Господин Экзамен
(2*sin(a)+3*cos(a))/(3*sin(a)-5*cos(a)) если a=2
Решение
Вы ввели
[src]
2*sin(a) + 3*cos(a) ------------------- 3*sin(a) - 5*cos(a)
$$\frac{2 \sin{\left(a \right)} + 3 \cos{\left(a \right)}}{3 \sin{\left(a \right)} - 5 \cos{\left(a \right)}}$$
(2*sin(a) + 3*cos(a))/(3*sin(a) - 5*cos(a))
Подстановка условия
[src]
(2*sin(a) + 3*cos(a))/(3*sin(a) - 5*cos(a)) при a = 2
подставляем
2*sin(a) + 3*cos(a) ------------------- 3*sin(a) - 5*cos(a)
$$\frac{2 \sin{\left(a \right)} + 3 \cos{\left(a \right)}}{3 \sin{\left(a \right)} - 5 \cos{\left(a \right)}}$$
2*sin(a) + 3*cos(a) -------------------- -5*cos(a) + 3*sin(a)
$$\frac{2 \sin{\left(a \right)} + 3 \cos{\left(a \right)}}{3 \sin{\left(a \right)} - 5 \cos{\left(a \right)}}$$
переменные
a = 2
$$a = 2$$
2*sin((2)) + 3*cos((2)) ------------------------ -5*cos((2)) + 3*sin((2))
$$\frac{2 \sin{\left((2) \right)} + 3 \cos{\left((2) \right)}}{3 \sin{\left((2) \right)} - 5 \cos{\left((2) \right)}}$$
2*sin(2) + 3*cos(2) -------------------- -5*cos(2) + 3*sin(2)
$$\frac{3 \cos{\left(2 \right)} + 2 \sin{\left(2 \right)}}{- 5 \cos{\left(2 \right)} + 3 \sin{\left(2 \right)}}$$
(2*sin(2) + 3*cos(2))/(-5*cos(2) + 3*sin(2))
Общий знаменатель
[src]
3 19*sin(a) - - - ---------------------- 5 -15*sin(a) + 25*cos(a)
$$- \frac{3}{5} - \frac{19 \sin{\left(a \right)}}{- 15 \sin{\left(a \right)} + 25 \cos{\left(a \right)}}$$
-3/5 - 19*sin(a)/(-15*sin(a) + 25*cos(a))
Комбинаторика
[src]
-(2*sin(a) + 3*cos(a)) ----------------------- -3*sin(a) + 5*cos(a)
$$- \frac{2 \sin{\left(a \right)} + 3 \cos{\left(a \right)}}{- 3 \sin{\left(a \right)} + 5 \cos{\left(a \right)}}$$
-(2*sin(a) + 3*cos(a))/(-3*sin(a) + 5*cos(a))
Численный ответ
[src]
(2.0*sin(a) + 3.0*cos(a))/(3.0*sin(a) - 5.0*cos(a))
(2.0*sin(a) + 3.0*cos(a))/(3.0*sin(a) - 5.0*cos(a))
Собрать выражение
[src]
2*sin(a) 3*cos(a) -------------------- + -------------------- -5*cos(a) + 3*sin(a) -5*cos(a) + 3*sin(a)
$$\frac{2 \sin{\left(a \right)}}{3 \sin{\left(a \right)} - 5 \cos{\left(a \right)}} + \frac{3 \cos{\left(a \right)}}{3 \sin{\left(a \right)} - 5 \cos{\left(a \right)}}$$
2*sin(a)/(-5*cos(a) + 3*sin(a)) + 3*cos(a)/(-5*cos(a) + 3*sin(a))
Степени
[src]
I*a -I*a
3*e 3*e / -I*a I*a\
------ + ------- - I*\- e + e /
2 2
-----------------------------------------
I*a -I*a / -I*a I*a\
5*e 5*e 3*I*\- e + e /
- ------ - ------- - --------------------
2 2 2
$$\frac{- i \left(e^{i a} - e^{- i a}\right) + \frac{3 e^{i a}}{2} + \frac{3 e^{- i a}}{2}}{- \frac{3 i \left(e^{i a} - e^{- i a}\right)}{2} - \frac{5 e^{i a}}{2} - \frac{5 e^{- i a}}{2}}$$
(3*exp(i*a)/2 + 3*exp(-i*a)/2 - i*(-exp(-i*a) + exp(i*a)))/(-5*exp(i*a)/2 - 5*exp(-i*a)/2 - 3*i*(-exp(-i*a) + exp(i*a))/2)
Рациональный знаменатель
[src]
2*sin(a) 3*cos(a) -------------------- + -------------------- -5*cos(a) + 3*sin(a) -5*cos(a) + 3*sin(a)
$$\frac{2 \sin{\left(a \right)}}{3 \sin{\left(a \right)} - 5 \cos{\left(a \right)}} + \frac{3 \cos{\left(a \right)}}{3 \sin{\left(a \right)} - 5 \cos{\left(a \right)}}$$
2*sin(a)/(-5*cos(a) + 3*sin(a)) + 3*cos(a)/(-5*cos(a) + 3*sin(a))
Тригонометрическая часть
[src]
-(2*sin(a) + 3*cos(a)) ----------------------- -3*sin(a) + 5*cos(a)
$$- \frac{2 \sin{\left(a \right)} + 3 \cos{\left(a \right)}}{- 3 \sin{\left(a \right)} + 5 \cos{\left(a \right)}}$$
/ pi\
2*sin(a) + 3*sin|a + --|
\ 2 /
--------------------------
/ pi\
- 5*sin|a + --| + 3*sin(a)
\ 2 /
$$\frac{2 \sin{\left(a \right)} + 3 \sin{\left(a + \frac{\pi}{2} \right)}}{3 \sin{\left(a \right)} - 5 \sin{\left(a + \frac{\pi}{2} \right)}}$$
/ pi\
2*cos|a - --| + 3*cos(a)
\ 2 /
-------------------------
/ pi\
