Господин Экзамен
Общий знаменатель (2*sin(a-3*pi)-cos(((-pi)/2)+a))/(5*sin(a-pi))
Решение
Вы ввели
[src]
/-pi \
2*sin(a - 3*pi) - cos|---- + a|
\ 2 /
-------------------------------
5*sin(a - pi)
$$\frac{2 \sin{\left(a - 3 \pi \right)} - \cos{\left(a + \frac{\left(-1\right) \pi}{2} \right)}}{5 \sin{\left(a - \pi \right)}}$$
(2*sin(a - 3*pi) - cos(-pi/2 + a))/((5*sin(a - pi)))
Собрать выражение
[src]
3*csc(a)*sin(a)
---------------
5
$$\frac{3 \sin{\left(a \right)} \csc{\left(a \right)}}{5}$$
3*csc(a)*sin(a)/5
Общий знаменатель
[src]
-3*sin(a) ------------- 5*sin(a - pi)
$$- \frac{3 \sin{\left(a \right)}}{5 \sin{\left(a - \pi \right)}}$$
-3*sin(a)/(5*sin(a - pi))
Объединение рациональных выражений
[src]
/ /-pi + 2*a\ \
-|- cos|---------| - 2*sin(a)|
\ \ 2 / /
-------------------------------
5*sin(a)
$$- \frac{- 2 \sin{\left(a \right)} - \cos{\left(\frac{2 a - \pi}{2} \right)}}{5 \sin{\left(a \right)}}$$
-(-cos((-pi + 2*a)/2) - 2*sin(a))/(5*sin(a))
Степени
[src]
3/5
$$\frac{3}{5}$$
sin(a) 2*sin(a - 3*pi)
- ------ + ---------------
5 5
--------------------------
sin(a - pi)
$$\frac{- \frac{\sin{\left(a \right)}}{5} + \frac{2 \sin{\left(a - 3 \pi \right)}}{5}}{\sin{\left(a - \pi \right)}}$$
/ / pi\ /pi \ \
| I*|a - --| I*|-- - a| |
| \ 2 / \2 / |
| e e / I*(-a + 3*pi) I*(a - 3*pi)\|
2*I*|- ----------- - ----------- - I*\- e + e /|
\ 2 2 /
------------------------------------------------------------------------
/ I*(pi - a) I*(a - pi)\
5*\- e + e /
$$\frac{2 i \left(- i \left(- e^{i \left(- a + 3 \pi\right)} + e^{i \left(a - 3 \pi\right)}\right) - \frac{e^{i \left(- a + \frac{\pi}{2}\right)}}{2} - \frac{e^{i \left(a - \frac{\pi}{2}\right)}}{2}\right)}{5 \left(- e^{i \left(- a + \pi\right)} + e^{i \left(a - \pi\right)}\right)}$$
2*i*(-exp(i*(a - pi/2))/2 - exp(i*(pi/2 - a))/2 - i*(-exp(i*(-a + 3*pi)) + exp(i*(a - 3*pi))))/(5*(-exp(i*(pi - a)) + exp(i*(a - pi))))
Тригонометрическая часть
[src]
3/5
$$\frac{3}{5}$$
/ / pi\ \
-|- cos|a - --| - 2*sin(a)|
\ \ 2 / /
----------------------------
5*sin(a)
$$- \frac{- 2 \sin{\left(a \right)} - \cos{\left(a - \frac{\pi}{2} \right)}}{5 \sin{\left(a \right)}}$$
/ / -pi \ \
-|- cos|a + ----| - 2*sin(a)|
\ \ 2 / /
------------------------------
5*sin(a)
$$- \frac{- 2 \sin{\left(a \right)} - \cos{\left(a + \frac{\left(-1\right) \pi}{2} \right)}}{5 \sin{\left(a \right)}}$$
/ / pi\\
-|-sin(a) - 2*cos|a - --||
\ \ 2 //
---------------------------
/ pi\
5*cos|a - --|
\ 2 /
$$- \frac{- \sin{\left(a \right)} - 2 \cos{\left(a - \frac{\pi}{2} \right)}}{5 \cos{\left(a - \frac{\pi}{2} \right)}}$$
/ 1 /a\\
3*|------ + tan|-||*sin(a)
| /a\ \2/|
|tan|-| |
\ \2/ /
--------------------------
10
$$\frac{3 \left(\tan{\left(\frac{a}{2} \right)} + \frac{1}{\tan{\left(\frac{a}{2} \right)}}\right) \sin{\left(a \right)}}{10}$$
/ 1 2 \
-|- ------ - -----------|*csc(pi - a)
\ csc(a) csc(pi - a)/
--------------------------------------
5
$$- \frac{\left(- \frac{2}{\csc{\left(- a + \pi \right)}} - \frac{1}{\csc{\left(a \right)}}\right) \csc{\left(- a + \pi \right)}}{5}$$
/ 1 2 \
-|- ----------- - ------|*csc(a)
| / pi\ csc(a)|
| sec|a - --| |
\ \ 2 / /
---------------------------------
5
$$- \frac{\left(- \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}} - \frac{2}{\csc{\left(a \right)}}\right) \csc{\left(a \right)}}{5}$$
/ 1 2 \
-|- ----------- - ------|*csc(a)
| /pi \ csc(a)|
| sec|-- - a| |
\ \2 / /
---------------------------------
5
$$- \frac{\left(- \frac{1}{\sec{\left(- a + \frac{\pi}{2} \right)}} - \frac{2}{\csc{\left(a \right)}}\right) \csc{\left(a \right)}}{5}$$
sin(a)
2 + 2*cos(a) + ------
/a\
tan|-|
\2/
---------------------
5*(1 + cos(a))
$$\frac{2 \cos{\left(a \right)} + \frac{\sin{\left(a \right)}}{\tan{\left(\frac{a}{2} \right)}} + 2}{5 \left(\cos{\left(a \right)} + 1\right)}$$
