cos(a - b)
-1 + ------------
2/ pi\
cos |a - --|
\ 2 /
$$\frac{b \cos{\left(a - b \right)}}{\cos^{2}{\left(a - \frac{\pi}{2} \right)}} - 1$$
/ 2/a b\\
b*|1 - tan |- - -||
\ \2 2//
-1 + --------------------------------------------------
4
/ 2/a b\\ / 2/a\\ 8/a\ 2/a\
4*|1 + tan |- - -||*|1 - tan |-|| *cos |-|*tan |-|
\ \2 2// \ \4// \4/ \2/
$$\frac{b \left(1 - \tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}\right)}{4 \left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{4} \left(\tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1\right) \cos^{8}{\left(\frac{a}{4} \right)} \tan^{2}{\left(\frac{a}{2} \right)}} - 1$$
2/a\ / 2/a b\\
b*cot |-|*|1 - tan |- - -||
\2/ \ \2 2//
-1 + -------------------------------------------
4
/ 2/a b\\ / 2/a\\ 8/a\
4*|1 + tan |- - -||*|-1 + cot |-|| *sin |-|
\ \2 2// \ \4// \4/
$$\frac{b \left(1 - \tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}\right) \cot^{2}{\left(\frac{a}{2} \right)}}{4 \left(\tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{a}{4} \right)}} - 1$$
/ 2/a pi b\\
| cos |- - -- - -||
| \2 2 2/|
b*|1 - ----------------|
| 2/a b\ |
| cos |- - -| |
\ \2 2/ /
-1 + ---------------------------------------------
/ 2/a pi b\\
| cos |- - -- - -||
| \2 2 2/| 2/a\ 2/a pi\
4*|1 + ----------------|*cos |-|*cos |- - --|
| 2/a b\ | \2/ \2 2 /
| cos |- - -| |
\ \2 2/ /
$$\frac{b \left(1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{b}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}\right)}{4 \left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{b}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}\right) \cos^{2}{\left(\frac{a}{2} \right)} \cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1$$
2 -1 + b*csc (a)*cos(a - b)
$$b \cos{\left(a - b \right)} \csc^{2}{\left(a \right)} - 1$$
/ 2 \
| / 1 \ 2 b |
2*|-2 + |1 - -----------| + ----------- + ---------------|
| | /pi \| /pi \ / pi \|
| | csc|-- - a|| csc|-- - a| csc|b + -- - a||
\ \ \2 // \2 / \ 2 //
-----------------------------------------------------------
1
1 - -------------
/pi \
csc|-- - 2*a|
\2 /
$$\frac{2 \left(\frac{b}{\csc{\left(- a + b + \frac{\pi}{2} \right)}} + \left(1 - \frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}}\right)^{2} - 2 + \frac{2}{\csc{\left(- a + \frac{\pi}{2} \right)}}\right)}{1 - \frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}}}$$
2/pi \
b*sec |-- - a|
\2 /
-1 + --------------
sec(a - b)
$$\frac{b \sec^{2}{\left(- a + \frac{\pi}{2} \right)}}{\sec{\left(a - b \right)}} - 1$$
/ 2 \
| / 2/a\\ / 2/a\\ / 2/a b\\|
| | 1 - tan |-|| 2*|1 - tan |-|| b*|1 - tan |- - -|||
| | \2/| \ \2// \ \2 2//|
2*|-2 + |1 - -----------| + --------------- + -------------------|
| | 2/a\| 2/a\ 2/a b\ |
| | 1 + tan |-|| 1 + tan |-| 1 + tan |- - -| |
\ \ \2// \2/ \2 2/ /
-------------------------------------------------------------------
2
1 - tan (a)
1 - -----------
2
1 + tan (a)
$$\frac{2 \left(\frac{b \left(1 - \tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}\right)}{\tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1} + \frac{2 \left(1 - \tan^{2}{\left(\frac{a}{2} \right)}\right)}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + \left(- \frac{1 - \tan^{2}{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + 1\right)^{2} - 2\right)}{- \frac{1 - \tan^{2}{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1} + 1}$$
4
/ 2/a\\ 2/a\ / 1 \
b*|1 + cot |-|| *cot |-|*|1 - -----------|
\ \4// \2/ | 2/a b\|
| cot |- - -||
\ \2 2//
-1 + ------------------------------------------
4
/ 1 \ / 2/a\\
4*|1 + -----------|*|-1 + cot |-||
| 2/a b\| \ \4//
| cot |- - -||
\ \2 2//
$$\frac{b \left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}\right) \left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{4} \cot^{2}{\left(\frac{a}{2} \right)}}{4 \left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}\right) \left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{4}} - 1$$
/ pi \
b*sin|a + -- - b|
\ 2 /
-1 + -----------------
2
sin (a)
$$\frac{b \sin{\left(a - b + \frac{\pi}{2} \right)}}{\sin^{2}{\left(a \right)}} - 1$$
b*cos(a - b)
-1 + ------------
2
sin (a)
$$\frac{b \cos{\left(a - b \right)}}{\sin^{2}{\left(a \right)}} - 1$$
2
/ 2/a\\ / 2/a b\\
b*|1 + cot |-|| *|-1 + cot |- - -||
\ \2// \ \2 2//
-1 + -----------------------------------
/ 2/a b\\ 2/a\
4*|1 + cot |- - -||*cot |-|
\ \2 2// \2/
$$\frac{b \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} - 1\right)}{4 \left(\cot^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1\right) \cot^{2}{\left(\frac{a}{2} \right)}} - 1$$
/ 2/a b\ \
| sec |- - -| |
2/a\ 2/a pi\ | \2 2/ |
b*sec |-|*sec |- - --|*|1 - ----------------|
\2/ \2 2 / | 2/a pi b\|
| sec |- - -- - -||
\ \2 2 2//
-1 + ---------------------------------------------
/ 2/a b\ \
| sec |- - -| |
| \2 2/ |
4*|1 + ----------------|
| 2/a pi b\|
| sec |- - -- - -||
\ \2 2 2//
$$\frac{b \left(- \frac{\sec^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{b}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec^{2}{\left(\frac{a}{2} \right)} \sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{4 \left(\frac{\sec^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{b}{2} - \frac{\pi}{2} \right)}} + 1\right)} - 1$$
/ 2/a b\\
b*|1 - tan |- - -||
2 \ \2 2//
-2 + (-1 + cos(a)) + 2*cos(a) + -------------------
2/a b\
1 + tan |- - -|
\2 2/
----------------------------------------------------
2
sin (a)
$$\frac{\frac{b \left(1 - \tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)}\right)}{\tan^{2}{\left(\frac{a}{2} - \frac{b}{2} \right)} + 1} + \left(\cos{\left(a \right)} - 1\right)^{2} + 2 \cos{\left(a \right)} - 2}{\sin^{2}{\left(a \right)}}$$
(-2 + (-1 + cos(a))^2 + 2*cos(a) + b*(1 - tan(a/2 - b/2)^2)/(1 + tan(a/2 - b/2)^2))/sin(a)^2