-5*cos(a) + 3*cos|a - --|
\ 2 /
$$\frac{3 \cos{\left(a \right)} + 2 \cos{\left(a - \frac{\pi}{2} \right)}}{- 5 \cos{\left(a \right)} + 3 \cos{\left(a - \frac{\pi}{2} \right)}}$$
2 3
------ + ------
csc(a) sec(a)
-----------------
5 3
- ------ + ------
sec(a) csc(a)
$$\frac{\frac{3}{\sec{\left(a \right)}} + \frac{2}{\csc{\left(a \right)}}}{- \frac{5}{\sec{\left(a \right)}} + \frac{3}{\csc{\left(a \right)}}}$$
2 3
----------- + ------
/ pi\ sec(a)
sec|a - --|
\ 2 /
----------------------
5 3
- ------ + -----------
sec(a) / pi\
sec|a - --|
\ 2 /
$$\frac{\frac{2}{\sec{\left(a - \frac{\pi}{2} \right)}} + \frac{3}{\sec{\left(a \right)}}}{\frac{3}{\sec{\left(a - \frac{\pi}{2} \right)}} - \frac{5}{\sec{\left(a \right)}}}$$
2 3
----------- + ------
/pi \ sec(a)
sec|-- - a|
\2 /
----------------------
5 3
- ------ + -----------
sec(a) /pi \
sec|-- - a|
\2 /
$$\frac{\frac{2}{\sec{\left(- a + \frac{\pi}{2} \right)}} + \frac{3}{\sec{\left(a \right)}}}{\frac{3}{\sec{\left(- a + \frac{\pi}{2} \right)}} - \frac{5}{\sec{\left(a \right)}}}$$
2 3
------ + -----------
csc(a) /pi \
csc|-- - a|
\2 /
----------------------
5 3
- ----------- + ------
/pi \ csc(a)
csc|-- - a|
\2 /
$$\frac{\frac{3}{\csc{\left(- a + \frac{\pi}{2} \right)}} + \frac{2}{\csc{\left(a \right)}}}{- \frac{5}{\csc{\left(- a + \frac{\pi}{2} \right)}} + \frac{3}{\csc{\left(a \right)}}}$$
2 3
----------- + -----------
csc(pi - a) /pi \
csc|-- - a|
\2 /
---------------------------
5 3
- ----------- + -----------
/pi \ csc(pi - a)
csc|-- - a|
\2 /
$$\frac{\frac{3}{\csc{\left(- a + \frac{\pi}{2} \right)}} + \frac{2}{\csc{\left(- a + \pi \right)}}}{- \frac{5}{\csc{\left(- a + \frac{\pi}{2} \right)}} + \frac{3}{\csc{\left(- a + \pi \right)}}}$$
/a\
3*cos(a) + 2*(1 + cos(a))*tan|-|
\2/
---------------------------------
/a\
-5*cos(a) + 3*(1 + cos(a))*tan|-|
\2/
$$\frac{2 \left(\cos{\left(a \right)} + 1\right) \tan{\left(\frac{a}{2} \right)} + 3 \cos{\left(a \right)}}{3 \left(\cos{\left(a \right)} + 1\right) \tan{\left(\frac{a}{2} \right)} - 5 \cos{\left(a \right)}}$$
/ 2/a pi\\
3*cos(a) + |1 - cot |- + --||*(1 + sin(a))
\ \2 4 //
---------------------------------------------
/ 2/a pi\\
3*|1 - cot |- + --||*(1 + sin(a))
\ \2 4 //
-5*cos(a) + ---------------------------------
2
$$\frac{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(a \right)} + 1\right) + 3 \cos{\left(a \right)}}{\frac{3 \cdot \left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(a \right)} + 1\right)}{2} - 5 \cos{\left(a \right)}}$$
/ 2/a\\ /a\
3*|1 - tan |-|| 4*tan|-|
\ \2// \2/
--------------- + -----------
2/a\ 2/a\
1 + tan |-| 1 + tan |-|
\2/ \2/
-------------------------------
/ 2/a\\ /a\
5*|1 - tan |-|| 6*tan|-|
\ \2// \2/
- --------------- + -----------
2/a\ 2/a\
1 + tan |-| 1 + tan |-|
\2/ \2/
$$\frac{\frac{3 \cdot \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + \frac{4 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}}{- \frac{5 \cdot \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + \frac{6 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}}$$
/a\ /a pi\
4*cot|-| 6*tan|- + --|
\2/ \2 4 /
----------- + ----------------
2/a\ 2/a pi\
1 + cot |-| 1 + tan |- + --|
\2/ \2 4 /
--------------------------------
/a pi\ /a\
10*tan|- + --| 6*cot|-|
\2 4 / \2/
- ---------------- + -----------
2/a pi\ 2/a\
1 + tan |- + --| 1 + cot |-|
\2 4 / \2/
$$\frac{\frac{4 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} + \frac{6 \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}}{\frac{6 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} - \frac{10 \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}}$$
