Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
(r*cos(v)/(b-r*cos(v))-r^2*sin(v)^2/(b-r*cos(v))^2)/(1+r^2*sin(v)^2/(b-r*cos(v))^2)
¿Cómo vas a descomponer esta (r*cos(v)/(b-r*cos(v))-r^2*sin(v)^2/(b-r*cos(v))^2)/(1+r^2*sin(v)^2/(b-r*cos(v))^2) expresión en fracciones?
Solución
Ha introducido
[src]
2 2
r*cos(v) r *sin (v)
------------ - ---------------
b - r*cos(v) 2
(b - r*cos(v))
------------------------------
2 2
r *sin (v)
1 + ---------------
2
(b - r*cos(v))
$$\frac{- \frac{r^{2} \sin^{2}{\left(v \right)}}{\left(b - r \cos{\left(v \right)}\right)^{2}} + \frac{r \cos{\left(v \right)}}{b - r \cos{\left(v \right)}}}{\frac{r^{2} \sin^{2}{\left(v \right)}}{\left(b - r \cos{\left(v \right)}\right)^{2}} + 1}$$
((r*cos(v))/(b - r*cos(v)) - r^2*sin(v)^2/(b - r*cos(v))^2)/(1 + (r^2*sin(v)^2)/(b - r*cos(v))^2)
Simplificación general
[src]
r*(-r + b*cos(v)) ---------------------- 2 2 b + r - 2*b*r*cos(v)
$$\frac{r \left(b \cos{\left(v \right)} - r\right)}{b^{2} - 2 b r \cos{\left(v \right)} + r^{2}}$$
r*(-r + b*cos(v))/(b^2 + r^2 - 2*b*r*cos(v))
Respuesta numérica
[src]
(r*cos(v)/(b - r*cos(v)) - r^2*sin(v)^2/(b - r*cos(v))^2)/(1.0 + r^2*sin(v)^2/(b - r*cos(v))^2)
(r*cos(v)/(b - r*cos(v)) - r^2*sin(v)^2/(b - r*cos(v))^2)/(1.0 + r^2*sin(v)^2/(b - r*cos(v))^2)
Denominador racional
[src]
2 3 2 2 2
r*(b - r*cos(v)) *cos(v) + r *sin (v)*cos(v) - b*r *sin (v)
------------------------------------------------------------
/ 2 2 2 2 2 \
(b - r*cos(v))*\b + r *cos (v) + r *sin (v) - 2*b*r*cos(v)/
$$\frac{- b r^{2} \sin^{2}{\left(v \right)} + r^{3} \sin^{2}{\left(v \right)} \cos{\left(v \right)} + r \left(b - r \cos{\left(v \right)}\right)^{2} \cos{\left(v \right)}}{\left(b - r \cos{\left(v \right)}\right) \left(b^{2} - 2 b r \cos{\left(v \right)} + r^{2} \sin^{2}{\left(v \right)} + r^{2} \cos^{2}{\left(v \right)}\right)}$$
(r*(b - r*cos(v))^2*cos(v) + r^3*sin(v)^2*cos(v) - b*r^2*sin(v)^2)/((b - r*cos(v))*(b^2 + r^2*cos(v)^2 + r^2*sin(v)^2 - 2*b*r*cos(v)))
Combinatoria
[src]
/ 2 2 \ -r*\r*cos (v) + r*sin (v) - b*cos(v)/ ------------------------------------------- 2 2 2 2 2 b + r *cos (v) + r *sin (v) - 2*b*r*cos(v)
$$- \frac{r \left(- b \cos{\left(v \right)} + r \sin^{2}{\left(v \right)} + r \cos^{2}{\left(v \right)}\right)}{b^{2} - 2 b r \cos{\left(v \right)} + r^{2} \sin^{2}{\left(v \right)} + r^{2} \cos^{2}{\left(v \right)}}$$
-r*(r*cos(v)^2 + r*sin(v)^2 - b*cos(v))/(b^2 + r^2*cos(v)^2 + r^2*sin(v)^2 - 2*b*r*cos(v))
Potencias
[src]
/ I*v -I*v\
|e e | 2
r*|---- + -----| 2 / -I*v I*v\
\ 2 2 / r *\- e + e /
-------------------- + -------------------------
/ I*v -I*v\ 2
|e e | / / I*v -I*v\\
b - r*|---- + -----| | |e e ||
\ 2 2 / 4*|b - r*|---- + -----||
\ \ 2 2 //
------------------------------------------------
2
2 / -I*v I*v\
r *\- e + e /
1 - -------------------------
2
/ / I*v -I*v\\
| |e e ||
4*|b - r*|---- + -----||
\ \ 2 2 //
