Тригонометрическая часть
[src]
$$2 \csc{\left(t \right)}$$
$$\frac{2}{\sin{\left(t \right)}}$$
/ pi\
2*sec|t - --|
\ 2 /
$$2 \sec{\left(t - \frac{\pi}{2} \right)}$$
2
-----------
/ pi\
cos|t - --|
\ 2 /
$$\frac{2}{\cos{\left(t - \frac{\pi}{2} \right)}}$$
/t\
-2*tan|-|
\2/
-----------
-1 + cos(t)
$$- \frac{2 \tan{\left(\frac{t}{2} \right)}}{\cos{\left(t \right)} - 1}$$
2/t\
1 + tan |-|
\2/
-----------
/t\
tan|-|
\2/
$$\frac{\tan^{2}{\left(\frac{t}{2} \right)} + 1}{\tan{\left(\frac{t}{2} \right)}}$$
// zoo for t mod pi = 0\
|| |
2*|< 1 |
||------ otherwise |
\\sin(t) /
$$2 \left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\sin{\left(t \right)}} & \text{otherwise} \end{cases}\right)$$
/ pi\ / pi\
cos|t - --| cos|t - --|
\ 2 / \ 2 /
----------- + -----------
1 - cos(t) 1 + cos(t)
$$\frac{\cos{\left(t - \frac{\pi}{2} \right)}}{\cos{\left(t \right)} + 1} + \frac{\cos{\left(t - \frac{\pi}{2} \right)}}{- \cos{\left(t \right)} + 1}$$
sin(t) sin(t)
--------------- + ---------------
/ pi\ / pi\
1 - sin|t + --| 1 + sin|t + --|
\ 2 / \ 2 /
$$\frac{\sin{\left(t \right)}}{\sin{\left(t + \frac{\pi}{2} \right)} + 1} + \frac{\sin{\left(t \right)}}{- \sin{\left(t + \frac{\pi}{2} \right)} + 1}$$
1 1
------------------- + -------------------
/ 1 \ / 1 \
|1 + ------|*csc(t) |1 - ------|*csc(t)
\ sec(t)/ \ sec(t)/
$$\frac{1}{\left(1 + \frac{1}{\sec{\left(t \right)}}\right) \csc{\left(t \right)}} + \frac{1}{\left(1 - \frac{1}{\sec{\left(t \right)}}\right) \csc{\left(t \right)}}$$
// zoo for t mod pi = 0\
|| |
|| 2/t\ |
||1 + cot |-| |
2*|< \2/ |
||----------- otherwise |
|| /t\ |
|| 2*cot|-| |
\\ \2/ /
$$2 \left(\begin{cases} \tilde{\infty} & \text{for}\: t \bmod \pi = 0 \\\frac{\cot^{2}{\left(\frac{t}{2} \right)} + 1}{2 \cot{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
1 1
------------------------ + ------------------------
/ 1 \ / pi\ / 1 \ / pi\
|1 + ------|*sec|t - --| |1 - ------|*sec|t - --|
\ sec(t)/ \ 2 / \ sec(t)/ \ 2 /
$$\frac{1}{\left(1 + \frac{1}{\sec{\left(t \right)}}\right) \sec{\left(t - \frac{\pi}{2} \right)}} + \frac{1}{\left(1 - \frac{1}{\sec{\left(t \right)}}\right) \sec{\left(t - \frac{\pi}{2} \right)}}$$
1 1
------------------------ + ------------------------
/ 1 \ / 1 \
|1 + -----------|*csc(t) |1 - -----------|*csc(t)
| /pi \| | /pi \|
| csc|-- - t|| | csc|-- - t||
\ \2 // \ \2 //
$$\frac{1}{\left(1 + \frac{1}{\csc{\left(- t + \frac{\pi}{2} \right)}}\right) \csc{\left(t \right)}} + \frac{1}{\left(1 - \frac{1}{\csc{\left(- t + \frac{\pi}{2} \right)}}\right) \csc{\left(t \right)}}$$
1 1
------------------------ + ------------------------
/ 1 \ /pi \ / 1 \ /pi \
|1 + ------|*sec|-- - t| |1 - ------|*sec|-- - t|
\ sec(t)/ \2 / \ sec(t)/ \2 /
$$\frac{1}{\left(1 + \frac{1}{\sec{\left(t \right)}}\right) \sec{\left(- t + \frac{\pi}{2} \right)}} + \frac{1}{\left(1 - \frac{1}{\sec{\left(t \right)}}\right) \sec{\left(- t + \frac{\pi}{2} \right)}}$$
1 1
----------------------------- + -----------------------------
/ 1 \ / 1 \
|1 + -----------|*csc(pi - t) |1 - -----------|*csc(pi - t)
| /pi \| | /pi \|
| csc|-- - t|| | csc|-- - t||
\ \2 // \ \2 //
$$\frac{1}{\left(1 + \frac{1}{\csc{\left(- t + \frac{\pi}{2} \right)}}\right) \csc{\left(- t + \pi \right)}} + \frac{1}{\left(1 - \frac{1}{\csc{\left(- t + \frac{\pi}{2} \right)}}\right) \csc{\left(- t + \pi \right)}}$$
1 1 1 1
------ + ------- - ------------------- - --------------------
sin(t) 2 2/t pi\ 2 2/t pi\
sin (t) sin(t)*tan |- + --| sin (t)*tan |- + --|
\2 4 / \2 4 /
$$\frac{1}{\sin{\left(t \right)}} + \frac{1}{\sin^{2}{\left(t \right)}} - \frac{1}{\sin{\left(t \right)} \tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)}} - \frac{1}{\sin^{2}{\left(t \right)} \tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}$$
/t\ /t\
2*tan|-| 2*tan|-|
\2/ \2/
------------------------------- + -------------------------------
/ 2/t\\ / 2/t\\
| 1 - tan |-|| | 1 - tan |-||
/ 2/t\\ | \2/| / 2/t\\ | \2/|
|1 + tan |-||*|1 + -----------| |1 + tan |-||*|1 - -----------|
\ \2// | 2/t\| \ \2// | 2/t\|
| 1 + tan |-|| | 1 + tan |-||
\ \2// \ \2//
$$\frac{2 \tan{\left(\frac{t}{2} \right)}}{\left(\frac{- \tan^{2}{\left(\frac{t}{2} \right)} + 1}{\tan^{2}{\left(\frac{t}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)} + \frac{2 \tan{\left(\frac{t}{2} \right)}}{\left(- \frac{- \tan^{2}{\left(\frac{t}{2} \right)} + 1}{\tan^{2}{\left(\frac{t}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)}$$
/t\ /t\
2*tan|-| 2*tan|-|