/a\ /a pi\
4*tan|-| 6*tan|- + --|
\2/ \2 4 /
----------- + ----------------
2/a\ 2/a pi\
1 + tan |-| 1 + tan |- + --|
\2/ \2 4 /
--------------------------------
/a pi\ /a\
10*tan|- + --| 6*tan|-|
\2 4 / \2/
- ---------------- + -----------
2/a pi\ 2/a\
1 + tan |- + --| 1 + tan |-|
\2 4 / \2/
$$\frac{\frac{6 \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} + \frac{4 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}}{- \frac{10 \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} + \frac{6 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}}$$
/ 1 \
3*|1 - -------|
| 2/a\|
| cot |-||
\ \2// 4
--------------- + --------------------
1 / 1 \ /a\
1 + ------- |1 + -------|*cot|-|
2/a\ | 2/a\| \2/
cot |-| | cot |-||
\2/ \ \2//
----------------------------------------
/ 1 \
5*|1 - -------|
| 2/a\|
| cot |-||
\ \2// 6
- --------------- + --------------------
1 / 1 \ /a\
1 + ------- |1 + -------|*cot|-|
2/a\ | 2/a\| \2/
cot |-| | cot |-||
\2/ \ \2//
$$\frac{\frac{3 \cdot \left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)}{1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}} + \frac{4}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \cot{\left(\frac{a}{2} \right)}}}{- \frac{5 \cdot \left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)}{1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}} + \frac{6}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \cot{\left(\frac{a}{2} \right)}}}$$
/ 2/a pi\\ / 2/a\\
2*|-1 + tan |- + --|| 3*|-1 + cot |-||
\ \2 4 // \ \2//
--------------------- + ----------------
2/a pi\ 2/a\
1 + tan |- + --| 1 + cot |-|
\2 4 / \2/
------------------------------------------
/ 2/a\\ / 2/a pi\\
5*|-1 + cot |-|| 3*|-1 + tan |- + --||
\ \2// \ \2 4 //
- ---------------- + ---------------------
2/a\ 2/a pi\
1 + cot |-| 1 + tan |- + --|
\2/ \2 4 /
$$\frac{\frac{2 \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} + \frac{3 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}}{\frac{3 \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} - \frac{5 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}}$$
/ 2/a pi\\ / 2/a\\
2*|1 - cot |- + --|| 3*|1 - tan |-||
\ \2 4 // \ \2//
-------------------- + ---------------
2/a pi\ 2/a\
1 + cot |- + --| 1 + tan |-|
\2 4 / \2/
----------------------------------------
/ 2/a\\ / 2/a pi\\
5*|1 - tan |-|| 3*|1 - cot |- + --||
\ \2// \ \2 4 //
- --------------- + --------------------
2/a\ 2/a pi\
1 + tan |-| 1 + cot |- + --|
\2/ \2 4 /
$$\frac{\frac{3 \cdot \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + \frac{2 \cdot \left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}}{- \frac{5 \cdot \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + \frac{3 \cdot \left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}}$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\
2*|< | + 3*|< |
\\sin(a) otherwise / \\cos(a) otherwise /
-----------------------------------------------------------------
// 1 for a mod 2*pi = 0\ // 0 for a mod pi = 0\
- 5*|< | + 3*|< |
\\cos(a) otherwise / \\sin(a) otherwise /
$$\frac{\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)\right)}{\left(3 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)\right)}$$
// 0 for a mod pi = 0\
|| | // 1 for a mod 2*pi = 0\
2*|< / pi\ | + 3*|< |
||cos|a - --| otherwise | \\cos(a) otherwise /
\\ \ 2 / /
----------------------------------------------------------------------
// 0 for a mod pi = 0\
// 1 for a mod 2*pi = 0\ || |
- 5*|< | + 3*|< / pi\ |
\\cos(a) otherwise / ||cos|a - --| otherwise |
\\ \ 2 / /
