Господин Экзамен
Общий знаменатель (sin(a)/(1+cos(a)))-(sin(a)/(1-cos(a)))
Решение
Вы ввели
[src]
sin(a) sin(a) ---------- - ---------- 1 + cos(a) 1 - cos(a)
$$\frac{\sin{\left(a \right)}}{\cos{\left(a \right)} + 1} - \frac{\sin{\left(a \right)}}{- \cos{\left(a \right)} + 1}$$
sin(a)/(1 + cos(a)) - sin(a)/(1 - cos(a))
Собрать выражение
[src]
sin(a) sin(a) ---------- + ----------- 1 + cos(a) -1 + cos(a)
$$\frac{\sin{\left(a \right)}}{\cos{\left(a \right)} + 1} + \frac{\sin{\left(a \right)}}{\cos{\left(a \right)} - 1}$$
sin(a)/(1 + cos(a)) + sin(a)/(-1 + cos(a))
Общий знаменатель
[src]
2*cos(a)*sin(a)
---------------
2
-1 + cos (a)
$$\frac{2 \sin{\left(a \right)} \cos{\left(a \right)}}{\cos^{2}{\left(a \right)} - 1}$$
2*cos(a)*sin(a)/(-1 + cos(a)^2)
Тригонометрическая часть
[src]
-2*cot(a)
$$- 2 \cot{\left(a \right)}$$
-2 ------ tan(a)
$$- \frac{2}{\tan{\left(a \right)}}$$
-sin(2*a)
----------
2
sin (a)
$$- \frac{\sin{\left(2 a \right)}}{\sin^{2}{\left(a \right)}}$$
-2*cos(a) ----------- / pi\ cos|a - --| \ 2 /
$$- \frac{2 \cos{\left(a \right)}}{\cos{\left(a - \frac{\pi}{2} \right)}}$$
/ pi\
-2*sec|a - --|
\ 2 /
--------------
sec(a)
$$- \frac{2 \sec{\left(a - \frac{\pi}{2} \right)}}{\sec{\left(a \right)}}$$
-2*csc(a) ----------- /pi \ csc|-- - a| \2 /
$$- \frac{2 \csc{\left(a \right)}}{\csc{\left(- a + \frac{\pi}{2} \right)}}$$
/a\
2*cos(a)*tan|-|
\2/
---------------
-1 + cos(a)
$$\frac{2 \cos{\left(a \right)} \tan{\left(\frac{a}{2} \right)}}{\cos{\left(a \right)} - 1}$$
sin(a) sin(a)
--------------- - ---------------
/ pi\ / pi\
1 + sin|a + --| 1 - sin|a + --|
\ 2 / \ 2 /
$$\frac{\sin{\left(a \right)}}{\sin{\left(a + \frac{\pi}{2} \right)} + 1} - \frac{\sin{\left(a \right)}}{- \sin{\left(a + \frac{\pi}{2} \right)} + 1}$$
/ pi\ / pi\ cos|a - --| cos|a - --| \ 2 / \ 2 / ----------- - ----------- 1 + cos(a) 1 - cos(a)
$$\frac{\cos{\left(a - \frac{\pi}{2} \right)}}{\cos{\left(a \right)} + 1} - \frac{\cos{\left(a - \frac{\pi}{2} \right)}}{- \cos{\left(a \right)} + 1}$$
1 1 ------------------- - ------------------- / 1 \ / 1 \ |1 + ------|*csc(a) |1 - ------|*csc(a) \ sec(a)/ \ sec(a)/
$$\frac{1}{\left(1 + \frac{1}{\sec{\left(a \right)}}\right) \csc{\left(a \right)}} - \frac{1}{\left(1 - \frac{1}{\sec{\left(a \right)}}\right) \csc{\left(a \right)}}$$
1 1 ------------------------ - ------------------------ / 1 \ / pi\ / 1 \ / pi\ |1 + ------|*sec|a - --| |1 - ------|*sec|a - --| \ sec(a)/ \ 2 / \ sec(a)/ \ 2 /
$$\frac{1}{\left(1 + \frac{1}{\sec{\left(a \right)}}\right) \sec{\left(a - \frac{\pi}{2} \right)}} - \frac{1}{\left(1 - \frac{1}{\sec{\left(a \right)}}\right) \sec{\left(a - \frac{\pi}{2} \right)}}$$
1 1 ------------------------ - ------------------------ / 1 \ / 1 \ |1 + -----------|*csc(a) |1 - -----------|*csc(a) | /pi \| | /pi \| | csc|-- - a|| | csc|-- - a|| \ \2 // \ \2 //
$$\frac{1}{\left(1 + \frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}}\right) \csc{\left(a \right)}} - \frac{1}{\left(1 - \frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}}\right) \csc{\left(a \right)}}$$
1 1 ------------------------ - ------------------------ / 1 \ /pi \ / 1 \ /pi \ |1 + ------|*sec|-- - a| |1 - ------|*sec|-- - a| \ sec(a)/ \2 / \ sec(a)/ \2 /
$$\frac{1}{\left(1 + \frac{1}{\sec{\left(a \right)}}\right) \sec{\left(- a + \frac{\pi}{2} \right)}} - \frac{1}{\left(1 - \frac{1}{\sec{\left(a \right)}}\right) \sec{\left(- a + \frac{\pi}{2} \right)}}$$