$$\frac{\frac{r^{2} \left(e^{i v} - e^{- i v}\right)^{2}}{4 \left(b - r \left(\frac{e^{i v}}{2} + \frac{e^{- i v}}{2}\right)\right)^{2}} + \frac{r \left(\frac{e^{i v}}{2} + \frac{e^{- i v}}{2}\right)}{b - r \left(\frac{e^{i v}}{2} + \frac{e^{- i v}}{2}\right)}}{- \frac{r^{2} \left(e^{i v} - e^{- i v}\right)^{2}}{4 \left(b - r \left(\frac{e^{i v}}{2} + \frac{e^{- i v}}{2}\right)\right)^{2}} + 1}$$
(r*(exp(i*v)/2 + exp(-i*v)/2)/(b - r*(exp(i*v)/2 + exp(-i*v)/2)) + r^2*(-exp(-i*v) + exp(i*v))^2/(4*(b - r*(exp(i*v)/2 + exp(-i*v)/2))^2))/(1 - r^2*(-exp(-i*v) + exp(i*v))^2/(4*(b - r*(exp(i*v)/2 + exp(-i*v)/2))^2))
Unión de expresiones racionales
[src]
/ 2 \
r*\(b - r*cos(v))*cos(v) - r*sin (v)/
-------------------------------------
2 2 2
(b - r*cos(v)) + r *sin (v)
$$\frac{r \left(- r \sin^{2}{\left(v \right)} + \left(b - r \cos{\left(v \right)}\right) \cos{\left(v \right)}\right)}{r^{2} \sin^{2}{\left(v \right)} + \left(b - r \cos{\left(v \right)}\right)^{2}}$$
r*((b - r*cos(v))*cos(v) - r*sin(v)^2)/((b - r*cos(v))^2 + r^2*sin(v)^2)
Denominador común
[src]
2
- b + b*r*cos(v)
-1 - -------------------------------------------
2 2 2 2 2
b + r *cos (v) + r *sin (v) - 2*b*r*cos(v)
$$- \frac{- b^{2} + b r \cos{\left(v \right)}}{b^{2} - 2 b r \cos{\left(v \right)} + r^{2} \sin^{2}{\left(v \right)} + r^{2} \cos^{2}{\left(v \right)}} - 1$$
-1 - (-b^2 + b*r*cos(v))/(b^2 + r^2*cos(v)^2 + r^2*sin(v)^2 - 2*b*r*cos(v))
Parte trigonométrica
[src]
r*(-r + b*cos(v)) ---------------------- 2 2 b + r - 2*b*r*cos(v)
$$\frac{r \left(b \cos{\left(v \right)} - r\right)}{b^{2} - 2 b r \cos{\left(v \right)} + r^{2}}$$
2
r r
----------------------------- - --------------------------
/ r \ /pi \ 2
|b - -----------|*csc|-- - v| / r \ 2
| /pi \| \2 / |b - -----------| *csc (v)
| csc|-- - v|| | /pi \|
\ \2 // | csc|-- - v||
\ \2 //
----------------------------------------------------------
2
r
1 + --------------------------
2
/ r \ 2
|b - -----------| *csc (v)
| /pi \|
| csc|-- - v||
\ \2 //
$$\frac{- \frac{r^{2}}{\left(b - \frac{r}{\csc{\left(- v + \frac{\pi}{2} \right)}}\right)^{2} \csc^{2}{\left(v \right)}} + \frac{r}{\left(b - \frac{r}{\csc{\left(- v + \frac{\pi}{2} \right)}}\right) \csc{\left(- v + \frac{\pi}{2} \right)}}}{\frac{r^{2}}{\left(b - \frac{r}{\csc{\left(- v + \frac{\pi}{2} \right)}}\right)^{2} \csc^{2}{\left(v \right)}} + 1}$$
/ 2/v\\ 2 2/v\
r*|1 - tan |-|| 4*r *tan |-|
\ \2// \2/
----------------------------------- - -------------------------------------
/ / 2/v\\\ 2
| r*|1 - tan |-||| / / 2/v\\\
/ 2/v\\ | \ \2//| 2 | r*|1 - tan |-|||
|1 + tan |-||*|b - ---------------| / 2/v\\ | \ \2//|
\ \2// | 2/v\ | |1 + tan |-|| *|b - ---------------|
| 1 + tan |-| | \ \2// | 2/v\ |
\ \2/ / | 1 + tan |-| |
\ \2/ /
---------------------------------------------------------------------------
2 2/v\
4*r *tan |-|
\2/
1 + -------------------------------------