/ 1 2 \ / pi\
-|- ------ - -----------|*sec|a - --|
| csc(a) / pi\| \ 2 /
| sec|a - --||
\ \ 2 //
--------------------------------------
5
$$- \frac{\left(- \frac{2}{\sec{\left(a - \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(a \right)}}\right) \sec{\left(a - \frac{\pi}{2} \right)}}{5}$$
/9 2 cos(2*a)\
-|- + (1 - cos(a)) - 4*cos(a) - --------|
\2 2 /
-------------------------------------------
10*(-1 + cos(a))
$$- \frac{\left(- \cos{\left(a \right)} + 1\right)^{2} - 4 \cos{\left(a \right)} - \frac{\cos{\left(2 a \right)}}{2} + \frac{9}{2}}{10 \left(\cos{\left(a \right)} - 1\right)}$$
1 1 1 1
- - --------------- + --------- - ----------------------
2 2/a pi\ 10*sin(a) 2/a pi\
10*tan |- + --| 10*sin(a)*tan |- + --|
\2 4 / \2 4 /
$$\frac{1}{2} + \frac{1}{10 \sin{\left(a \right)}} - \frac{1}{10 \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} - \frac{1}{10 \sin{\left(a \right)} \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
/ /a\\
| tan|-||
| 1 /a\ \2/|
2*|------- + 3*tan|-| + ------|
| 2/a\ \2/ sin(a)|
|cos |-| |
\ \2/ /
-------------------------------
2
/ /a\\
5*|1 + tan|-||
\ \2//
$$\frac{2 \cdot \left(3 \tan{\left(\frac{a}{2} \right)} + \frac{\tan{\left(\frac{a}{2} \right)}}{\sin{\left(a \right)}} + \frac{1}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)}{5 \left(\tan{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
2 2
1 1 cos (a) cos (a)
- + --------- - --------------- - ----------------------
2 10*sin(a) 4/a pi\ 4/a pi\
40*sin |- + --| 40*sin(a)*sin |- + --|
\2 4 / \2 4 /
$$\frac{1}{2} + \frac{1}{10 \sin{\left(a \right)}} - \frac{\cos^{2}{\left(a \right)}}{40 \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} - \frac{\cos^{2}{\left(a \right)}}{40 \sin{\left(a \right)} \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
/ / 2/a pi\\ \
-|-sin(a) - |1 - cot |- + --||*(1 + sin(a))|
\ \ \2 4 // /
---------------------------------------------
/ 2/a pi\\ 2/a pi\
5*|1 - cot |- + --||*sin |- + --|
\ \2 4 // \2 4 /
$$- \frac{- \left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(a \right)} + 1\right) - \sin{\left(a \right)}}{5 \cdot \left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
2/a\ 2/a\
1 + tan |-| 1 + tan |-|
1 1 \2/ \2/
- - --------------- + ----------- - ----------------------
2 2/a pi\ /a\ /a\ 2/a pi\
10*tan |- + --| 20*tan|-| 20*tan|-|*tan |- + --|
\2 4 / \2/ \2/ \2 4 /
$$\frac{\tan^{2}{\left(\frac{a}{2} \right)} + 1}{20 \tan{\left(\frac{a}{2} \right)}} + \frac{1}{2} - \frac{\tan^{2}{\left(\frac{a}{2} \right)} + 1}{20 \tan{\left(\frac{a}{2} \right)} \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} - \frac{1}{10 \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
2/a pi\ 2/a pi\
csc |- + --| csc |- + --|*csc(a)
1 csc(a) \2 4 / \2 4 /
- + ------ - ----------------- - -------------------
2 10 2/ a pi\ 2/ a pi\
10*csc |- - + --| 10*csc |- - + --|
\ 2 4 / \ 2 4 /
$$\frac{\csc{\left(a \right)}}{10} - \frac{\csc{\left(a \right)} \csc^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{10 \csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{4} \right)}} + \frac{1}{2} - \frac{\csc^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{10 \csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{4} \right)}}$$
/ 2/a pi\ /a\ \
| 1 - cot |- + --| 4*tan|-| |
/ 2/a\\ | \2 4 / \2/ |
-|1 + tan |-||*|- ---------------- - -----------|
\ \2// | 2/a pi\ 2/a\|
| 1 + cot |- + --| 1 + tan |-||
\ \2 4 / \2//
--------------------------------------------------
/a\
10*tan|-|
\2/
$$- \frac{\left(- \frac{- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} - \frac{4 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}{10 \tan{\left(\frac{a}{2} \right)}}$$
2/a pi\ 2/a pi\
cos |- + --| cos |- + --|
1 1 \2 4 / \2 4 /
- + -------------- - --------------- - ---------------------------