1 1 ----------------------------- - ----------------------------- / 1 \ / 1 \ |1 + -----------|*csc(pi - a) |1 - -----------|*csc(pi - a) | /pi \| | /pi \| | csc|-- - a|| | csc|-- - a|| \ \2 // \ \2 //
$$\frac{1}{\left(1 + \frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}}\right) \csc{\left(- a + \pi \right)}} - \frac{1}{\left(1 - \frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}}\right) \csc{\left(- a + \pi \right)}}$$
/ 2/a pi\\ / 2/a pi\\
|1 - cot |- + --||*(1 + sin(a)) |1 - cot |- + --||*(1 + sin(a))
\ \2 4 // \ \2 4 //
------------------------------- - -------------------------------
2*(1 + cos(a)) 2*(1 - cos(a))
$$\frac{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(a \right)} + 1\right)}{2 \left(\cos{\left(a \right)} + 1\right)} - \frac{\left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(a \right)} + 1\right)}{2 \cdot \left(- \cos{\left(a \right)} + 1\right)}$$
/a\ /a\
2*tan|-| 2*tan|-|
\2/ \2/
- ------------------------------- + -------------------------------
/ 2/a\\ / 2/a\\
| 1 - tan |-|| | 1 - tan |-||
/ 2/a\\ | \2/| / 2/a\\ | \2/|
|1 + tan |-||*|1 - -----------| |1 + tan |-||*|1 + -----------|
\ \2// | 2/a\| \ \2// | 2/a\|
| 1 + tan |-|| | 1 + tan |-||
\ \2// \ \2//
$$\frac{2 \tan{\left(\frac{a}{2} \right)}}{\left(\frac{- \tan^{2}{\left(\frac{a}{2} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)} - \frac{2 \tan{\left(\frac{a}{2} \right)}}{\left(- \frac{- \tan^{2}{\left(\frac{a}{2} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
/a\ /a\
2*cot|-| 2*cot|-|
\2/ \2/
- ------------------------------------ + ------------------------------------
/ /a pi\ \ / /a pi\ \
| 2*tan|- + --| | | 2*tan|- + --| |
/ 2/a\\ | \2 4 / | / 2/a\\ | \2 4 / |
|1 + cot |-||*|1 - ----------------| |1 + cot |-||*|1 + ----------------|
\ \2// | 2/a pi\| \ \2// | 2/a pi\|
| 1 + tan |- + --|| | 1 + tan |- + --||
\ \2 4 // \ \2 4 //
$$\frac{2 \cot{\left(\frac{a}{2} \right)}}{\left(1 + \frac{2 \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)} - \frac{2 \cot{\left(\frac{a}{2} \right)}}{\left(1 - \frac{2 \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
/a\ /a\
2*tan|-| 2*tan|-|
\2/ \2/
- ------------------------------------ + ------------------------------------
/ /a pi\ \ / /a pi\ \
| 2*tan|- + --| | | 2*tan|- + --| |
/ 2/a\\ | \2 4 / | / 2/a\\ | \2 4 / |
|1 + tan |-||*|1 - ----------------| |1 + tan |-||*|1 + ----------------|
\ \2// | 2/a pi\| \ \2// | 2/a pi\|
| 1 + tan |- + --|| | 1 + tan |- + --||
\ \2 4 // \ \2 4 //
$$\frac{2 \tan{\left(\frac{a}{2} \right)}}{\left(1 + \frac{2 \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)} - \frac{2 \tan{\left(\frac{a}{2} \right)}}{\left(1 - \frac{2 \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}\right) \left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
2/a\ 2/a\
4*sin |-|*(-1 + cos(a)) 4*sin |-|*(-1 + cos(a))
\2/ \2/
- ---------------------------- - ------------------------------------------------
/ 2 4/a\\ / 4/a\\
|sin (a) + 4*sin |-||*sin(a) | 4*sin |-||
\ \2// | \2/| / 2 \
|1 + ---------|*\-2 + sin (a) + 2*cos(a)/*sin(a)
| 2 |
\ sin (a) /
$$- \frac{4 \left(\cos{\left(a \right)} - 1\right) \sin^{2}{\left(\frac{a}{2} \right)}}{\left(4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}\right) \sin{\left(a \right)}} - \frac{4 \left(\cos{\left(a \right)} - 1\right) \sin^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \left(\sin^{2}{\left(a \right)} + 2 \cos{\left(a \right)} - 2\right) \sin{\left(a \right)}}$$