\2/ \2/
------------------------------------ + ------------------------------------
/ /t pi\ \ / /t pi\ \
| 2*tan|- + --| | | 2*tan|- + --| |
/ 2/t\\ | \2 4 / | / 2/t\\ | \2 4 / |
|1 + tan |-||*|1 - ----------------| |1 + tan |-||*|1 + ----------------|
\ \2// | 2/t pi\| \ \2// | 2/t pi\|
| 1 + tan |- + --|| | 1 + tan |- + --||
\ \2 4 // \ \2 4 //
$$\frac{2 \tan{\left(\frac{t}{2} \right)}}{\left(1 + \frac{2 \tan{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)} + \frac{2 \tan{\left(\frac{t}{2} \right)}}{\left(1 - \frac{2 \tan{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{t}{2} \right)} + 1\right)}$$
/t\ /t\
2*cot|-| 2*cot|-|
\2/ \2/
------------------------------------ + ------------------------------------
/ /t pi\ \ / /t pi\ \
| 2*tan|- + --| | | 2*tan|- + --| |
/ 2/t\\ | \2 4 / | / 2/t\\ | \2 4 / |
|1 + cot |-||*|1 - ----------------| |1 + cot |-||*|1 + ----------------|
\ \2// | 2/t pi\| \ \2// | 2/t pi\|
| 1 + tan |- + --|| | 1 + tan |- + --||
\ \2 4 // \ \2 4 //
$$\frac{2 \cot{\left(\frac{t}{2} \right)}}{\left(1 + \frac{2 \tan{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)} + \frac{2 \cot{\left(\frac{t}{2} \right)}}{\left(1 - \frac{2 \tan{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{t}{2} \right)} + 1\right)}$$
2/t\ 2/t\
4*sin |-|*(-1 + cos(t)) 4*sin |-|*(-1 + cos(t))
\2/ \2/
- ---------------------------- + ------------------------------------------------
/ 2 4/t\\ / 4/t\\
|sin (t) + 4*sin |-||*sin(t) | 4*sin |-||
\ \2// | \2/| / 2 \
|1 + ---------|*\-2 + sin (t) + 2*cos(t)/*sin(t)
| 2 |
\ sin (t) /
$$- \frac{4 \left(\cos{\left(t \right)} - 1\right) \sin^{2}{\left(\frac{t}{2} \right)}}{\left(4 \sin^{4}{\left(\frac{t}{2} \right)} + \sin^{2}{\left(t \right)}\right) \sin{\left(t \right)}} + \frac{4 \left(\cos{\left(t \right)} - 1\right) \sin^{2}{\left(\frac{t}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1\right) \left(\sin^{2}{\left(t \right)} + 2 \cos{\left(t \right)} - 2\right) \sin{\left(t \right)}}$$
2 2
-------------------------------------- + --------------------------------------
/ 1 \ / 1 \
| 1 - -------| | 1 - -------|
| 2/t\| | 2/t\|
| cot |-|| | cot |-||
/ 1 \ | \2/| /t\ / 1 \ | \2/| /t\
|1 + -------|*|1 + -----------|*cot|-| |1 + -------|*|1 - -----------|*cot|-|
| 2/t\| | 1 | \2/ | 2/t\| | 1 | \2/
| cot |-|| | 1 + -------| | cot |-|| | 1 + -------|
\ \2// | 2/t\| \ \2// | 2/t\|
| cot |-|| | cot |-||
\ \2// \ \2//
$$\frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}\right) \left(\frac{1 - \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}}{1 + \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}} + 1\right) \cot{\left(\frac{t}{2} \right)}} + \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}\right) \left(- \frac{1 - \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}}{1 + \frac{1}{\cot^{2}{\left(\frac{t}{2} \right)}}} + 1\right) \cot{\left(\frac{t}{2} \right)}}$$
2/t pi\ 2/t pi\
-1 + tan |- + --| -1 + tan |- + --|
\2 4 / \2 4 /
------------------------------------- + -------------------------------------
/ 2/t\\ / 2/t\\
| -1 + cot |-|| | -1 + cot |-||
/ 2/t pi\\ | \2/| / 2/t pi\\ | \2/|
|1 + tan |- + --||*|1 + ------------| |1 + tan |- + --||*|1 - ------------|
\ \2 4 // | 2/t\ | \ \2 4 // | 2/t\ |
| 1 + cot |-| | | 1 + cot |-| |
\ \2/ / \ \2/ /
$$\frac{\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} - 1}{\left(\frac{\cot^{2}{\left(\frac{t}{2} \right)} - 1}{\cot^{2}{\left(\frac{t}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)} + \frac{\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} - 1}{\left(- \frac{\cot^{2}{\left(\frac{t}{2} \right)} - 1}{\cot^{2}{\left(\frac{t}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)}$$
2/t pi\ 2/t pi\
1 - cot |- + --| 1 - cot |- + --|
\2 4 / \2 4 /
------------------------------------ + ------------------------------------
/ 2/t\\ / 2/t\\
| 1 - tan |-|| | 1 - tan |-||
/ 2/t pi\\ | \2/| / 2/t pi\\ | \2/|
|1 + cot |- + --||*|1 + -----------| |1 + cot |- + --||*|1 - -----------|
\ \2 4 // | 2/t\| \ \2 4 // | 2/t\|
| 1 + tan |-|| | 1 + tan |-||
\ \2// \ \2//
$$\frac{- \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1}{\left(\frac{- \tan^{2}{\left(\frac{t}{2} \right)} + 1}{\tan^{2}{\left(\frac{t}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)} + \frac{- \cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1}{\left(- \frac{- \tan^{2}{\left(\frac{t}{2} \right)} + 1}{\tan^{2}{\left(\frac{t}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1\right)}$$