2 / pi\ 2/a pi\ / pi\ 2/a pi\
10*cos|a - --| 10*cos |- - --| 10*cos|a - --|*cos |- - --|
\ 2 / \2 4 / \ 2 / \2 4 /
$$\frac{1}{2} - \frac{\cos^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{10 \cos^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)}} + \frac{1}{10 \cos{\left(a - \frac{\pi}{2} \right)}} - \frac{\cos^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{10 \cos^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)} \cos{\left(a - \frac{\pi}{2} \right)}}$$
/ pi\ 2/a pi\ 2/a pi\ / pi\
sec|a - --| sec |- - --| sec |- - --|*sec|a - --|
1 \ 2 / \2 4 / \2 4 / \ 2 /
- + ----------- - --------------- - ------------------------
2 10 2/a pi\ 2/a pi\
10*sec |- + --| 10*sec |- + --|
\2 4 / \2 4 /
$$- \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)} \sec{\left(a - \frac{\pi}{2} \right)}}{10 \sec^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} + \frac{\sec{\left(a - \frac{\pi}{2} \right)}}{10} - \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}{10 \sec^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} + \frac{1}{2}$$
/ 1 \
| 1 - ------------ |
| 2/a pi\ /a\ |
| tan |- + --| 4*tan|-| |
/ 2/a\\ | \2 4 / \2/ |
-|1 + tan |-||*|- ---------------- - -----------|
\ \2// | 1 2/a\|
| 1 + ------------ 1 + tan |-||
| 2/a pi\ \2/|
| tan |- + --| |
\ \2 4 / /
--------------------------------------------------
/a\
10*tan|-|
\2/
$$- \frac{\left(- \frac{1 - \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}} - \frac{4 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}{10 \tan{\left(\frac{a}{2} \right)}}$$
/ 2/a pi\ \
| 1 - cot |- + --| |
/ 1 \ | \2 4 / 4 | /a\
-|1 + -------|*|- ---------------- - --------------------|*cot|-|
| 2/a\| | 2/a pi\ / 1 \ /a\| \2/
| cot |-|| | 1 + cot |- + --| |1 + -------|*cot|-||
\ \2// | \2 4 / | 2/a\| \2/|
| | cot |-|| |
\ \ \2// /
------------------------------------------------------------------
10
$$- \frac{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \left(- \frac{- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} - \frac{4}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \cot{\left(\frac{a}{2} \right)}}\right) \cot{\left(\frac{a}{2} \right)}}{10}$$
/ /a\ / 2/a pi\\\
| 2*cot|-| 2*|-1 + tan |- + --|||
/ 2/a pi\\ | \2/ \ \2 4 //|
-|1 + tan |- + --||*|- ----------- - ---------------------|
\ \2 4 // | 2/a\ 2/a pi\ |
| 1 + cot |-| 1 + tan |- + --| |
\ \2/ \2 4 / /
------------------------------------------------------------
/ 2/a pi\\
5*|-1 + tan |- + --||
\ \2 4 //
$$- \frac{\left(- \frac{2 \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} - \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)}{5 \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)}$$
/ / 2/a pi\\ /a\ \
| 2*|1 - cot |- + --|| 2*tan|-| |
/ 2/a pi\\ | \ \2 4 // \2/ |
-|1 + cot |- + --||*|- -------------------- - -----------|
\ \2 4 // | 2/a pi\ 2/a\|
| 1 + cot |- + --| 1 + tan |-||
\ \2 4 / \2//
-----------------------------------------------------------
/ 2/a pi\\
5*|1 - cot |- + --||
\ \2 4 //
$$- \frac{\left(- \frac{2 \cdot \left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} - \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)}{5 \cdot \left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)}$$
/ // / 3*pi\ \ \
| || 1 for |a + ----| mod 2*pi = 0| // 0 for a mod pi = 0\| // zoo for a mod pi = 0\
-|- |< \ 2 / | - 2*|< ||*|< |
| || | \\sin(a) otherwise /| \\csc(a) otherwise /
\ \\sin(a) otherwise / /
--------------------------------------------------------------------------------------------------------
5
$$- \frac{\left(\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\csc{\left(a \right)} & \text{otherwise} \end{cases}\right)}{5}$$
/ // / 3*pi\ \ \ // zoo for a mod pi = 0\