$$\frac{\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)\right)}{\left(3 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)\right)}$$
// 1 for a mod 2*pi = 0\
// 0 for a mod pi = 0\ || |
2*|< | + 3*|< / pi\ |
\\sin(a) otherwise / ||sin|a + --| otherwise |
\\ \ 2 / /
----------------------------------------------------------------------
// 1 for a mod 2*pi = 0\
|| | // 0 for a mod pi = 0\
- 5*|< / pi\ | + 3*|< |
||sin|a + --| otherwise | \\sin(a) otherwise /
\\ \ 2 / /
$$\frac{\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\sin{\left(a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right)}{\left(3 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\sin{\left(a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right)}$$
2/a\
8*sin |-|*sin(a)
6*(-1 - cos(2*a) + 2*cos(a)) \2/
------------------------------ + -------------------
2 2 4/a\
1 - cos(2*a) + 2*(1 - cos(a)) sin (a) + 4*sin |-|
\2/
------------------------------------------------------
2/a\
12*sin |-|*sin(a)
10*(-1 - cos(2*a) + 2*cos(a)) \2/
- ------------------------------ + -------------------
2 2 4/a\
1 - cos(2*a) + 2*(1 - cos(a)) sin (a) + 4*sin |-|
\2/
$$\frac{\frac{8 \sin^{2}{\left(\frac{a}{2} \right)} \sin{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}} + \frac{6 \cdot \left(2 \cos{\left(a \right)} - \cos{\left(2 a \right)} - 1\right)}{2 \left(- \cos{\left(a \right)} + 1\right)^{2} - \cos{\left(2 a \right)} + 1}}{\frac{12 \sin^{2}{\left(\frac{a}{2} \right)} \sin{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}} - \frac{10 \cdot \left(2 \cos{\left(a \right)} - \cos{\left(2 a \right)} - 1\right)}{2 \left(- \cos{\left(a \right)} + 1\right)^{2} - \cos{\left(2 a \right)} + 1}}$$
// / 3*pi\ \
|| 1 for |a + ----| mod 2*pi = 0| // 1 for a mod 2*pi = 0\
2*|< \ 2 / | + 3*|< |
|| | \\cos(a) otherwise /
\\sin(a) otherwise /
----------------------------------------------------------------------------
// / 3*pi\ \
// 1 for a mod 2*pi = 0\ || 1 for |a + ----| mod 2*pi = 0|
- 5*|< | + 3*|< \ 2 / |
\\cos(a) otherwise / || |
\\sin(a) otherwise /
$$\frac{\left(3 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right)}{\left(- 5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right)}$$
// 0 for a mod pi = 0\
|| | // 1 for a mod 2*pi = 0\
|| 1 | || |
2*|<----------- otherwise | + 3*|< 1 |
|| / pi\ | ||------ otherwise |
||sec|a - --| | \\sec(a) /
\\ \ 2 / /
----------------------------------------------------------------------
// 0 for a mod pi = 0\
// 1 for a mod 2*pi = 0\ || |
|| | || 1 |
- 5*|< 1 | + 3*|<----------- otherwise |
||------ otherwise | || / pi\ |
\\sec(a) / ||sec|a - --| |
\\ \ 2 / /
$$\frac{\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right)}{\left(3 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right)}$$
// 1 for a mod 2*pi = 0\
// 0 for a mod pi = 0\ || |
|| | || 1 |
2*|< 1 | + 3*|<----------- otherwise |
||------ otherwise | || /pi \ |
\\csc(a) / ||csc|-- - a| |
\\ \2 / /
----------------------------------------------------------------------
// 1 for a mod 2*pi = 0\
|| | // 0 for a mod pi = 0\
|| 1 | || |
- 5*|<----------- otherwise | + 3*|< 1 |
|| /pi \ | ||------ otherwise |
||csc|-- - a| | \\csc(a) /
\\ \2 / /
$$\frac{\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right)}{\left(3 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right)}$$
// 0 for a mod pi = 0\
|| |
||1 - cos(a) | // 1 for a mod 2*pi = 0\
2*|<---------- otherwise | + 3*|< |
|| /a\ | \\cos(a) otherwise /
|| tan|-| |
\\ \2/ /
---------------------------------------------------------------------
// 0 for a mod pi = 0\
|| |
// 1 for a mod 2*pi = 0\ ||1 - cos(a) |