/ 0 for t mod pi = 0 / 0 for t mod pi = 0
< <
\sin(t) otherwise \sin(t) otherwise
--------------------------------- + ---------------------------------
// 1 for t mod 2*pi = 0\ // 1 for t mod 2*pi = 0\
1 - |< | 1 + |< |
\\cos(t) otherwise / \\cos(t) otherwise /
$$\left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin{\left(t \right)} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos{\left(t \right)} & \text{otherwise} \end{cases}\right) + 1}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin{\left(t \right)} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos{\left(t \right)} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for t mod pi = 0 / 0 for t mod pi = 0
| |
< / pi\ < / pi\
|cos|t - --| otherwise |cos|t - --| otherwise
\ \ 2 / \ \ 2 /
--------------------------------- + ---------------------------------
// 1 for t mod 2*pi = 0\ // 1 for t mod 2*pi = 0\
1 - |< | 1 + |< |
\\cos(t) otherwise / \\cos(t) otherwise /
$$\left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\cos{\left(t - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos{\left(t \right)} & \text{otherwise} \end{cases}\right) + 1}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\cos{\left(t - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos{\left(t \right)} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for t mod pi = 0 / 0 for t mod pi = 0
< <
\sin(t) otherwise \sin(t) otherwise
-------------------------------------- + --------------------------------------
// 1 for t mod 2*pi = 0\ // 1 for t mod 2*pi = 0\
|| | || |
1 - |< / pi\ | 1 + |< / pi\ |
||sin|t + --| otherwise | ||sin|t + --| otherwise |
\\ \ 2 / / \\ \ 2 / /
$$\left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin{\left(t \right)} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\sin{\left(t + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + 1}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin{\left(t \right)} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\sin{\left(t + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ / 3*pi\ / / 3*pi\
| 1 for |t + ----| mod 2*pi = 0 | 1 for |t + ----| mod 2*pi = 0
< \ 2 / < \ 2 /
| |
\sin(t) otherwise \sin(t) otherwise
------------------------------------ + ------------------------------------
// 1 for t mod 2*pi = 0\ // 1 for t mod 2*pi = 0\
1 - |< | 1 + |< |
\\cos(t) otherwise / \\cos(t) otherwise /
$$\left(\frac{\begin{cases} 1 & \text{for}\: \left(t + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(t \right)} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos{\left(t \right)} & \text{otherwise} \end{cases}\right) + 1}\right) + \left(\frac{\begin{cases} 1 & \text{for}\: \left(t + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(t \right)} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos{\left(t \right)} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for t mod pi = 0 / 0 for t mod pi = 0
| |
| 1 | 1
<----------- otherwise <----------- otherwise
| / pi\ | / pi\
|sec|t - --| |sec|t - --|
\ \ 2 / \ \ 2 /
--------------------------------- + ---------------------------------
// 1 for t mod 2*pi = 0\ // 1 for t mod 2*pi = 0\
|| | || |
1 - |< 1 | 1 + |< 1 |
||------ otherwise | ||------ otherwise |
\\sec(t) / \\sec(t) /
$$\left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\sec{\left(t - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(t \right)}} & \text{otherwise} \end{cases}\right) + 1}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\sec{\left(t - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(t \right)}} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for t mod pi = 0 / 0 for t mod pi = 0
| |
< 1 < 1
|------ otherwise |------ otherwise
\csc(t) \csc(t)
-------------------------------------- + --------------------------------------
// 1 for t mod 2*pi = 0\ // 1 for t mod 2*pi = 0\
|| | || |
|| 1 | || 1 |
1 - |<----------- otherwise | 1 + |<----------- otherwise |
|| /pi \ | || /pi \ |
||csc|-- - t| | ||csc|-- - t| |
\\ \2 / / \\ \2 / /
$$\left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\csc{\left(t \right)}} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- t + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{1}{\csc{\left(t \right)}} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- t + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for t mod pi = 0 / 0 for t mod pi = 0
| |
|1 - cos(t) |1 - cos(t)
<---------- otherwise <---------- otherwise