2
/ / 2/v\\\
2 | r*|1 - tan |-|||
/ 2/v\\ | \ \2//|
|1 + tan |-|| *|b - ---------------|
\ \2// | 2/v\ |
| 1 + tan |-| |
\ \2/ /
$$\frac{- \frac{4 r^{2} \tan^{2}{\left(\frac{v}{2} \right)}}{\left(b - \frac{r \left(1 - \tan^{2}{\left(\frac{v}{2} \right)}\right)}{\tan^{2}{\left(\frac{v}{2} \right)} + 1}\right)^{2} \left(\tan^{2}{\left(\frac{v}{2} \right)} + 1\right)^{2}} + \frac{r \left(1 - \tan^{2}{\left(\frac{v}{2} \right)}\right)}{\left(b - \frac{r \left(1 - \tan^{2}{\left(\frac{v}{2} \right)}\right)}{\tan^{2}{\left(\frac{v}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{v}{2} \right)} + 1\right)}}{\frac{4 r^{2} \tan^{2}{\left(\frac{v}{2} \right)}}{\left(b - \frac{r \left(1 - \tan^{2}{\left(\frac{v}{2} \right)}\right)}{\tan^{2}{\left(\frac{v}{2} \right)} + 1}\right)^{2} \left(\tan^{2}{\left(\frac{v}{2} \right)} + 1\right)^{2}} + 1}$$
/ b \
r*|-r + -----------|
| /pi \|
| csc|-- - v||
\ \2 //
---------------------
2 2 2*b*r
b + r - -----------
/pi \
csc|-- - v|
\2 /
$$\frac{r \left(\frac{b}{\csc{\left(- v + \frac{\pi}{2} \right)}} - r\right)}{b^{2} - \frac{2 b r}{\csc{\left(- v + \frac{\pi}{2} \right)}} + r^{2}}$$
2
r r
------------------- - ---------------------
/ r \ 2
|b - ------|*sec(v) / r \ 2
\ sec(v)/ |b - ------| *csc (v)
\ sec(v)/
-------------------------------------------
2
r
1 + ---------------------
2
/ r \ 2
|b - ------| *csc (v)
\ sec(v)/
$$\frac{- \frac{r^{2}}{\left(b - \frac{r}{\sec{\left(v \right)}}\right)^{2} \csc^{2}{\left(v \right)}} + \frac{r}{\left(b - \frac{r}{\sec{\left(v \right)}}\right) \sec{\left(v \right)}}}{\frac{r^{2}}{\left(b - \frac{r}{\sec{\left(v \right)}}\right)^{2} \csc^{2}{\left(v \right)}} + 1}$$
/ r r 2*b 2*r \
r*|b - ------ - ------------ - ------------ + -------------------|
| sec(v) 2/ pi\ 2/v pi\ 2/v pi\|
| sec |v - --| sec |- - --| sec(v)*sec |- - --||
\ \ 2 / \2 2 / \2 2 //
------------------------------------------------------------------
2 2 2*b*r
b + r - ------
sec(v)
$$\frac{r \left(b - \frac{2 b}{\sec^{2}{\left(\frac{v}{2} - \frac{\pi}{2} \right)}} - \frac{r}{\sec^{2}{\left(v - \frac{\pi}{2} \right)}} - \frac{r}{\sec{\left(v \right)}} + \frac{2 r}{\sec{\left(v \right)} \sec^{2}{\left(\frac{v}{2} - \frac{\pi}{2} \right)}}\right)}{b^{2} - \frac{2 b r}{\sec{\left(v \right)}} + r^{2}}$$
/ / 2/v\\ 2/v\ 2/v\ 2/v\ / 2/v\\ \
| r*|1 - tan |-|| 8*b*tan |-| 4*r*tan |-| 8*r*tan |-|*|1 - tan |-|| |
| \ \2// \4/ \2/ \4/ \ \2// |
r*|b - --------------- - -------------- - -------------- + ----------------------------|
| 2/v\ 2 2 2|
| 1 + tan |-| / 2/v\\ / 2/v\\ / 2/v\\ / 2/v\\ |
| \2/ |1 + tan |-|| |1 + tan |-|| |1 + tan |-||*|1 + tan |-|| |
\ \ \4// \ \2// \ \2// \ \4// /
----------------------------------------------------------------------------------------
/ 2/v\\
2*b*r*|1 - tan |-||
2 2 \ \2//
b + r - -------------------
2/v\
1 + tan |-|
\2/