- 5*|< | + 3*|<---------- otherwise |
\\cos(a) otherwise / || /a\ |
|| tan|-| |
\\ \2/ /
$$\frac{\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{- \cos{\left(a \right)} + 1}{\tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)\right)}{\left(3 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{- \cos{\left(a \right)} + 1}{\tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)\right)}$$
/ 4/a\\
| 4*sin |-||
| \2/|
3*|1 - ---------| 2/a\
| 2 | 8*sin |-|
\ sin (a) / \2/
----------------- + ----------------------
4/a\ / 4/a\\
4*sin |-| | 4*sin |-||
\2/ | \2/|
1 + --------- |1 + ---------|*sin(a)
2 | 2 |
sin (a) \ sin (a) /
--------------------------------------------
/ 4/a\\
| 4*sin |-||
| \2/|
5*|1 - ---------| 2/a\
| 2 | 12*sin |-|
\ sin (a) / \2/
- ----------------- + ----------------------
4/a\ / 4/a\\
4*sin |-| | 4*sin |-||
\2/ | \2/|
1 + --------- |1 + ---------|*sin(a)
2 | 2 |
sin (a) \ sin (a) /
$$\frac{\frac{3 \left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)}{\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1} + \frac{8 \sin^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \sin{\left(a \right)}}}{- \frac{5 \left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)}{\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1} + \frac{12 \sin^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \sin{\left(a \right)}}}$$
// / pi\ \
|| 0 for |a + --| mod pi = 0|
// 0 for a mod pi = 0\ || \ 2 / |
2*|< | + 3*|< |
\\sin(a) otherwise / || /a pi\ |
||(1 + sin(a))*cot|- + --| otherwise |
\\ \2 4 / /
----------------------------------------------------------------------------------------
// / pi\ \
|| 0 for |a + --| mod pi = 0|
|| \ 2 / | // 0 for a mod pi = 0\
- 5*|< | + 3*|< |
|| /a pi\ | \\sin(a) otherwise /
||(1 + sin(a))*cot|- + --| otherwise |
\\ \2 4 / /
$$\frac{\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(a \right)} + 1\right) \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right)}{\left(3 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(a \right)} + 1\right) \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right)}$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
|| /a\ | || 2/a\ |
|| 2*cot|-| | ||-1 + cot |-| |
2*|< \2/ | + 3*|< \2/ |
||----------- otherwise | ||------------ otherwise |
|| 2/a\ | || 2/a\ |
||1 + cot |-| | ||1 + cot |-| |
\\ \2/ / \\ \2/ /
----------------------------------------------------------------------------
// 1 for a mod 2*pi = 0\ // 0 for a mod pi = 0\
|| | || |
|| 2/a\ | || /a\ |
||-1 + cot |-| | || 2*cot|-| |
- 5*|< \2/ | + 3*|< \2/ |
||------------ otherwise | ||----------- otherwise |
|| 2/a\ | || 2/a\ |
||1 + cot |-| | ||1 + cot |-| |
\\ \2/ / \\ \2/ /
$$\frac{\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)}{\left(3 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)}$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
|| /a\ | || 2/a\ |
|| 2*tan|-| | ||1 - tan |-| |
2*|< \2/ | + 3*|< \2/ |
||----------- otherwise | ||----------- otherwise |
|| 2/a\ | || 2/a\ |
||1 + tan |-| | ||1 + tan |-| |
\\ \2/ / \\ \2/ /
---------------------------------------------------------------------------
// 1 for a mod 2*pi = 0\ // 0 for a mod pi = 0\
|| | || |
|| 2/a\ | || /a\ |
||1 - tan |-| | || 2*tan|-| |
- 5*|< \2/ | + 3*|< \2/ |
||----------- otherwise | ||----------- otherwise |
|| 2/a\ | || 2/a\ |
||1 + tan |-| | ||1 + tan |-| |
\\ \2/ / \\ \2/ /
$$\frac{\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{a}{2} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)}{\left(3 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{a}{2} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)}$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