| /t\ | /t\
| tan|-| | tan|-|
\ \2/ \ \2/
--------------------------------- + ---------------------------------
// 1 for t mod 2*pi = 0\ // 1 for t mod 2*pi = 0\
1 - |< | 1 + |< |
\\cos(t) otherwise / \\cos(t) otherwise /
$$\left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{- \cos{\left(t \right)} + 1}{\tan{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos{\left(t \right)} & \text{otherwise} \end{cases}\right) + 1}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{- \cos{\left(t \right)} + 1}{\tan{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos{\left(t \right)} & \text{otherwise} \end{cases}\right) + 1}\right)$$
2/t\ 2/t\
4*sin |-| 4*sin |-|
\2/ \2/
------------------------------------------ + ------------------------------------------
/ 4/t\\ / 4/t\\
| 4*sin |-|| | 4*sin |-||
| \2/| | \2/|
| 1 - ---------| / 4/t\\ | 1 - ---------| / 4/t\\
| 2 | | 4*sin |-|| | 2 | | 4*sin |-||
| sin (t) | | \2/| | sin (t) | | \2/|
|1 + -------------|*|1 + ---------|*sin(t) |1 - -------------|*|1 + ---------|*sin(t)
| 4/t\| | 2 | | 4/t\| | 2 |
| 4*sin |-|| \ sin (t) / | 4*sin |-|| \ sin (t) /
| \2/| | \2/|
| 1 + ---------| | 1 + ---------|
| 2 | | 2 |
\ sin (t) / \ sin (t) /
$$\frac{4 \sin^{2}{\left(\frac{t}{2} \right)}}{\left(\frac{- \frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1}{\frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1\right) \sin{\left(t \right)}} + \frac{4 \sin^{2}{\left(\frac{t}{2} \right)}}{\left(- \frac{- \frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1}{\frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{t}{2} \right)}}{\sin^{2}{\left(t \right)}} + 1\right) \sin{\left(t \right)}}$$
/ 0 for t mod pi = 0 / 0 for t mod pi = 0
< <
\sin(t) otherwise \sin(t) otherwise
-------------------------------------------------------- + --------------------------------------------------------
// / pi\ \ // / pi\ \
|| 0 for |t + --| mod pi = 0| || 0 for |t + --| mod pi = 0|
|| \ 2 / | || \ 2 / |
1 - |< | 1 + |< |
|| /t pi\ | || /t pi\ |
||(1 + sin(t))*cot|- + --| otherwise | ||(1 + sin(t))*cot|- + --| otherwise |
\\ \2 4 / / \\ \2 4 / /
$$\left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin{\left(t \right)} & \text{otherwise} \end{cases}}{\left(\begin{cases} 0 & \text{for}\: \left(t + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(t \right)} + 1\right) \cot{\left(\frac{t}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right) + 1}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin{\left(t \right)} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 0 & \text{for}\: \left(t + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(t \right)} + 1\right) \cot{\left(\frac{t}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for t mod pi = 0 / 0 for t mod pi = 0
| |
| /t\ | /t\
| 2*cot|-| | 2*cot|-|
< \2/ < \2/
|----------- otherwise |----------- otherwise
| 2/t\ | 2/t\
|1 + cot |-| |1 + cot |-|
\ \2/ \ \2/
--------------------------------------- + ---------------------------------------
// 1 for t mod 2*pi = 0\ // 1 for t mod 2*pi = 0\
|| | || |
|| 2/t\ | || 2/t\ |
||-1 + cot |-| | ||-1 + cot |-| |
1 - |< \2/ | 1 + |< \2/ |
||------------ otherwise | ||------------ otherwise |
|| 2/t\ | || 2/t\ |
||1 + cot |-| | ||1 + cot |-| |
\\ \2/ / \\ \2/ /
$$\left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{t}{2} \right)}}{\cot^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{t}{2} \right)} - 1}{\cot^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{t}{2} \right)}}{\cot^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{t}{2} \right)} - 1}{\cot^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for t mod pi = 0 / 0 for t mod pi = 0
| |
| /t\ | /t\
| 2*tan|-| | 2*tan|-|
< \2/ < \2/
|----------- otherwise |----------- otherwise
| 2/t\ | 2/t\
|1 + tan |-| |1 + tan |-|
\ \2/ \ \2/
-------------------------------------- + --------------------------------------
// 1 for t mod 2*pi = 0\ // 1 for t mod 2*pi = 0\
|| | || |
|| 2/t\ | || 2/t\ |
||1 - tan |-| | ||1 - tan |-| |
1 - |< \2/ | 1 + |< \2/ |
||----------- otherwise | ||----------- otherwise |
|| 2/t\ | || 2/t\ |
||1 + tan |-| | ||1 + tan |-| |
\\ \2/ / \\ \2/ /
$$\left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{t}{2} \right)}}{\tan^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{t}{2} \right)} + 1}{\tan^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{t}{2} \right)}}{\tan^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{t}{2} \right)} + 1}{\tan^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for t mod pi = 0 / 0 for t mod pi = 0