$$\frac{r \left(b - \frac{8 b \tan^{2}{\left(\frac{v}{4} \right)}}{\left(\tan^{2}{\left(\frac{v}{4} \right)} + 1\right)^{2}} - \frac{r \left(1 - \tan^{2}{\left(\frac{v}{2} \right)}\right)}{\tan^{2}{\left(\frac{v}{2} \right)} + 1} + \frac{8 r \left(1 - \tan^{2}{\left(\frac{v}{2} \right)}\right) \tan^{2}{\left(\frac{v}{4} \right)}}{\left(\tan^{2}{\left(\frac{v}{4} \right)} + 1\right)^{2} \left(\tan^{2}{\left(\frac{v}{2} \right)} + 1\right)} - \frac{4 r \tan^{2}{\left(\frac{v}{2} \right)}}{\left(\tan^{2}{\left(\frac{v}{2} \right)} + 1\right)^{2}}\right)}{b^{2} - \frac{2 b r \left(1 - \tan^{2}{\left(\frac{v}{2} \right)}\right)}{\tan^{2}{\left(\frac{v}{2} \right)} + 1} + r^{2}}$$
/ / 2/v\\ 2/v\ 2/v\ 2/v\ / 2/v\\ \
| r*|-1 + cot |-|| 8*b*cot |-| 4*r*cot |-| 8*r*cot |-|*|-1 + cot |-|| |
| \ \2// \4/ \2/ \4/ \ \2// |
r*|b - ---------------- - -------------- - -------------- + ----------------------------|
| 2/v\ 2 2 2|
| 1 + cot |-| / 2/v\\ / 2/v\\ / 2/v\\ / 2/v\\ |
| \2/ |1 + cot |-|| |1 + cot |-|| |1 + cot |-||*|1 + cot |-|| |
\ \ \4// \ \2// \ \2// \ \4// /
-----------------------------------------------------------------------------------------
/ 2/v\\
2*b*r*|-1 + cot |-||
2 2 \ \2//
b + r - --------------------
2/v\
1 + cot |-|
\2/
$$\frac{r \left(b - \frac{8 b \cot^{2}{\left(\frac{v}{4} \right)}}{\left(\cot^{2}{\left(\frac{v}{4} \right)} + 1\right)^{2}} - \frac{r \left(\cot^{2}{\left(\frac{v}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{v}{2} \right)} + 1} - \frac{4 r \cot^{2}{\left(\frac{v}{2} \right)}}{\left(\cot^{2}{\left(\frac{v}{2} \right)} + 1\right)^{2}} + \frac{8 r \left(\cot^{2}{\left(\frac{v}{2} \right)} - 1\right) \cot^{2}{\left(\frac{v}{4} \right)}}{\left(\cot^{2}{\left(\frac{v}{4} \right)} + 1\right)^{2} \left(\cot^{2}{\left(\frac{v}{2} \right)} + 1\right)}\right)}{b^{2} - \frac{2 b r \left(\cot^{2}{\left(\frac{v}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{v}{2} \right)} + 1} + r^{2}}$$
/ / 2/v\\\
| b*|1 - tan |-|||
| \ \2//|
r*|-r + ---------------|
| 2/v\ |
| 1 + tan |-| |
\ \2/ /
-----------------------------
/ 2/v\\
2*b*r*|1 - tan |-||
2 2 \ \2//
b + r - -------------------
2/v\
1 + tan |-|
\2/
$$\frac{r \left(\frac{b \left(1 - \tan^{2}{\left(\frac{v}{2} \right)}\right)}{\tan^{2}{\left(\frac{v}{2} \right)} + 1} - r\right)}{b^{2} - \frac{2 b r \left(1 - \tan^{2}{\left(\frac{v}{2} \right)}\right)}{\tan^{2}{\left(\frac{v}{2} \right)} + 1} + r^{2}}$$
/ pi\
r*sin|v + --| 2 2
\ 2 / r *sin (v)
----------------- - --------------------
/ pi\ 2
b - r*sin|v + --| / / pi\\
\ 2 / |b - r*sin|v + --||
\ \ 2 //
----------------------------------------
2 2
r *sin (v)
1 + --------------------
2
/ / pi\\
|b - r*sin|v + --||
\ \ 2 //
$$\frac{- \frac{r^{2} \sin^{2}{\left(v \right)}}{\left(b - r \sin{\left(v + \frac{\pi}{2} \right)}\right)^{2}} + \frac{r \sin{\left(v + \frac{\pi}{2} \right)}}{b - r \sin{\left(v + \frac{\pi}{2} \right)}}}{\frac{r^{2} \sin^{2}{\left(v \right)}}{\left(b - r \sin{\left(v + \frac{\pi}{2} \right)}\right)^{2}} + 1}$$