2*|
$$\frac{\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)}{\left(3 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)}$$
/ 2/a\ \
| sec |-| |
| \2/ |
3*|1 - ------------|
| 2/a pi\| /a\
| sec |- - --|| 4*sec|-|
\ \2 2 // \2/
-------------------- + ------------------------------
2/a\ / 2/a\ \
sec |-| | sec |-| |
\2/ | \2/ | /a pi\
1 + ------------ |1 + ------------|*sec|- - --|
2/a pi\ | 2/a pi\| \2 2 /
sec |- - --| | sec |- - --||
\2 2 / \ \2 2 //
-------------------------------------------------------
/ 2/a\ \
| sec |-| |
| \2/ |
5*|1 - ------------|
| 2/a pi\| /a\
| sec |- - --|| 6*sec|-|
\ \2 2 // \2/
- -------------------- + ------------------------------
2/a\ / 2/a\ \
sec |-| | sec |-| |
\2/ | \2/ | /a pi\
1 + ------------ |1 + ------------|*sec|- - --|
2/a pi\ | 2/a pi\| \2 2 /
sec |- - --| | sec |- - --||
\2 2 / \ \2 2 //
$$\frac{\frac{3 \left(- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)}{\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1} + \frac{4 \sec{\left(\frac{a}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}}{- \frac{5 \left(- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)}{\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1} + \frac{6 \sec{\left(\frac{a}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}}$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 1 |
|| | ||-1 + ------- |
|| 2 | || 2/a\ |
||-------------------- otherwise | || tan |-| |
2*|
$$\frac{\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)}{\left(3 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)}$$
/ 2/a pi\\
| cos |- - --||
| \2 2 /|
3*|1 - ------------|
| 2/a\ | /a pi\
| cos |-| | 4*cos|- - --|
\ \2/ / \2 2 /
-------------------- + -------------------------
2/a pi\ / 2/a pi\\
cos |- - --| | cos |- - --||
\2 2 / | \2 2 /| /a\
1 + ------------ |1 + ------------|*cos|-|
2/a\ | 2/a\ | \2/
cos |-| | cos |-| |
\2/ \ \2/ /
--------------------------------------------------
/ 2/a pi\\
| cos |- - --||
| \2 2 /|
5*|1 - ------------|
| 2/a\ | /a pi\
| cos |-| | 6*cos|- - --|
\ \2/ / \2 2 /
- -------------------- + -------------------------
2/a pi\ / 2/a pi\\
cos |- - --| | cos |- - --||
\2 2 / | \2 2 /| /a\
1 + ------------ |1 + ------------|*cos|-|
2/a\ | 2/a\ | \2/
cos |-| | cos |-| |
\2/ \ \2/ /
$$\frac{\frac{3 \cdot \left(1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)}{1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}} + \frac{4 \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \cos{\left(\frac{a}{2} \right)}}}{- \frac{5 \cdot \left(1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)}{1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}} + \frac{6 \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \cos{\left(\frac{a}{2} \right)}}}$$
/ 2/pi a\\
| csc |-- - -||
| \2 2/|
3*|1 - ------------|
| 2/a\ | /pi a\
| csc |-| | 4*csc|-- - -|
\ \2/ / \2 2/
-------------------- + -------------------------
2/pi a\ / 2/pi a\\
csc |-- - -| | csc |-- - -||
\2 2/ | \2 2/| /a\
1 + ------------ |1 + ------------|*csc|-|
2/a\ | 2/a\ | \2/
csc |-| | csc |-| |
\2/ \ \2/ /
--------------------------------------------------
/ 2/pi a\\
| csc |-- - -||
| \2 2/|
5*|1 - ------------|
| 2/a\ | /pi a\
| csc |-| | 6*csc|-- - -|
\ \2/ / \2 2/
- -------------------- + -------------------------
2/pi a\ / 2/pi a\\
csc |-- - -| | csc |-- - -||
\2 2/ | \2 2/| /a\
1 + ------------ |1 + ------------|*csc|-|
2/a\ | 2/a\ | \2/
csc |-| | csc |-| |
\2/ \ \2/ /