| || 1 for |a + ----| mod 2*pi = 0| // 0 for a mod pi = 0\| || |
-|- |< \ 2 / | - 2*|< ||*|< 1 |
| || | \\sin(a) otherwise /| ||------ otherwise |
\ \\sin(a) otherwise / / \\sin(a) /
--------------------------------------------------------------------------------------------------------
5
$$- \frac{\left(\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sin{\left(a \right)}} & \text{otherwise} \end{cases}\right)}{5}$$
/ zoo for a mod pi = 0 // zoo for a mod pi = 0\
| 2/a pi\ || |
2/a pi\ < 1 cot |- + --|*|< 1 |
cot |- + --| |------ otherwise \2 4 / ||------ otherwise |
1 \2 4 / \sin(a) \\sin(a) /
- - ------------ + ------------------------- - ----------------------------------------
2 10 10 10
$$\left(- \frac{\left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sin{\left(a \right)}} & \text{otherwise} \end{cases}\right) \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{10}\right) - \frac{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{10} + \left(\frac{\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sin{\left(a \right)}} & \text{otherwise} \end{cases}}{10}\right) + \frac{1}{2}$$
/ // / 3*pi\ \ \
| || 1 for |a + ----| mod 2*pi = 0| // 0 for a mod pi = 0\|
| || \ 2 / | || || // zoo for a mod pi = 0\
-|- |< | - 2*|< 1 ||*|< |
| || 1 | ||------ otherwise || \\csc(a) otherwise /
| ||------ otherwise | \\csc(a) /|
\ \\csc(a) / /
--------------------------------------------------------------------------------------------------------
5
$$- \frac{\left(\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\csc{\left(a \right)} & \text{otherwise} \end{cases}\right)}{5}$$
/ // / 3*pi\ \ \ // zoo for a mod pi = 0\
| || 1 for |a + ----| mod 2*pi = 0| // 0 for a mod pi = 0\| || |
| || \ 2 / | || || || 1 |
-|- |< | - 2*|< / pi\ ||*|<----------- otherwise |
| || / pi\ | ||cos|a - --| otherwise || || / pi\ |
| ||cos|a - --| otherwise | \\ \ 2 / /| ||cos|a - --| |
\ \\ \ 2 / / / \\ \ 2 / /
-----------------------------------------------------------------------------------------------------------------------
5
$$- \frac{\left(\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\cos{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\cos{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)}{5}$$
/ // / 3*pi\ \ \
| || 1 for |a + ----| mod 2*pi = 0| // 0 for a mod pi = 0\|
| || \ 2 / | || || // zoo for a mod pi = 0\
| || | || 1 || || |
-|- |< 1 | - 2*|<----------- otherwise ||*|< / pi\ |
| ||----------- otherwise | || / pi\ || ||sec|a - --| otherwise |
| || / pi\ | ||sec|a - --| || \\ \ 2 / /
| ||sec|a - --| | \\ \ 2 / /|
\ \\ \ 2 / / /
-----------------------------------------------------------------------------------------------------------------------
5
$$- \frac{\left(\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\sec{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)}{5}$$
// / 3*pi\ \
|| 1 for |a + ----| mod 2*pi = 0|
|| \ 2 / |
/ // / 3*pi\ \\ || |
| // 0 for a mod pi = 0\ || 1 for |a + ----| mod 2*pi = 0|| || 1 /a\ |
-|- |< | - 2*|< \ 2 / ||*|<------ + tan|-| |
| \\sin(a) otherwise / || || || /a\ \2/ |
\ \\sin(a) otherwise // ||tan|-| |
|| \2/ |
||--------------- otherwise |
\\ 2 /
----------------------------------------------------------------------------------------------------------------------------
5
$$- \frac{\left(\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right)\right) \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan{\left(\frac{a}{2} \right)} + \frac{1}{\tan{\left(\frac{a}{2} \right)}}}{2} & \text{otherwise} \end{cases}\right)}{5}$$
/ 2 \
| cos (a) |
| 1 - -------------- |
/ 4/a\\ | 4/a pi\ 2/a\ |
| 4*sin |-|| | 4*sin |- + --| 8*sin |-| |
| \2/| | \2 4 / \2/ |
-|1 + ---------|*|- ------------------ - ----------------------|*sin(a)
| 2 | | 2 / 4/a\\ |