2 2
- -------------------------------------- + --------------------------------------
/ 1 \ / 1 \
| 1 - -------| | 1 - -------|
| 2/a\| | 2/a\|
| cot |-|| | cot |-||
/ 1 \ | \2/| /a\ / 1 \ | \2/| /a\
|1 + -------|*|1 - -----------|*cot|-| |1 + -------|*|1 + -----------|*cot|-|
| 2/a\| | 1 | \2/ | 2/a\| | 1 | \2/
| cot |-|| | 1 + -------| | cot |-|| | 1 + -------|
\ \2// | 2/a\| \ \2// | 2/a\|
| cot |-|| | cot |-||
\ \2// \ \2//
$$\frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \left(\frac{1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}} + 1\right) \cot{\left(\frac{a}{2} \right)}} - \frac{2}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right) \left(- \frac{1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}} + 1\right) \cot{\left(\frac{a}{2} \right)}}$$
2/a pi\ 2/a pi\
-1 + tan |- + --| -1 + tan |- + --|
\2 4 / \2 4 /
------------------------------------- - -------------------------------------
/ 2/a\\ / 2/a\\
| -1 + cot |-|| | -1 + cot |-||
/ 2/a pi\\ | \2/| / 2/a pi\\ | \2/|
|1 + tan |- + --||*|1 + ------------| |1 + tan |- + --||*|1 - ------------|
\ \2 4 // | 2/a\ | \ \2 4 // | 2/a\ |
| 1 + cot |-| | | 1 + cot |-| |
\ \2/ / \ \2/ /
$$\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\left(\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)} - \frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\left(- \frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} + 1\right) \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)}$$
2/a pi\ 2/a pi\
1 - cot |- + --| 1 - cot |- + --|
\2 4 / \2 4 /
------------------------------------ - ------------------------------------
/ 2/a\\ / 2/a\\
| 1 - tan |-|| | 1 - tan |-||
/ 2/a pi\\ | \2/| / 2/a pi\\ | \2/|
|1 + cot |- + --||*|1 + -----------| |1 + cot |- + --||*|1 - -----------|
\ \2 4 // | 2/a\| \ \2 4 // | 2/a\|
| 1 + tan |-|| | 1 + tan |-||
\ \2// \ \2//
$$\frac{- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}{\left(\frac{- \tan^{2}{\left(\frac{a}{2} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)} - \frac{- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1}{\left(- \frac{- \tan^{2}{\left(\frac{a}{2} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)}$$
/ 0 for a mod pi = 0 / 0 for a mod pi = 0
< <
\sin(a) otherwise \sin(a) otherwise
--------------------------------- - ---------------------------------
// 1 for a mod 2*pi = 0\ // 1 for a mod 2*pi = 0\
1 + |< | 1 - |< |
\\cos(a) otherwise / \\cos(a) otherwise /
$$\left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for a mod pi = 0 / 0 for a mod pi = 0
< <
\sin(a) otherwise \sin(a) otherwise
-------------------------------------- - --------------------------------------
// 1 for a mod 2*pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
1 + |< / pi\ | 1 - |< / pi\ |
||sin|a + --| otherwise | ||sin|a + --| otherwise |
\\ \ 2 / / \\ \ 2 / /
$$\left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\sin{\left(a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + 1}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\sin{\left(a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for a mod pi = 0 / 0 for a mod pi = 0
| |
< / pi\ < / pi\
|cos|a - --| otherwise |cos|a - --| otherwise
\ \ 2 / \ \ 2 /
--------------------------------- - ---------------------------------
// 1 for a mod 2*pi = 0\ // 1 for a mod 2*pi = 0\