| |
0 for t mod pi = 0 0 for t mod pi = 0
|< otherwise |< otherwise
\\sin(t) otherwise \\sin(t) otherwise
------------------------------------------------------ + ------------------------------------------------------
// 1 for t mod 2*pi = 0\ // 1 for t mod 2*pi = 0\
|| | || |
1 - | 1 for t mod 2*pi = 0 | 1 + | 1 for t mod 2*pi = 0 |
||< otherwise | ||< otherwise |
\\\cos(t) otherwise / \\\cos(t) otherwise /
$$\left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin{\left(t \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos{\left(t \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\sin{\left(t \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\cos{\left(t \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for t mod pi = 0 / 0 for t mod pi = 0
| |
| 2 | 2
|-------------------- otherwise |-------------------- otherwise
1 \ /t\ 1 \ /t\
||1 + -------|*tan|-| ||1 + -------|*tan|-|
|| 2/t\| \2/ || 2/t\| \2/
|| tan |-|| || tan |-||
\\ \2// \\ \2//
--------------------------------------- + ---------------------------------------
// 1 for t mod 2*pi = 0\ // 1 for t mod 2*pi = 0\
|| | || |
|| 1 | || 1 |
||-1 + ------- | ||-1 + ------- |
|| 2/t\ | || 2/t\ |
|| tan |-| | || tan |-| |
1 - |< \2/ | 1 + |< \2/ |
||------------ otherwise | ||------------ otherwise |
|| 1 | || 1 |
||1 + ------- | ||1 + ------- |
|| 2/t\ | || 2/t\ |
|| tan |-| | || tan |-| |
\\ \2/ / \\ \2/ /
$$\left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} \right)}}\right) \tan{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} \right)}}} & \text{otherwise} \end{cases}\right) + 1}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} \right)}}\right) \tan{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{t}{2} \right)}}} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for t mod pi = 0 / 0 for t mod pi = 0
| |
| /t\ | /t\
| 2*cot|-| | 2*cot|-|
< \2/ < \2/
|----------- otherwise |----------- otherwise
| 2/t\ | 2/t\
|1 + cot |-| |1 + cot |-|
\ \2/ \ \2/
------------------------------------------------ + ------------------------------------------------
// / pi\ \ // / pi\ \
|| 0 for |t + --| mod pi = 0| || 0 for |t + --| mod pi = 0|
|| \ 2 / | || \ 2 / |
|| | || |
|| /t pi\ | || /t pi\ |
1 - |< 2*cot|- + --| | 1 + |< 2*cot|- + --| |
|| \2 4 / | || \2 4 / |
||---------------- otherwise | ||---------------- otherwise |
|| 2/t pi\ | || 2/t pi\ |
||1 + cot |- + --| | ||1 + cot |- + --| |
\\ \2 4 / / \\ \2 4 / /
$$\left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{t}{2} \right)}}{\cot^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases}}{\left(\begin{cases} 0 & \text{for}\: \left(t + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{t}{2} \right)}}{\cot^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 0 & \text{for}\: \left(t + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{t}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/t pi\ /t pi\
2*cos|- - --| 2*cos|- - --|
\2 2 / \2 2 /
------------------------------------------------ + ------------------------------------------------
/ 2/t pi\\ / 2/t pi\\
| cos |- - --|| | cos |- - --||
| \2 2 /| | \2 2 /|
| 1 - ------------| | 1 - ------------|
| 2/t\ | / 2/t pi\\ / 2/t pi\\ | 2/t\ |
| cos |-| | | cos |- - --|| | cos |- - --|| | cos |-| |
| \2/ | | \2 2 /| /t\ | \2 2 /| | \2/ | /t\
|1 + ----------------|*|1 + ------------|*cos|-| |1 + ------------|*|1 - ----------------|*cos|-|
| 2/t pi\| | 2/t\ | \2/ | 2/t\ | | 2/t pi\| \2/
| cos |- - --|| | cos |-| | | cos |-| | | cos |- - --||
| \2 2 /| \ \2/ / \ \2/ / | \2 2 /|
| 1 + ------------| | 1 + ------------|
| 2/t\ | | 2/t\ |
| cos |-| | | cos |-| |
\ \2/ / \ \2/ /
$$\frac{2 \cos{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} \right)}}\right) \left(\frac{1 - \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} \right)}}}{1 + \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} \right)}}} + 1\right) \cos{\left(\frac{t}{2} \right)}} + \frac{2 \cos{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} \right)}}\right) \left(- \frac{1 - \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} \right)}}}{1 + \frac{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} \right)}}} + 1\right) \cos{\left(\frac{t}{2} \right)}}$$
/t\ /t\
2*sec|-| 2*sec|-|
\2/ \2/
----------------------------------------------------- + -----------------------------------------------------