\ sin (a) / | cos (a) | 4*sin |-|| |
| 1 + -------------- | \2/| |
| 4/a pi\ |1 + ---------|*sin(a)|
| 4*sin |- + --| | 2 | |
\ \2 4 / \ sin (a) / /
------------------------------------------------------------------------
2/a\
20*sin |-|
\2/
$$- \frac{\left(- \frac{1 - \frac{\cos^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}}{1 + \frac{\cos^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}} - \frac{8 \sin^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \sin{\left(a \right)}}\right) \left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \sin{\left(a \right)}}{20 \sin^{2}{\left(\frac{a}{2} \right)}}$$
/ // 0 for a mod pi = 0\\ // zoo for a mod pi = 0\
| // / 3*pi\ \ || || || |
| || 1 for |a + ----| mod 2*pi = 0| ||1 - cos(a) || || /a\ |
-|- |< \ 2 / | - 2*|<---------- otherwise ||*|< -tan|-| |
| || | || /a\ || || \2/ |
| \\sin(a) otherwise / || tan|-| || ||----------- otherwise |
\ \\ \2/ // \\-1 + cos(a) /
-----------------------------------------------------------------------------------------------------------------
5
$$- \frac{\left(\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{- \cos{\left(a \right)} + 1}{\tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\- \frac{\tan{\left(\frac{a}{2} \right)}}{\cos{\left(a \right)} - 1} & \text{otherwise} \end{cases}\right)}{5}$$
/ zoo for a mod pi = 0 // zoo for a mod pi = 0\
| || |
| 2/a\ || 2/a\ |
|1 + cot |-| 2/a pi\ ||1 + cot |-| |
< \2/ cot |- + --|*|< \2/ |
|----------- otherwise \2 4 / ||----------- otherwise |
2/a pi\ | /a\ || /a\ |
cot |- + --| | 2*cot|-| || 2*cot|-| |
1 \2 4 / \ \2/ \\ \2/ /
- - ------------ + ------------------------------ - ---------------------------------------------
2 10 10 10
$$\left(- \frac{\left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} + 1}{2 \cot{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{10}\right) - \frac{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{10} + \left(\frac{\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} + 1}{2 \cot{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}}{10}\right) + \frac{1}{2}$$
/ // / 3*pi\ \ \
| || 1 for |a + ----| mod 2*pi = 0| // 0 for a mod pi = 0\| // zoo for a mod pi = 0\
| || \ 2 / | || || || |
| || | || /a\ || || 2/a\ |
| || /a\ | || 2*tan|-| || ||1 + tan |-| |
-|- |< 2*tan|-| | - 2*|< \2/ ||*|< \2/ |
| || \2/ | ||----------- otherwise || ||----------- otherwise |
| ||----------- otherwise | || 2/a\ || || /a\ |
| || 2/a\ | ||1 + tan |-| || || 2*tan|-| |
| ||1 + tan |-| | \\ \2/ /| \\ \2/ /
\ \\ \2/ / /
-----------------------------------------------------------------------------------------------------------------------
5
$$- \frac{\left(\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} \right)} + 1}{2 \tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)}{5}$$
/ // / 3*pi\ \ \
| || 1 for |a + ----| mod 2*pi = 0| // 0 for a mod pi = 0\| // zoo for a mod pi = 0\
| || \ 2 / | || || || |
| || | || /a\ || || 2/a\ |
| || 2/a pi\ | || 2*cot|-| || ||1 + cot |-| |
-|- |<-1 + tan |- + --| | - 2*|< \2/ ||*|< \2/ |
| || \2 4 / | ||----------- otherwise || ||----------- otherwise |
| ||----------------- otherwise | || 2/a\ || || /a\ |
| || 2/a pi\ | ||1 + cot |-| || || 2*cot|-| |
| || 1 + tan |- + --| | \\ \2/ /| \\ \2/ /
\ \\ \2 4 / / /
-----------------------------------------------------------------------------------------------------------------------------
5
$$- \frac{\left(\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} + 1}{2 \cot{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)}{5}$$
/ 2/a pi\ \
| cos |- + --| |
| \2 4 / |
| 1 - ------------ |
/ 2/a pi\\ | 2/a pi\ /a pi\ |
| cos |- - --|| | cos |- - --| 4*cos|- - --| |
| \2 2 /| | \2 4 / \2 2 / | /a\
-|1 + ------------|*|- ---------------- - -------------------------|*cos|-|