/ / 2/v\\\
| b*|-1 + cot |-|||
| \ \2//|
r*|-r + ----------------|
| 2/v\ |
| 1 + cot |-| |
\ \2/ /
------------------------------
/ 2/v\\
2*b*r*|-1 + cot |-||
2 2 \ \2//
b + r - --------------------
2/v\
1 + cot |-|
\2/
$$\frac{r \left(\frac{b \left(\cot^{2}{\left(\frac{v}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{v}{2} \right)} + 1} - r\right)}{b^{2} - \frac{2 b r \left(\cot^{2}{\left(\frac{v}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{v}{2} \right)} + 1} + r^{2}}$$
/ / pi\\
r*|-r + b*sin|v + --||
\ \ 2 //
---------------------------
2 2 / pi\
b + r - 2*b*r*sin|v + --|
\ 2 /
$$\frac{r \left(b \sin{\left(v + \frac{\pi}{2} \right)} - r\right)}{b^{2} - 2 b r \sin{\left(v + \frac{\pi}{2} \right)} + r^{2}}$$
/ 2/v\\ 2 2/v\
r*|-1 + cot |-|| 4*r *cot |-|
\ \2// \2/
------------------------------------ - --------------------------------------
/ / 2/v\\\ 2
| r*|-1 + cot |-||| / / 2/v\\\
/ 2/v\\ | \ \2//| 2 | r*|-1 + cot |-|||
|1 + cot |-||*|b - ----------------| / 2/v\\ | \ \2//|
\ \2// | 2/v\ | |1 + cot |-|| *|b - ----------------|
| 1 + cot |-| | \ \2// | 2/v\ |
\ \2/ / | 1 + cot |-| |
\ \2/ /
-----------------------------------------------------------------------------
2 2/v\
4*r *cot |-|
\2/
1 + --------------------------------------
2
/ / 2/v\\\
2 | r*|-1 + cot |-|||
/ 2/v\\ | \ \2//|
|1 + cot |-|| *|b - ----------------|
\ \2// | 2/v\ |
| 1 + cot |-| |
\ \2/ /
$$\frac{- \frac{4 r^{2} \cot^{2}{\left(\frac{v}{2} \right)}}{\left(b - \frac{r \left(\cot^{2}{\left(\frac{v}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{v}{2} \right)} + 1}\right)^{2} \left(\cot^{2}{\left(\frac{v}{2} \right)} + 1\right)^{2}} + \frac{r \left(\cot^{2}{\left(\frac{v}{2} \right)} - 1\right)}{\left(b - \frac{r \left(\cot^{2}{\left(\frac{v}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{v}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{v}{2} \right)} + 1\right)}}{\frac{4 r^{2} \cot^{2}{\left(\frac{v}{2} \right)}}{\left(b - \frac{r \left(\cot^{2}{\left(\frac{v}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{v}{2} \right)} + 1}\right)^{2} \left(\cot^{2}{\left(\frac{v}{2} \right)} + 1\right)^{2}} + 1}$$
2
r r
------------------- - --------------------------
/ r \ 2
|b - ------|*sec(v) / r \ 2/ pi\
\ sec(v)/ |b - ------| *sec |v - --|
\ sec(v)/ \ 2 /
------------------------------------------------
2
r
1 + --------------------------
2
/ r \ 2/ pi\
|b - ------| *sec |v - --|
\ sec(v)/ \ 2 /
$$\frac{- \frac{r^{2}}{\left(b - \frac{r}{\sec{\left(v \right)}}\right)^{2} \sec^{2}{\left(v - \frac{\pi}{2} \right)}} + \frac{r}{\left(b - \frac{r}{\sec{\left(v \right)}}\right) \sec{\left(v \right)}}}{\frac{r^{2}}{\left(b - \frac{r}{\sec{\left(v \right)}}\right)^{2} \sec^{2}{\left(v - \frac{\pi}{2} \right)}} + 1}$$
/ b \