1 + |< | 1 - |< |
\\cos(a) otherwise / \\cos(a) otherwise /
$$\left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ / 3*pi\ / / 3*pi\
| 1 for |a + ----| mod 2*pi = 0 | 1 for |a + ----| mod 2*pi = 0
< \ 2 / < \ 2 /
| |
\sin(a) otherwise \sin(a) otherwise
------------------------------------ - ------------------------------------
// 1 for a mod 2*pi = 0\ // 1 for a mod 2*pi = 0\
1 + |< | 1 - |< |
\\cos(a) otherwise / \\cos(a) otherwise /
$$\left(\frac{\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}\right) - \left(\frac{\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for a mod pi = 0 / 0 for a mod pi = 0
| |
| 1 | 1
<----------- otherwise <----------- otherwise
| / pi\ | / pi\
|sec|a - --| |sec|a - --|
\ \ 2 / \ \ 2 /
--------------------------------- - ---------------------------------
// 1 for a mod 2*pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
1 + |< 1 | 1 - |< 1 |
||------ otherwise | ||------ otherwise |
\\sec(a) / \\sec(a) /
$$\left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(a \right)}} & \text{otherwise} \end{cases}\right) + 1}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(a \right)}} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for a mod pi = 0 / 0 for a mod pi = 0
| |
< 1 < 1
|------ otherwise |------ otherwise
\csc(a) \csc(a)
-------------------------------------- - --------------------------------------
// 1 for a mod 2*pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
|| 1 | || 1 |
1 + |<----------- otherwise | 1 - |<----------- otherwise |
|| /pi \ | || /pi \ |
||csc|-- - a| | ||csc|-- - a| |
\\ \2 / / \\ \2 / /
$$\left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc{\left(a \right)}} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc{\left(a \right)}} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for a mod pi = 0 / 0 for a mod pi = 0
| |
|1 - cos(a) |1 - cos(a)
<---------- otherwise <---------- otherwise
| /a\ | /a\
| tan|-| | tan|-|
\ \2/ \ \2/
--------------------------------- - ---------------------------------
// 1 for a mod 2*pi = 0\ // 1 for a mod 2*pi = 0\
1 + |< | 1 - |< |
\\cos(a) otherwise / \\cos(a) otherwise /
$$\left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{- \cos{\left(a \right)} + 1}{\tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{- \cos{\left(a \right)} + 1}{\tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) + 1}\right)$$
2/a\ 2/a\
4*sin |-| 4*sin |-|
\2/ \2/
- ------------------------------------------ + ------------------------------------------
/ 4/a\\ / 4/a\\
| 4*sin |-|| | 4*sin |-||
| \2/| | \2/|
| 1 - ---------| / 4/a\\ | 1 - ---------| / 4/a\\
| 2 | | 4*sin |-|| | 2 | | 4*sin |-||
| sin (a) | | \2/| | sin (a) | | \2/|
|1 - -------------|*|1 + ---------|*sin(a) |1 + -------------|*|1 + ---------|*sin(a)
| 4/a\| | 2 | | 4/a\| | 2 |
| 4*sin |-|| \ sin (a) / | 4*sin |-|| \ sin (a) /
| \2/| | \2/|
| 1 + ---------| | 1 + ---------|
| 2 | | 2 |
\ sin (a) / \ sin (a) /
$$\frac{4 \sin^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1}{\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \sin{\left(a \right)}} - \frac{4 \sin^{2}{\left(\frac{a}{2} \right)}}{\left(- \frac{- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1}{\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \sin{\left(a \right)}}$$
/ 0 for a mod pi = 0 / 0 for a mod pi = 0
< <
\sin(a) otherwise \sin(a) otherwise