| 2/a\ | | 2/a pi\ / 2/a pi\\ | \2/
| cos |-| | | cos |- + --| | cos |- - --|| |
\ \2/ / | \2 4 / | \2 2 /| /a\|
| 1 + ------------ |1 + ------------|*cos|-||
| 2/a pi\ | 2/a\ | \2/|
| cos |- - --| | cos |-| | |
\ \2 4 / \ \2/ / /
----------------------------------------------------------------------------
/a pi\
10*cos|- - --|
\2 2 /
$$- \frac{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \left(- \frac{1 - \frac{\cos^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}}{1 + \frac{\cos^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}} - \frac{4 \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \cos{\left(\frac{a}{2} \right)}}\right) \cos{\left(\frac{a}{2} \right)}}{10 \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}$$
/ 2/a pi\ \
| sec |- - --| |
| \2 4 / |
| 1 - ------------ |
/ 2/a\ \ | 2/a pi\ /a\ |
| sec |-| | | sec |- + --| 4*sec|-| |
| \2/ | | \2 4 / \2/ | /a pi\
-|1 + ------------|*|- ---------------- - ------------------------------|*sec|- - --|
| 2/a pi\| | 2/a pi\ / 2/a\ \ | \2 2 /
| sec |- - --|| | sec |- - --| | sec |-| | |
\ \2 2 // | \2 4 / | \2/ | /a pi\|
| 1 + ------------ |1 + ------------|*sec|- - --||
| 2/a pi\ | 2/a pi\| \2 2 /|
| sec |- + --| | sec |- - --|| |
\ \2 4 / \ \2 2 // /
--------------------------------------------------------------------------------------
/a\
10*sec|-|
\2/
$$- \frac{\left(- \frac{- \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}{\sec^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} + 1}{\frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}{\sec^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} + 1} - \frac{4 \sec{\left(\frac{a}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}\right) \left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{10 \sec{\left(\frac{a}{2} \right)}}$$
/ // / 3*pi\ \ \
| || 1 for |a + ----| mod 2*pi = 0| |
| || \ 2 / | |
| || | // 0 for a mod pi = 0\| // zoo for a mod pi = 0\
| || 1 | || || || |
| ||-1 + ------------ | || /a\ || || 2/a\ |
| || 2/a pi\ | || 2*cot|-| || ||1 + cot |-| |
-|- |< cot |- + --| | - 2*|< \2/ ||*|< \2/ |
| || \2 4 / | ||----------- otherwise || ||----------- otherwise |
| ||----------------- otherwise | || 2/a\ || || /a\ |
| || 1 | ||1 + cot |-| || || 2*cot|-| |
| || 1 + ------------ | \\ \2/ /| \\ \2/ /
| || 2/a pi\ | |
| || cot |- + --| | |
\ \\ \2 4 / / /
-----------------------------------------------------------------------------------------------------------------------------
5
$$- \frac{\left(\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}}{1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} + 1}{2 \cot{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)}{5}$$
/ 2/a pi\ \
| csc |- + --| |
| \2 4 / |
| 1 - -------------- |
/ 2/pi a\\ | 2/ a pi\ /pi a\ |
| csc |-- - -|| | csc |- - + --| 4*csc|-- - -| |
| \2 2/| | \ 2 4 / \2 2/ | /a\
-|1 + ------------|*|- ------------------ - -------------------------|*csc|-|
| 2/a\ | | 2/a pi\ / 2/pi a\\ | \2/
| csc |-| | | csc |- + --| | csc |-- - -|| |
\ \2/ / | \2 4 / | \2 2/| /a\|
| 1 + -------------- |1 + ------------|*csc|-||
| 2/ a pi\ | 2/a\ | \2/|
| csc |- - + --| | csc |-| | |
\ \ 2 4 / \ \2/ / /
------------------------------------------------------------------------------
/pi a\
10*csc|-- - -|
\2 2/
$$- \frac{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \left(- \frac{1 - \frac{\csc^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{4} \right)}}}{1 + \frac{\csc^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{4} \right)}}} - \frac{4 \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \csc{\left(\frac{a}{2} \right)}}\right) \csc{\left(\frac{a}{2} \right)}}{10 \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}$$
/ // / 3*pi\ \ \ // zoo for a mod pi = 0\