/ 2/t\ \ / 2/t\ \
| sec |-| | | sec |-| |
| \2/ | | \2/ |
| 1 - ------------| | 1 - ------------|
| 2/t pi\| / 2/t\ \ / 2/t\ \ | 2/t pi\|
| sec |- - --|| | sec |-| | | sec |-| | | sec |- - --||
| \2 2 /| | \2/ | /t pi\ | \2/ | | \2 2 /| /t pi\
|1 + ----------------|*|1 + ------------|*sec|- - --| |1 + ------------|*|1 - ----------------|*sec|- - --|
| 2/t\ | | 2/t pi\| \2 2 / | 2/t pi\| | 2/t\ | \2 2 /
| sec |-| | | sec |- - --|| | sec |- - --|| | sec |-| |
| \2/ | \ \2 2 // \ \2 2 // | \2/ |
| 1 + ------------| | 1 + ------------|
| 2/t pi\| | 2/t pi\|
| sec |- - --|| | sec |- - --||
\ \2 2 // \ \2 2 //
$$\frac{2 \sec{\left(\frac{t}{2} \right)}}{\left(\frac{- \frac{\sec^{2}{\left(\frac{t}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1}{\frac{\sec^{2}{\left(\frac{t}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1} + 1\right) \left(\frac{\sec^{2}{\left(\frac{t}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + \frac{2 \sec{\left(\frac{t}{2} \right)}}{\left(- \frac{- \frac{\sec^{2}{\left(\frac{t}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1}{\frac{\sec^{2}{\left(\frac{t}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1} + 1\right) \left(\frac{\sec^{2}{\left(\frac{t}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}$$
/pi t\ /pi t\
2*csc|-- - -| 2*csc|-- - -|
\2 2/ \2 2/
------------------------------------------------ + ------------------------------------------------
/ 2/pi t\\ / 2/pi t\\
| csc |-- - -|| | csc |-- - -||
| \2 2/| | \2 2/|
| 1 - ------------| | 1 - ------------|
| 2/t\ | / 2/pi t\\ / 2/pi t\\ | 2/t\ |
| csc |-| | | csc |-- - -|| | csc |-- - -|| | csc |-| |
| \2/ | | \2 2/| /t\ | \2 2/| | \2/ | /t\
|1 + ----------------|*|1 + ------------|*csc|-| |1 + ------------|*|1 - ----------------|*csc|-|
| 2/pi t\| | 2/t\ | \2/ | 2/t\ | | 2/pi t\| \2/
| csc |-- - -|| | csc |-| | | csc |-| | | csc |-- - -||
| \2 2/| \ \2/ / \ \2/ / | \2 2/|
| 1 + ------------| | 1 + ------------|
| 2/t\ | | 2/t\ |
| csc |-| | | csc |-| |
\ \2/ / \ \2/ /
$$\frac{2 \csc{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} \right)}}\right) \left(\frac{1 - \frac{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} \right)}}}{1 + \frac{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} \right)}}} + 1\right) \csc{\left(\frac{t}{2} \right)}} + \frac{2 \csc{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} \right)}}\right) \left(- \frac{1 - \frac{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} \right)}}}{1 + \frac{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{t}{2} \right)}}} + 1\right) \csc{\left(\frac{t}{2} \right)}}$$
/ / 3*pi\ / / 3*pi\
| 1 for |t + ----| mod 2*pi = 0 | 1 for |t + ----| mod 2*pi = 0
| \ 2 / | \ 2 /
| |
| 2/t pi\ | 2/t pi\
<-1 + tan |- + --| <-1 + tan |- + --|
| \2 4 / | \2 4 /
|----------------- otherwise |----------------- otherwise
| 2/t pi\ | 2/t pi\
| 1 + tan |- + --| | 1 + tan |- + --|
\ \2 4 / \ \2 4 /
----------------------------------------------- + -----------------------------------------------
// 1 for t mod 2*pi = 0\ // 1 for t mod 2*pi = 0\
|| | || |
|| 2/t\ | || 2/t\ |
||-1 + cot |-| | ||-1 + cot |-| |
1 - |< \2/ | 1 + |< \2/ |
||------------ otherwise | ||------------ otherwise |
|| 2/t\ | || 2/t\ |
||1 + cot |-| | ||1 + cot |-| |
\\ \2/ / \\ \2/ /
$$\left(\frac{\begin{cases} 1 & \text{for}\: \left(t + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{t}{2} \right)} - 1}{\cot^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}\right) + \left(\frac{\begin{cases} 1 & \text{for}\: \left(t + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{t}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{t}{2} \right)} - 1}{\cot^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for t mod pi = 0 / 0 for t mod pi = 0
| |
| 2*sin(t) | 2*sin(t)
|---------------------------- otherwise |---------------------------- otherwise
| / 2 \ | / 2 \
< | sin (t) | < | sin (t) |
|(1 - cos(t))*|1 + ---------| |(1 - cos(t))*|1 + ---------|
| | 4/t\| | | 4/t\|
| | 4*sin |-|| | | 4*sin |-||
| \ \2// | \ \2//
\ \
------------------------------------------------------ + ------------------------------------------------------
// 1 for t mod 2*pi = 0\ // 1 for t mod 2*pi = 0\
|| | || |
|| 2 | || 2 |
1 - |< -4 + 4*sin (t) + 4*cos(t) | 1 + |< -4 + 4*sin (t) + 4*cos(t) |
||--------------------------- otherwise | ||--------------------------- otherwise |
|| 2 2 | || 2 2 |