-------------------------------------------------------- - --------------------------------------------------------
// / pi\ \ // / pi\ \
|| 0 for |a + --| mod pi = 0| || 0 for |a + --| mod pi = 0|
|| \ 2 / | || \ 2 / |
1 + |< | 1 - |< |
|| /a pi\ | || /a pi\ |
||(1 + sin(a))*cot|- + --| otherwise | ||(1 + sin(a))*cot|- + --| otherwise |
\\ \2 4 / / \\ \2 4 / /
$$\left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}}{\left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(a \right)} + 1\right) \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right) + 1}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(a \right)} + 1\right) \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for a mod pi = 0 / 0 for a mod pi = 0
| |
| /a\ | /a\
| 2*cot|-| | 2*cot|-|
< \2/ < \2/
|----------- otherwise |----------- otherwise
| 2/a\ | 2/a\
|1 + cot |-| |1 + cot |-|
\ \2/ \ \2/
--------------------------------------- - ---------------------------------------
// 1 for a mod 2*pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
|| 2/a\ | || 2/a\ |
||-1 + cot |-| | ||-1 + cot |-| |
1 + |< \2/ | 1 - |< \2/ |
||------------ otherwise | ||------------ otherwise |
|| 2/a\ | || 2/a\ |
||1 + cot |-| | ||1 + cot |-| |
\\ \2/ / \\ \2/ /
$$\left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for a mod pi = 0 / 0 for a mod pi = 0
| |
| /a\ | /a\
| 2*tan|-| | 2*tan|-|
< \2/ < \2/
|----------- otherwise |----------- otherwise
| 2/a\ | 2/a\
|1 + tan |-| |1 + tan |-|
\ \2/ \ \2/
-------------------------------------- - --------------------------------------
// 1 for a mod 2*pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
|| 2/a\ | || 2/a\ |
||1 - tan |-| | ||1 - tan |-| |
1 + |< \2/ | 1 - |< \2/ |
||----------- otherwise | ||----------- otherwise |
|| 2/a\ | || 2/a\ |
||1 + tan |-| | ||1 + tan |-| |
\\ \2/ / \\ \2/ /
$$\left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{a}{2} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{a}{2} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for a mod pi = 0 / 0 for a mod pi = 0
| |
$$\left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for a mod pi = 0 / 0 for a mod pi = 0
| |
| 2 | 2
|-------------------- otherwise |-------------------- otherwise
$$\left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) + 1}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for a mod pi = 0 / 0 for a mod pi = 0
| |
| /a\ | /a\
| 2*cot|-| | 2*cot|-|
< \2/ < \2/
|----------- otherwise |----------- otherwise
| 2/a\ | 2/a\
|1 + cot |-| |1 + cot |-|
\ \2/ \ \2/
------------------------------------------------ - ------------------------------------------------
// / pi\ \ // / pi\ \
|| 0 for |a + --| mod pi = 0| || 0 for |a + --| mod pi = 0|
|| \ 2 / | || \ 2 / |
|| | || |
|| /a pi\ | || /a pi\ |
1 + |< 2*cot|- + --| | 1 - |< 2*cot|- + --| |
|| \2 4 / | || \2 4 / |
||---------------- otherwise | ||---------------- otherwise |
|| 2/a pi\ | || 2/a pi\ |
||1 + cot |- + --| | ||1 + cot |- + --| |
\\ \2 4 / / \\ \2 4 / /
$$\left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}}{\left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/a pi\ /a pi\
2*cos|- - --| 2*cos|- - --|
\2 2 / \2 2 /
- ------------------------------------------------ + ------------------------------------------------
/ 2/a pi\\ / 2/a pi\\
| cos |- - --|| | cos |- - --||
| \2 2 /| | \2 2 /|
| 1 - ------------| | 1 - ------------|