$$\frac{\frac{3 \cdot \left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)}{1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}} + \frac{4 \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \csc{\left(\frac{a}{2} \right)}}}{- \frac{5 \cdot \left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)}{1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}} + \frac{6 \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \csc{\left(\frac{a}{2} \right)}}}$$
// / pi\ \
// 0 for a mod pi = 0\ || 0 for |a + --| mod pi = 0|
|| | || \ 2 / |
|| /a\ | || |
|| 2*cot|-| | || /a pi\ |
2*|< \2/ | + 3*|< 2*cot|- + --| |
||----------- otherwise | || \2 4 / |
|| 2/a\ | ||---------------- otherwise |
||1 + cot |-| | || 2/a pi\ |
\\ \2/ / ||1 + cot |- + --| |
\\ \2 4 / /
-------------------------------------------------------------------------------------
// / pi\ \
|| 0 for |a + --| mod pi = 0| // 0 for a mod pi = 0\
|| \ 2 / | || |
|| | || /a\ |
|| /a pi\ | || 2*cot|-| |
- 5*|< 2*cot|- + --| | + 3*|< \2/ |
|| \2 4 / | ||----------- otherwise |
||---------------- otherwise | || 2/a\ |
|| 2/a pi\ | ||1 + cot |-| |
||1 + cot |- + --| | \\ \2/ /
\\ \2 4 / /
$$\frac{\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)}{\left(3 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)}$$
// / 3*pi\ \
|| 1 for |a + ----| mod 2*pi = 0| // 1 for a mod 2*pi = 0\
|| \ 2 / | || |
|| | || 2/a\ |
|| 2/a pi\ | ||-1 + cot |-| |
2*|<-1 + tan |- + --| | + 3*|< \2/ |
|| \2 4 / | ||------------ otherwise |
||----------------- otherwise | || 2/a\ |
|| 2/a pi\ | ||1 + cot |-| |
|| 1 + tan |- + --| | \\ \2/ /
\\ \2 4 / /
---------------------------------------------------------------------------------------------
// / 3*pi\ \
// 1 for a mod 2*pi = 0\ || 1 for |a + ----| mod 2*pi = 0|
|| | || \ 2 / |
|| 2/a\ | || |
||-1 + cot |-| | || 2/a pi\ |
- 5*|< \2/ | + 3*|<-1 + tan |- + --| |
||------------ otherwise | || \2 4 / |
|| 2/a\ | ||----------------- otherwise |
||1 + cot |-| | || 2/a pi\ |
\\ \2/ / || 1 + tan |- + --| |
\\ \2 4 / /
$$\frac{\left(3 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)}{\left(- 5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)}$$
// 0 for a mod pi = 0\
|| |
|| 2*sin(a) | // 1 for a mod 2*pi = 0\
||---------------------------- otherwise | || |
|| / 2 \ | || 2 |
2*|< | sin (a) | | + 3*|< -4 + 4*sin (a) + 4*cos(a) |
||(1 - cos(a))*|1 + ---------| | ||--------------------------- otherwise |
|| | 4/a\| | || 2 2 |
|| | 4*sin |-|| | \\2*(1 - cos(a)) + 2*sin (a) /
|| \ \2// |
\\ /
------------------------------------------------------------------------------------------------------------
// 0 for a mod pi = 0\
|| |
// 1 for a mod 2*pi = 0\ || 2*sin(a) |
|| | ||---------------------------- otherwise |
|| 2 | || / 2 \ |
- 5*|< -4 + 4*sin (a) + 4*cos(a) | + 3*|< | sin (a) | |
||--------------------------- otherwise | ||(1 - cos(a))*|1 + ---------| |
|| 2 2 | || | 4/a\| |
\\2*(1 - cos(a)) + 2*sin (a) / || | 4*sin |-|| |
|| \ \2// |
\\ /
$$\frac{\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \left(- \cos{\left(a \right)} + 1\right)} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{4 \sin^{2}{\left(a \right)} + 4 \cos{\left(a \right)} - 4}{2 \left(- \cos{\left(a \right)} + 1\right)^{2} + 2 \sin^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right)}{\left(3 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \left(- \cos{\left(a \right)} + 1\right)} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{4 \sin^{2}{\left(a \right)} + 4 \cos{\left(a \right)} - 4}{2 \left(- \cos{\left(a \right)} + 1\right)^{2} + 2 \sin^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right)}$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 2 |