\\2*(1 - cos(t)) + 2*sin (t) / \\2*(1 - cos(t)) + 2*sin (t) /
$$\left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{2 \sin{\left(t \right)}}{\left(1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}\right) \left(- \cos{\left(t \right)} + 1\right)} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{4 \sin^{2}{\left(t \right)} + 4 \cos{\left(t \right)} - 4}{2 \left(- \cos{\left(t \right)} + 1\right)^{2} + 2 \sin^{2}{\left(t \right)}} & \text{otherwise} \end{cases}\right) + 1}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{2 \sin{\left(t \right)}}{\left(1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}\right) \left(- \cos{\left(t \right)} + 1\right)} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{4 \sin^{2}{\left(t \right)} + 4 \cos{\left(t \right)} - 4}{2 \left(- \cos{\left(t \right)} + 1\right)^{2} + 2 \sin^{2}{\left(t \right)}} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for t mod pi = 0 / 0 for t mod pi = 0
| |
| sin(t) | sin(t)
|----------------------- otherwise |----------------------- otherwise
|/ 2 \ |/ 2 \
<| sin (t) | 2/t\ <| sin (t) | 2/t\
||1 + ---------|*sin |-| ||1 + ---------|*sin |-|
|| 4/t\| \2/ || 4/t\| \2/
|| 4*sin |-|| || 4*sin |-||
|\ \2// |\ \2//
\ \
------------------------------------------ + ------------------------------------------
// 1 for t mod 2*pi = 0\ // 1 for t mod 2*pi = 0\
|| | || |
|| 2 | || 2 |
|| sin (t) | || sin (t) |
||-1 + --------- | ||-1 + --------- |
|| 4/t\ | || 4/t\ |
|| 4*sin |-| | || 4*sin |-| |
1 - |< \2/ | 1 + |< \2/ |
||-------------- otherwise | ||-------------- otherwise |
|| 2 | || 2 |
|| sin (t) | || sin (t) |
||1 + --------- | ||1 + --------- |
|| 4/t\ | || 4/t\ |
|| 4*sin |-| | || 4*sin |-| |
\\ \2/ / \\ \2/ /
$$\left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{\sin{\left(t \right)}}{\left(1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}\right) \sin^{2}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}}{1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}} & \text{otherwise} \end{cases}\right) + 1}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{\sin{\left(t \right)}}{\left(1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}\right) \sin^{2}{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}}{1 + \frac{\sin^{2}{\left(t \right)}}{4 \sin^{4}{\left(\frac{t}{2} \right)}}} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for t mod pi = 0 / 0 for t mod pi = 0
| |
|/ 0 for t mod pi = 0 |/ 0 for t mod pi = 0
|| ||
|| /t\ || /t\
<| 2*cot|-| <| 2*cot|-|
|< \2/ otherwise |< \2/ otherwise
||----------- otherwise ||----------- otherwise
|| 2/t\ || 2/t\
||1 + cot |-| ||1 + cot |-|
\\ \2/ \\ \2/
------------------------------------------------------------ + ------------------------------------------------------------
// 1 for t mod 2*pi = 0\ // 1 for t mod 2*pi = 0\
|| | || |
||/ 1 for t mod 2*pi = 0 | ||/ 1 for t mod 2*pi = 0 |
||| | ||| |
||| 2/t\ | ||| 2/t\ |
1 - |<|-1 + cot |-| | 1 + |<|-1 + cot |-| |
||< \2/ otherwise | ||< \2/ otherwise |
|||------------ otherwise | |||------------ otherwise |
||| 2/t\ | ||| 2/t\ |
|||1 + cot |-| | |||1 + cot |-| |
\\\ \2/ / \\\ \2/ /
$$\left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{t}{2} \right)}}{\cot^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{t}{2} \right)} - 1}{\cot^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{t}{2} \right)}}{\cot^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{t}{2} \right)} - 1}{\cot^{2}{\left(\frac{t}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for t mod pi = 0 / 0 for t mod pi = 0
| |
| /t\ | /t\
| 2*cos|-| | 2*cos|-|
| \2/ | \2/
|------------------------------ otherwise |------------------------------ otherwise
2/t\ \ 2/t\ \
|| cos |-| | || cos |-| |
|| \2/ | /t pi\ || \2/ | /t pi\
||1 + ------------|*cos|- - --| ||1 + ------------|*cos|- - --|
|| 2/t pi\| \2 2 / || 2/t pi\| \2 2 /
|| cos |- - --|| || cos |- - --||
\\ \2 2 // \\ \2 2 //
------------------------------------------------- + -------------------------------------------------
// 1 for t mod 2*pi = 0\ // 1 for t mod 2*pi = 0\
|| | || |
|| 2/t\ | || 2/t\ |
|| cos |-| | || cos |-| |
|| \2/ | || \2/ |
||-1 + ------------ | ||-1 + ------------ |
|| 2/t pi\ | || 2/t pi\ |
|| cos |- - --| | || cos |- - --| |
1 - |< \2 2 / | 1 + |< \2 2 / |
||----------------- otherwise | ||----------------- otherwise |
|| 2/t\ | || 2/t\ |
|| cos |-| | || cos |-| |
|| \2/ | || \2/ |