r*|-r + ------|
\ sec(v)/
----------------
2 2 2*b*r
b + r - ------
sec(v)
$$\frac{r \left(\frac{b}{\sec{\left(v \right)}} - r\right)}{b^{2} - \frac{2 b r}{\sec{\left(v \right)}} + r^{2}}$$
/ 2 / pi\ 2/v\ 2/v\ / pi\\
r*|b - r*sin (v) - r*sin|v + --| - 2*b*sin |-| + 2*r*sin |-|*sin|v + --||
\ \ 2 / \2/ \2/ \ 2 //
-------------------------------------------------------------------------
2 2 / pi\
b + r - 2*b*r*sin|v + --|
\ 2 /
$$\frac{r \left(- 2 b \sin^{2}{\left(\frac{v}{2} \right)} + b + 2 r \sin^{2}{\left(\frac{v}{2} \right)} \sin{\left(v + \frac{\pi}{2} \right)} - r \sin^{2}{\left(v \right)} - r \sin{\left(v + \frac{\pi}{2} \right)}\right)}{b^{2} - 2 b r \sin{\left(v + \frac{\pi}{2} \right)} + r^{2}}$$
/ 2 2/v\ 2/v\ \
r*|b - r*sin (v) - r*cos(v) - 2*b*sin |-| + 2*r*sin |-|*cos(v)|
\ \2/ \2/ /
---------------------------------------------------------------
2 2
b + r - 2*b*r*cos(v)
$$\frac{r \left(- 2 b \sin^{2}{\left(\frac{v}{2} \right)} + b + 2 r \sin^{2}{\left(\frac{v}{2} \right)} \cos{\left(v \right)} - r \sin^{2}{\left(v \right)} - r \cos{\left(v \right)}\right)}{b^{2} - 2 b r \cos{\left(v \right)} + r^{2}}$$
2 2/ pi\
r *cos |v - --|
r*cos(v) \ 2 /
------------ - ---------------
b - r*cos(v) 2
(b - r*cos(v))
------------------------------
2 2/ pi\
r *cos |v - --|
\ 2 /
1 + ---------------
2
(b - r*cos(v))
$$\frac{- \frac{r^{2} \cos^{2}{\left(v - \frac{\pi}{2} \right)}}{\left(b - r \cos{\left(v \right)}\right)^{2}} + \frac{r \cos{\left(v \right)}}{b - r \cos{\left(v \right)}}}{\frac{r^{2} \cos^{2}{\left(v - \frac{\pi}{2} \right)}}{\left(b - r \cos{\left(v \right)}\right)^{2}} + 1}$$
/ 2/ pi\ 2/v pi\ 2/v pi\ \
r*|b - r*cos |v - --| - r*cos(v) - 2*b*cos |- - --| + 2*r*cos |- - --|*cos(v)|
\ \ 2 / \2 2 / \2 2 / /
------------------------------------------------------------------------------
2 2
b + r - 2*b*r*cos(v)
$$\frac{r \left(- 2 b \cos^{2}{\left(\frac{v}{2} - \frac{\pi}{2} \right)} + b + 2 r \cos{\left(v \right)} \cos^{2}{\left(\frac{v}{2} - \frac{\pi}{2} \right)} - r \cos{\left(v \right)} - r \cos^{2}{\left(v - \frac{\pi}{2} \right)}\right)}{b^{2} - 2 b r \cos{\left(v \right)} + r^{2}}$$
/ r r 2*b 2*r \
r*|b - ------- - ----------- - ------- + -------------------|
| 2 /pi \ 2/v\ 2/v\ /pi \|
| csc (v) csc|-- - v| csc |-| csc |-|*csc|-- - v||
\ \2 / \2/ \2/ \2 //
-------------------------------------------------------------
2 2 2*b*r
b + r - -----------
/pi \
csc|-- - v|
\2 /
$$\frac{r \left(b - \frac{2 b}{\csc^{2}{\left(\frac{v}{2} \right)}} - \frac{r}{\csc{\left(- v + \frac{\pi}{2} \right)}} - \frac{r}{\csc^{2}{\left(v \right)}} + \frac{2 r}{\csc^{2}{\left(\frac{v}{2} \right)} \csc{\left(- v + \frac{\pi}{2} \right)}}\right)}{b^{2} - \frac{2 b r}{\csc{\left(- v + \frac{\pi}{2} \right)}} + r^{2}}$$
r*(b - r/csc(v)^2 - r/csc(pi/2 - v) - 2*b/csc(v/2)^2 + 2*r/(csc(v/2)^2*csc(pi/2 - v)))/(b^2 + r^2 - 2*b*r/csc(pi/2 - v))