/ 2/a pi\\ | 2/a\ | | 2/a\ | / 2/a pi\\
| cos |- - --|| | cos |-| | | cos |-| | | cos |- - --||
| \2 2 /| | \2/ | /a\ | \2/ | | \2 2 /| /a\
|1 + ------------|*|1 - ----------------|*cos|-| |1 + ----------------|*|1 + ------------|*cos|-|
| 2/a\ | | 2/a pi\| \2/ | 2/a pi\| | 2/a\ | \2/
| cos |-| | | cos |- - --|| | cos |- - --|| | cos |-| |
\ \2/ / | \2 2 /| | \2 2 /| \ \2/ /
| 1 + ------------| | 1 + ------------|
| 2/a\ | | 2/a\ |
| cos |-| | | cos |-| |
\ \2/ / \ \2/ /
$$\frac{2 \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \left(\frac{1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}} + 1\right) \cos{\left(\frac{a}{2} \right)}} - \frac{2 \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \left(- \frac{1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}} + 1\right) \cos{\left(\frac{a}{2} \right)}}$$
/a\ /a\
2*sec|-| 2*sec|-|
\2/ \2/
- ----------------------------------------------------- + -----------------------------------------------------
/ 2/a\ \ / 2/a\ \
| sec |-| | | sec |-| |
| \2/ | | \2/ |
| 1 - ------------| | 1 - ------------|
/ 2/a\ \ | 2/a pi\| | 2/a pi\| / 2/a\ \
| sec |-| | | sec |- - --|| | sec |- - --|| | sec |-| |
| \2/ | | \2 2 /| /a pi\ | \2 2 /| | \2/ | /a pi\
|1 + ------------|*|1 - ----------------|*sec|- - --| |1 + ----------------|*|1 + ------------|*sec|- - --|
| 2/a pi\| | 2/a\ | \2 2 / | 2/a\ | | 2/a pi\| \2 2 /
| sec |- - --|| | sec |-| | | sec |-| | | sec |- - --||
\ \2 2 // | \2/ | | \2/ | \ \2 2 //
| 1 + ------------| | 1 + ------------|
| 2/a pi\| | 2/a pi\|
| sec |- - --|| | sec |- - --||
\ \2 2 // \ \2 2 //
$$\frac{2 \sec{\left(\frac{a}{2} \right)}}{\left(\frac{- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1}{\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1} + 1\right) \left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - \frac{2 \sec{\left(\frac{a}{2} \right)}}{\left(- \frac{- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1}{\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1} + 1\right) \left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}$$
/pi a\ /pi a\
2*csc|-- - -| 2*csc|-- - -|
\2 2/ \2 2/
- ------------------------------------------------ + ------------------------------------------------
/ 2/pi a\\ / 2/pi a\\
| csc |-- - -|| | csc |-- - -||
| \2 2/| | \2 2/|
| 1 - ------------| | 1 - ------------|
/ 2/pi a\\ | 2/a\ | | 2/a\ | / 2/pi a\\
| csc |-- - -|| | csc |-| | | csc |-| | | csc |-- - -||
| \2 2/| | \2/ | /a\ | \2/ | | \2 2/| /a\
|1 + ------------|*|1 - ----------------|*csc|-| |1 + ----------------|*|1 + ------------|*csc|-|
| 2/a\ | | 2/pi a\| \2/ | 2/pi a\| | 2/a\ | \2/
| csc |-| | | csc |-- - -|| | csc |-- - -|| | csc |-| |
\ \2/ / | \2 2/| | \2 2/| \ \2/ /
| 1 + ------------| | 1 + ------------|
| 2/a\ | | 2/a\ |
| csc |-| | | csc |-| |
\ \2/ / \ \2/ /
$$\frac{2 \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \left(\frac{1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}} + 1\right) \csc{\left(\frac{a}{2} \right)}} - \frac{2 \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \left(- \frac{1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}} + 1\right) \csc{\left(\frac{a}{2} \right)}}$$
/ / 3*pi\ / / 3*pi\
| 1 for |a + ----| mod 2*pi = 0 | 1 for |a + ----| mod 2*pi = 0
| \ 2 / | \ 2 /
| |
| 2/a pi\ | 2/a pi\
<-1 + tan |- + --| <-1 + tan |- + --|
| \2 4 / | \2 4 /
|----------------- otherwise |----------------- otherwise