| || 1 for |a + ----| mod 2*pi = 0| // 0 for a mod pi = 0\| || |
| || \ 2 / | || || ||/ zoo for a mod pi = 0 |
-|- |< | - 2*|
$$- \frac{\left(\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sin{\left(a \right)}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)}{5}$$
/ // / 3*pi\ \ \
| || 1 for |a + ----| mod 2*pi = 0| // 0 for a mod pi = 0\| // zoo for a mod pi = 0\
| || \ 2 / | || || || |
| || | || 2 || ||/ 1 \ /a\ |
| || 2/a pi\ | ||-------------------- otherwise || |||1 + -------|*tan|-| |
-|- |<-1 + tan |- + --| | - 2*|
$$- \frac{\left(\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \tan{\left(\frac{a}{2} \right)}}{2} & \text{otherwise} \end{cases}\right)}{5}$$
/ // / 3*pi\ \\ // / 3*pi\ \
| // 0 for a mod pi = 0\ || 1 for |a + ----| mod 2*pi = 0|| || 1 for |a + ----| mod 2*pi = 0|
| || | || \ 2 / || || \ 2 / |
| || /a\ | || || || |
| || 2*cot|-| | || 2/a pi\ || || 2/a pi\ |
-|- |< \2/ | - 2*|<-1 + tan |- + --| ||*|< 1 + tan |- + --| |
| ||----------- otherwise | || \2 4 / || || \2 4 / |
| || 2/a\ | ||----------------- otherwise || ||----------------- otherwise |
| ||1 + cot |-| | || 2/a pi\ || || 2/a pi\ |
| \\ \2/ / || 1 + tan |- + --| || ||-1 + tan |- + --| |
\ \\ \2 4 / // \\ \2 4 / /
----------------------------------------------------------------------------------------------------------------------------------------------
5
$$- \frac{\left(\left(- \begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)\right) \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1} & \text{otherwise} \end{cases}\right)}{5}$$
/ // 0 for a mod pi = 0\\
| // / 3*pi\ \ || || // zoo for a mod pi = 0\
| || 1 for |a + ----| mod 2*pi = 0| || 2*sin(a) || || |
| || \ 2 / | ||---------------------------- otherwise || || 2/a\ |
| || | || / 2 \ || || sin |-| |
-|- |< / 2\ | - 2*|< | sin (a) | ||*|< 1 \2/ |
| ||/1 sin(a)\ | (1 + sin(a)) | | ||(1 - cos(a))*|1 + ---------| || ||-------- + ------- otherwise |
| |||- - ------|*|-1 + -------------| otherwise | || | 4/a\| || || /a\ sin(a) |
| ||\2 2 / | 2 | | || | 4*sin |-|| || ||2*tan|-| |
| \\ \ cos (a) / / || \ \2// || \\ \2/ /
\ \\ //
---------------------------------------------------------------------------------------------------------------------------------------------------------------------
5
$$- \frac{\left(\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \left(- \cos{\left(a \right)} + 1\right)} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\left(- \frac{\sin{\left(a \right)}}{2} + \frac{1}{2}\right) \left(\frac{\left(\sin{\left(a \right)} + 1\right)^{2}}{\cos^{2}{\left(a \right)}} - 1\right) & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\sin^{2}{\left(\frac{a}{2} \right)}}{\sin{\left(a \right)}} + \frac{1}{2 \tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)}{5}$$
/ // / 3*pi\ \ \
| || 1 for |a + ----| mod 2*pi = 0| // 0 for a mod pi = 0\| // zoo for a mod pi = 0\
| || \ 2 / | || || || |
| || | ||/ 0 for a mod pi = 0 || ||/ zoo for a mod pi = 0 |
| ||/ 0 for a mod pi = 0 | ||| || ||| |
| ||| | ||| /a\ || ||| 2/a\ |
-|- |<| /a\ | - 2*|<| 2*cot|-| ||*|<|1 + cot |-| |
| ||| 2*cot|-| | ||< \2/ otherwise || ||< \2/ otherwise |
| ||< \2/ otherwise | |||----------- otherwise || |||----------- otherwise |
| |||----------- otherwise | ||| 2/a\ || ||| /a\ |
| ||| 2/a\ | |||1 + cot |-| || ||| 2*cot|-| |
| |||1 + cot |-| | \\\ \2/ /| \\\ \2/ /
\ \\\ \2/ / /
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
5
$$- \frac{\left(\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} + 1}{2 \cot{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)}{5}$$
/ // / 3*pi\ \ \
| || 1 for |a + ----| mod 2*pi = 0| |
| || \ 2 / | |
| || | // 0 for a mod pi = 0\| // zoo for a mod pi = 0\
| || 4/a pi\ | || || || |
| || 4*sin |- + --| | || sin(a) || || / 2 \ |