|| 1 + ------------ | || 1 + ------------ |
|| 2/t pi\ | || 2/t pi\ |
|| cos |- - --| | || cos |- - --| |
\\ \2 2 / / \\ \2 2 / /
$$\left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{t}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{t}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{t}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{t}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) + 1}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{t}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{t}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{t}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{t}{2} \right)}}{\cos^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for t mod pi = 0 / 0 for t mod pi = 0
| |
| /t pi\ | /t pi\
| 2*sec|- - --| | 2*sec|- - --|
| \2 2 / | \2 2 /
|------------------------- otherwise |------------------------- otherwise
2/t pi\\ 2/t pi\\
|| sec |- - --|| || sec |- - --||
|| \2 2 /| /t\ || \2 2 /| /t\
||1 + ------------|*sec|-| ||1 + ------------|*sec|-|
|| 2/t\ | \2/ || 2/t\ | \2/
|| sec |-| | || sec |-| |
\\ \2/ / \\ \2/ /
-------------------------------------------- + --------------------------------------------
// 1 for t mod 2*pi = 0\ // 1 for t mod 2*pi = 0\
|| | || |
|| 2/t pi\ | || 2/t pi\ |
|| sec |- - --| | || sec |- - --| |
|| \2 2 / | || \2 2 / |
||-1 + ------------ | ||-1 + ------------ |
|| 2/t\ | || 2/t\ |
|| sec |-| | || sec |-| |
1 - |< \2/ | 1 + |< \2/ |
||----------------- otherwise | ||----------------- otherwise |
|| 2/t pi\ | || 2/t pi\ |
|| sec |- - --| | || sec |- - --| |
|| \2 2 / | || \2 2 / |
|| 1 + ------------ | || 1 + ------------ |
|| 2/t\ | || 2/t\ |
|| sec |-| | || sec |-| |
\\ \2/ / \\ \2/ /
$$\left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} \right)}}\right) \sec{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} \right)}}} & \text{otherwise} \end{cases}\right) + 1}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} \right)}}\right) \sec{\left(\frac{t}{2} \right)}} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{t}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{t}{2} \right)}}} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for t mod pi = 0 / 0 for t mod pi = 0
| |
| /t\ | /t\
| 2*csc|-| | 2*csc|-|
| \2/ | \2/
|------------------------------ otherwise |------------------------------ otherwise
2/t\ \ 2/t\ \
|| csc |-| | || csc |-| |
|| \2/ | /pi t\ || \2/ | /pi t\
||1 + ------------|*csc|-- - -| ||1 + ------------|*csc|-- - -|
|| 2/pi t\| \2 2/ || 2/pi t\| \2 2/
|| csc |-- - -|| || csc |-- - -||
\\ \2 2// \\ \2 2//
------------------------------------------------- + -------------------------------------------------
// 1 for t mod 2*pi = 0\ // 1 for t mod 2*pi = 0\
|| | || |
|| 2/t\ | || 2/t\ |
|| csc |-| | || csc |-| |
|| \2/ | || \2/ |
||-1 + ------------ | ||-1 + ------------ |
|| 2/pi t\ | || 2/pi t\ |
|| csc |-- - -| | || csc |-- - -| |
1 - |< \2 2/ | 1 + |< \2 2/ |
||----------------- otherwise | ||----------------- otherwise |
|| 2/t\ | || 2/t\ |
|| csc |-| | || csc |-| |
|| \2/ | || \2/ |
|| 1 + ------------ | || 1 + ------------ |
|| 2/pi t\ | || 2/pi t\ |
|| csc |-- - -| | || csc |-- - -| |
\\ \2 2/ / \\ \2 2/ /
$$\left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{t}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{t}{2} \right)}}{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{t}{2} \right)}}{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{t}{2} \right)}}{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) + 1}\right) + \left(\frac{\begin{cases} 0 & \text{for}\: t \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{t}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{t}{2} \right)}}{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: t \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{t}{2} \right)}}{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{t}{2} \right)}}{\csc^{2}{\left(- \frac{t}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) + 1}\right)$$
Piecewise((0, Mod(t = pi, 0)), (2*csc(t/2)/((1 + csc(t/2)^2/csc(pi/2 - t/2)^2)*csc(pi/2 - t/2)), True))/(1 - Piecewise((1, Mod(t = 2*pi, 0)), ((-1 + csc(t/2)^2/csc(pi/2 - t/2)^2)/(1 + csc(t/2)^2/csc(pi/2 - t/2)^2), True))) + Piecewise((0, Mod(t = pi, 0)), (2*csc(t/2)/((1 + csc(t/2)^2/csc(pi/2 - t/2)^2)*csc(pi/2 - t/2)), True))/(1 + Piecewise((1, Mod(t = 2*pi, 0)), ((-1 + csc(t/2)^2/csc(pi/2 - t/2)^2)/(1 + csc(t/2)^2/csc(pi/2 - t/2)^2), True)))