| 2/a pi\ | 2/a pi\
| 1 + tan |- + --| | 1 + tan |- + --|
\ \2 4 / \ \2 4 /
----------------------------------------------- - -----------------------------------------------
// 1 for a mod 2*pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
|| 2/a\ | || 2/a\ |
||-1 + cot |-| | ||-1 + cot |-| |
1 + |< \2/ | 1 - |< \2/ |
||------------ otherwise | ||------------ otherwise |
|| 2/a\ | || 2/a\ |
||1 + cot |-| | ||1 + cot |-| |
\\ \2/ / \\ \2/ /
$$\left(\frac{\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}\right) - \left(\frac{\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for a mod pi = 0 / 0 for a mod pi = 0
| |
| 2*sin(a) | 2*sin(a)
|---------------------------- otherwise |---------------------------- otherwise
| / 2 \ | / 2 \
< | sin (a) | < | sin (a) |
|(1 - cos(a))*|1 + ---------| |(1 - cos(a))*|1 + ---------|
| | 4/a\| | | 4/a\|
| | 4*sin |-|| | | 4*sin |-||
| \ \2// | \ \2//
\ \
------------------------------------------------------ - ------------------------------------------------------
// 1 for a mod 2*pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
|| 2 | || 2 |
1 + |< -4 + 4*sin (a) + 4*cos(a) | 1 - |< -4 + 4*sin (a) + 4*cos(a) |
||--------------------------- otherwise | ||--------------------------- otherwise |
|| 2 2 | || 2 2 |
\\2*(1 - cos(a)) + 2*sin (a) / \\2*(1 - cos(a)) + 2*sin (a) /
$$\left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \left(- \cos{\left(a \right)} + 1\right)} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{4 \sin^{2}{\left(a \right)} + 4 \cos{\left(a \right)} - 4}{2 \left(- \cos{\left(a \right)} + 1\right)^{2} + 2 \sin^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right) + 1}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \left(- \cos{\left(a \right)} + 1\right)} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{4 \sin^{2}{\left(a \right)} + 4 \cos{\left(a \right)} - 4}{2 \left(- \cos{\left(a \right)} + 1\right)^{2} + 2 \sin^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for a mod pi = 0 / 0 for a mod pi = 0
| |
| sin(a) | sin(a)
|----------------------- otherwise |----------------------- otherwise
|/ 2 \ |/ 2 \
<| sin (a) | 2/a\ <| sin (a) | 2/a\
||1 + ---------|*sin |-| ||1 + ---------|*sin |-|
|| 4/a\| \2/ || 4/a\| \2/
|| 4*sin |-|| || 4*sin |-||
|\ \2// |\ \2//
\ \
------------------------------------------ - ------------------------------------------
// 1 for a mod 2*pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
|| 2 | || 2 |
|| sin (a) | || sin (a) |
||-1 + --------- | ||-1 + --------- |
|| 4/a\ | || 4/a\ |
|| 4*sin |-| | || 4*sin |-| |
1 + |< \2/ | 1 - |< \2/ |
||-------------- otherwise | ||-------------- otherwise |
|| 2 | || 2 |
|| sin (a) | || sin (a) |
||1 + --------- | ||1 + --------- |
|| 4/a\ | || 4/a\ |
|| 4*sin |-| | || 4*sin |-| |
\\ \2/ / \\ \2/ /
$$\left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) + 1}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for a mod pi = 0 / 0 for a mod pi = 0
| |
|/ 0 for a mod pi = 0 |/ 0 for a mod pi = 0
|| ||
|| /a\ || /a\
<| 2*cot|-| <| 2*cot|-|
|< \2/ otherwise |< \2/ otherwise
||----------- otherwise ||----------- otherwise
|| 2/a\ || 2/a\
||1 + cot |-| ||1 + cot |-|
\\ \2/ \\ \2/
------------------------------------------------------------ - ------------------------------------------------------------
// 1 for a mod 2*pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