|| | || sin (a) |
|| sin(a) | ||-1 + --------- |
||----------------------- otherwise | || 4/a\ |
||/ 2 \ | || 4*sin |-| |
2*|<| sin (a) | 2/a\ | + 3*|< \2/ |
|||1 + ---------|*sin |-| | ||-------------- otherwise |
||| 4/a\| \2/ | || 2 |
||| 4*sin |-|| | || sin (a) |
||\ \2// | ||1 + --------- |
\\ / || 4/a\ |
|| 4*sin |-| |
\\ \2/ /
------------------------------------------------------------------------------------------
// 1 for a mod 2*pi = 0\
|| |
|| 2 | // 0 for a mod pi = 0\
|| sin (a) | || |
||-1 + --------- | || sin(a) |
|| 4/a\ | ||----------------------- otherwise |
|| 4*sin |-| | ||/ 2 \ |
- 5*|< \2/ | + 3*|<| sin (a) | 2/a\ |
||-------------- otherwise | |||1 + ---------|*sin |-| |
|| 2 | ||| 4/a\| \2/ |
|| sin (a) | ||| 4*sin |-|| |
||1 + --------- | ||\ \2// |
|| 4/a\ | \\ /
|| 4*sin |-| |
\\ \2/ /
$$\frac{\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)}{\left(3 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)}$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
||/ 0 for a mod pi = 0 | ||/ 1 for a mod 2*pi = 0 |
||| | ||| |
||| /a\ | ||| 2/a\ |
2*|<| 2*cot|-| | + 3*|<|-1 + cot |-| |
||< \2/ otherwise | ||< \2/ otherwise |
|||----------- otherwise | |||------------ otherwise |
||| 2/a\ | ||| 2/a\ |
|||1 + cot |-| | |||1 + cot |-| |
\\\ \2/ / \\\ \2/ /
--------------------------------------------------------------------------------------------------------------------
// 1 for a mod 2*pi = 0\ // 0 for a mod pi = 0\
|| | || |
||/ 1 for a mod 2*pi = 0 | ||/ 0 for a mod pi = 0 |
||| | ||| |
||| 2/a\ | ||| /a\ |
- 5*|<|-1 + cot |-| | + 3*|<| 2*cot|-| |
||< \2/ otherwise | ||< \2/ otherwise |
|||------------ otherwise | |||----------- otherwise |
||| 2/a\ | ||| 2/a\ |
|||1 + cot |-| | |||1 + cot |-| |
\\\ \2/ / \\\ \2/ /
$$\frac{\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)}{\left(3 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)}$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 2/a\ |
|| | || cos |-| |
|| /a\ | || \2/ |
|| 2*cos|-| | ||-1 + ------------ |
|| \2/ | || 2/a pi\ |
||------------------------------ otherwise | || cos |- - --| |
2*|
$$\frac{\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right)}{\left(3 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right)}$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 2/a pi\ |
|| | || sec |- - --| |
|| /a pi\ | || \2 2 / |
|| 2*sec|- - --| | ||-1 + ------------ |
|| \2 2 / | || 2/a\ |
||------------------------- otherwise | || sec |-| |
2*|
$$\frac{\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right) \sec{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)}{\left(3 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right) \sec{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)}$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 2/a\ |
|| | || csc |-| |
|| /a\ | || \2/ |
|| 2*csc|-| | ||-1 + ------------ |
|| \2/ | || 2/pi a\ |
||------------------------------ otherwise | || csc |-- - -| |
2*|
$$\frac{\left(2 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right)}{\left(3 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right)}$$
(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)) + 3*Piecewise((1, Mod(a = 2*pi, 0)), ((-1 + csc(a/2)^2/csc(pi/2 - a/2)^2)/(1 + csc(a/2)^2/csc(pi/2 - a/2)^2), True)))/(-5*Piecewise((1, Mod(a = 2*pi, 0)), ((-1 + csc(a/2)^2/csc(pi/2 - a/2)^2)/(1 + csc(a/2)^2/csc(pi/2 - a/2)^2), True)) + 3*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)))