| || \2 4 / | ||----------------------- otherwise || || 2/a\ | sin (a) | |
| ||-1 + -------------- | ||/ 2 \ || ||sin |-|*|1 + ---------| |
-|- |< 2 | - 2*|<| sin (a) | 2/a\ ||*|< \2/ | 4/a\| |
| || cos (a) | |||1 + ---------|*sin |-| || || | 4*sin |-|| |
| ||------------------- otherwise | ||| 4/a\| \2/ || || \ \2// |
| || 4/a pi\ | ||| 4*sin |-|| || ||----------------------- otherwise |
| || 4*sin |- + --| | ||\ \2// || || sin(a) |
| || \2 4 / | \\ /| \\ /
| || 1 + -------------- | |
| || 2 | |
\ \\ cos (a) / /
-------------------------------------------------------------------------------------------------------------------------------------------------------
5
$$- \frac{\left(\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\frac{4 \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cos^{2}{\left(a \right)}} - 1}{\frac{4 \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cos^{2}{\left(a \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)}}{\sin{\left(a \right)}} & \text{otherwise} \end{cases}\right)}{5}$$
/ // / 3*pi\ \ \
| || 1 for |a + ----| mod 2*pi = 0| |
| || \ 2 / | |
| || | // 0 for a mod pi = 0\| // zoo for a mod pi = 0\
| || 2/a pi\ | || || || |
| || cos |- - --| | || /a\ || ||/ 2/a\ \ |
| || \2 4 / | || 2*cos|-| || ||| cos |-| | |
| ||-1 + ------------ | || \2/ || ||| \2/ | /a pi\ |
| || 2/a pi\ | ||------------------------------ otherwise || |||1 + ------------|*cos|- - --| |
-|- |< cos |- + --| | - 2*|
$$- \frac{\left(\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}{\cos^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}{\cos^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{2 \cos{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)}{5}$$
/ // / 3*pi\ \ \
| || 1 for |a + ----| mod 2*pi = 0| |
| || \ 2 / | |
| || | // 0 for a mod pi = 0\| // zoo for a mod pi = 0\
| || 2/a pi\ | || || || |
| || sec |- + --| | || /a pi\ || ||/ 2/a pi\\ |
| || \2 4 / | || 2*sec|- - --| || ||| sec |- - --|| |
| ||-1 + ------------ | || \2 2 / || ||| \2 2 /| /a\ |
| || 2/a pi\ | ||------------------------- otherwise || |||1 + ------------|*sec|-| |
-|- |< sec |- - --| | - 2*|
$$- \frac{\left(\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right) \sec{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right) \sec{\left(\frac{a}{2} \right)}}{2 \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)}{5}$$
/ // / 3*pi\ \ \
| || 1 for |a + ----| mod 2*pi = 0| |
| || \ 2 / | |
| || | // 0 for a mod pi = 0\| // zoo for a mod pi = 0\
| || 2/ a pi\ | || || || |
| || csc |- - + --| | || /a\ || ||/ 2/a\ \ |
| || \ 2 4 / | || 2*csc|-| || ||| csc |-| | |
| ||-1 + -------------- | || \2/ || ||| \2/ | /pi a\ |
| || 2/a pi\ | ||------------------------------ otherwise || |||1 + ------------|*csc|-- - -| |
-|- |< csc |- + --| | - 2*|
$$- \frac{\left(\left(- 2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{4} \right)}}{\csc^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} - 1}{\frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{4} \right)}}{\csc^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) \left(\begin{cases} \tilde{\infty} & \text{for}\: a \bmod \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{2 \csc{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)}{5}$$
-(-Piecewise((1, Mod(a + 3*pi/2 = 2*pi, 0)), ((-1 + csc(-a/2 + pi/4)^2/csc(a/2 + pi/4)^2)/(1 + csc(-a/2 + pi/4)^2/csc(a/2 + pi/4)^2), True)) - 2*Piecewise((0, Mod(a = pi, 0)), (2*csc(a/2)/((1 + csc(a/2)^2/csc(pi/2 - a/2)^2)*csc(pi/2 - a/2)), True)))*Piecewise((±oo, Mod(a = pi, 0)), ((1 + csc(a/2)^2/csc(pi/2 - a/2)^2)*csc(pi/2 - a/2)/(2*csc(a/2)), True))/5
Численный ответ
[src]
0.2*(-cos(-pi/2 + a) + 2.0*sin(a - 3*pi))/sin(a - pi)
0.2*(-cos(-pi/2 + a) + 2.0*sin(a - 3*pi))/sin(a - pi)