||/ 1 for a mod 2*pi = 0 | ||/ 1 for a mod 2*pi = 0 |
||| | ||| |
||| 2/a\ | ||| 2/a\ |
1 + |<|-1 + cot |-| | 1 - |<|-1 + cot |-| |
||< \2/ otherwise | ||< \2/ otherwise |
|||------------ otherwise | |||------------ otherwise |
||| 2/a\ | ||| 2/a\ |
|||1 + cot |-| | |||1 + cot |-| |
\\\ \2/ / \\\ \2/ /
$$\left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for a mod pi = 0 / 0 for a mod pi = 0
| |
| /a\ | /a\
| 2*cos|-| | 2*cos|-|
| \2/ | \2/
|------------------------------ otherwise |------------------------------ otherwise
$$\left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) + 1}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for a mod pi = 0 / 0 for a mod pi = 0
| |
| /a pi\ | /a pi\
| 2*sec|- - --| | 2*sec|- - --|
| \2 2 / | \2 2 /
|------------------------- otherwise |------------------------- otherwise
$$\left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right) \sec{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) + 1}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right) \sec{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right) + 1}\right)$$
/ 0 for a mod pi = 0 / 0 for a mod pi = 0
| |
| /a\ | /a\
| 2*csc|-| | 2*csc|-|
| \2/ | \2/
|------------------------------ otherwise |------------------------------ otherwise
$$\left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) + 1}\right) - \left(\frac{\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{\left(- \begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) + 1}\right)$$
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))/(1 + 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))) - 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))/(1 - 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)))
Численный ответ
[src]
sin(a)/(1.0 + cos(a)) - sin(a)/(1.0 - cos(a))
sin(a)/(1.0 + cos(a)) - sin(a)/(1.0 - cos(a))
Рациональный знаменатель
[src]
(1 - cos(a))*sin(a) - (1 + cos(a))*sin(a)
-----------------------------------------
(1 - cos(a))*(1 + cos(a))
$$\frac{\left(- \cos{\left(a \right)} + 1\right) \sin{\left(a \right)} - \left(\cos{\left(a \right)} + 1\right) \sin{\left(a \right)}}{\left(- \cos{\left(a \right)} + 1\right) \left(\cos{\left(a \right)} + 1\right)}$$
((1 - cos(a))*sin(a) - (1 + cos(a))*sin(a))/((1 - cos(a))*(1 + cos(a)))
Степени
[src]
/ -I*a I*a\ / -I*a I*a\ I*\- e + e / I*\- e + e / -------------------- - -------------------- / I*a -I*a\ / I*a -I*a\ | e e | | e e | 2*|1 - ---- - -----| 2*|1 + ---- + -----| \ 2 2 / \ 2 2 /
$$- \frac{i \left(e^{i a} - e^{- i a}\right)}{2 \left(\frac{e^{i a}}{2} + 1 + \frac{e^{- i a}}{2}\right)} + \frac{i \left(e^{i a} - e^{- i a}\right)}{2 \left(- \frac{e^{i a}}{2} + 1 - \frac{e^{- i a}}{2}\right)}$$
i*(-exp(-i*a) + exp(i*a))/(2*(1 - exp(i*a)/2 - exp(-i*a)/2)) - i*(-exp(-i*a) + exp(i*a))/(2*(1 + exp(i*a)/2 + exp(-i*a)/2))
Комбинаторика
[src]
2*cos(a)*sin(a) -------------------------- (1 + cos(a))*(-1 + cos(a))
$$\frac{2 \sin{\left(a \right)} \cos{\left(a \right)}}{\left(\cos{\left(a \right)} - 1\right) \left(\cos{\left(a \right)} + 1\right)}$$
2*cos(a)*sin(a)/((1 + cos(a))*(-1 + cos(a)))
Объединение рациональных выражений
[src]
-2*cos(a)*sin(a) ------------------------- (1 - cos(a))*(1 + cos(a))
$$- \frac{2 \sin{\left(a \right)} \cos{\left(a \right)}}{\left(- \cos{\left(a \right)} + 1\right) \left(\cos{\left(a \right)} + 1\right)}$$
-2*cos(a)*sin(a)/((1 - cos(a))*(1 + cos(a)))