Тригонометрическая часть
[src]
/a\ /a \
-2*sin|-|*sin|--|
\3/ \24/
$$- 2 \sin{\left(\frac{a}{24} \right)} \sin{\left(\frac{a}{3} \right)}$$
2/7*a\ 2/3*a\
- 2*cos |---| + 2*cos |---|
\ 48/ \ 16/
$$- 2 \cos^{2}{\left(\frac{7 a}{48} \right)} + 2 \cos^{2}{\left(\frac{3 a}{16} \right)}$$
1 1
-------- - --------
/3*a\ /7*a\
sec|---| sec|---|
\ 8 / \ 24/
$$\frac{1}{\sec{\left(\frac{3 a}{8} \right)}} - \frac{1}{\sec{\left(\frac{7 a}{24} \right)}}$$
/pi 7*a\ /pi 3*a\
- sin|-- + ---| + sin|-- + ---|
\2 24/ \2 8 /
$$- \sin{\left(\frac{7 a}{24} + \frac{\pi}{2} \right)} + \sin{\left(\frac{3 a}{8} + \frac{\pi}{2} \right)}$$
1 1
------------- - -------------
/pi 3*a\ /pi 7*a\
csc|-- - ---| csc|-- - ---|
\2 8 / \2 24/
$$- \frac{1}{\csc{\left(- \frac{7 a}{24} + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(- \frac{3 a}{8} + \frac{\pi}{2} \right)}}$$
2/3*a\ 2/7*a\
-1 + cot |---| -1 + cot |---|
\ 16/ \ 48/
-------------- - --------------
2/3*a\ 2/7*a\
1 + cot |---| 1 + cot |---|
\ 16/ \ 48/
$$- \frac{\cot^{2}{\left(\frac{7 a}{48} \right)} - 1}{\cot^{2}{\left(\frac{7 a}{48} \right)} + 1} + \frac{\cot^{2}{\left(\frac{3 a}{16} \right)} - 1}{\cot^{2}{\left(\frac{3 a}{16} \right)} + 1}$$
2/3*a\ 2/7*a\
1 - tan |---| 1 - tan |---|
\ 16/ \ 48/
------------- - -------------
2/3*a\ 2/7*a\
1 + tan |---| 1 + tan |---|
\ 16/ \ 48/
$$- \frac{- \tan^{2}{\left(\frac{7 a}{48} \right)} + 1}{\tan^{2}{\left(\frac{7 a}{48} \right)} + 1} + \frac{- \tan^{2}{\left(\frac{3 a}{16} \right)} + 1}{\tan^{2}{\left(\frac{3 a}{16} \right)} + 1}$$
1 1
1 - --------- 1 - ---------
2/3*a\ 2/7*a\
cot |---| cot |---|
\ 16/ \ 48/
------------- - -------------
1 1
1 + --------- 1 + ---------
2/3*a\ 2/7*a\
cot |---| cot |---|
\ 16/ \ 48/
$$- \frac{1 - \frac{1}{\cot^{2}{\left(\frac{7 a}{48} \right)}}}{1 + \frac{1}{\cot^{2}{\left(\frac{7 a}{48} \right)}}} + \frac{1 - \frac{1}{\cot^{2}{\left(\frac{3 a}{16} \right)}}}{1 + \frac{1}{\cot^{2}{\left(\frac{3 a}{16} \right)}}}$$
/pi 7*a\ /pi 3*a\
2*tan|-- + ---| 2*tan|-- + ---|
\4 48/ \4 16/
- ------------------ + ------------------
2/pi 7*a\ 2/pi 3*a\
1 + tan |-- + ---| 1 + tan |-- + ---|
\4 48/ \4 16/
$$\frac{2 \tan{\left(\frac{3 a}{16} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{3 a}{16} + \frac{\pi}{4} \right)} + 1} - \frac{2 \tan{\left(\frac{7 a}{48} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{7 a}{48} + \frac{\pi}{4} \right)} + 1}$$
// 7*a \ // 3*a \
|| 1 for --- mod 2*pi = 0| || 1 for --- mod 2*pi = 0|
|| 24 | || 8 |
- |< | + |< |
|| /7*a\ | || /3*a\ |
||cos|---| otherwise | ||cos|---| otherwise |
\\ \ 24/ / \\ \ 8 / /
$$\left(- \begin{cases} 1 & \text{for}\: \frac{7 a}{24} \bmod 2 \pi = 0 \\\cos{\left(\frac{7 a}{24} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \frac{3 a}{8} \bmod 2 \pi = 0 \\\cos{\left(\frac{3 a}{8} \right)} & \text{otherwise} \end{cases}\right)$$
// 7*a \ // 3*a \
|| 1 for --- mod 2*pi = 0| || 1 for --- mod 2*pi = 0|
|| 24 | || 8 |
|| | || |
- |< 1 | + |< 1 |
||-------- otherwise | ||-------- otherwise |
|| /7*a\ | || /3*a\ |
||sec|---| | ||sec|---| |
\\ \ 24/ / \\ \ 8 / /
$$\left(- \begin{cases} 1 & \text{for}\: \frac{7 a}{24} \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(\frac{7 a}{24} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \frac{3 a}{8} \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(\frac{3 a}{8} \right)}} & \text{otherwise} \end{cases}\right)$$
// 7*a \ // 3*a \
|| 1 for --- mod 2*pi = 0| || 1 for --- mod 2*pi = 0|
|| 24 | || 8 |
- |< | + |< |
|| /pi 7*a\ | || /pi 3*a\ |
||sin|-- + ---| otherwise | ||sin|-- + ---| otherwise |
\\ \2 24/ / \\ \2 8 / /
$$\left(- \begin{cases} 1 & \text{for}\: \frac{7 a}{24} \bmod 2 \pi = 0 \\\sin{\left(\frac{7 a}{24} + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \frac{3 a}{8} \bmod 2 \pi = 0 \\\sin{\left(\frac{3 a}{8} + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 7*a \ // 3*a \
|| 1 for --- mod 2*pi = 0| || 1 for --- mod 2*pi = 0|
|| 24 | || 8 |
|| | || |
- |< 1 | + |< 1 |
||------------- otherwise | ||------------- otherwise |
|| /pi 7*a\ | || /pi 3*a\ |
||csc|-- - ---| | ||csc|-- - ---| |
\\ \2 24/ / \\ \2 8 / /
$$\left(- \begin{cases} 1 & \text{for}\: \frac{7 a}{24} \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- \frac{7 a}{24} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \frac{3 a}{8} \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- \frac{3 a}{8} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
4/3*a\ 4/7*a\
4*sin |---| 4*sin |---|
\ 16/ \ 48/
1 - ----------- 1 - -----------
2/3*a\ 2/7*a\
sin |---| sin |---|
\ 8 / \ 24/
--------------- - ---------------
4/3*a\ 4/7*a\
4*sin |---| 4*sin |---|
\ 16/ \ 48/
1 + ----------- 1 + -----------
2/3*a\ 2/7*a\
sin |---| sin |---|
\ 8 / \ 24/
$$- \frac{- \frac{4 \sin^{4}{\left(\frac{7 a}{48} \right)}}{\sin^{2}{\left(\frac{7 a}{24} \right)}} + 1}{\frac{4 \sin^{4}{\left(\frac{7 a}{48} \right)}}{\sin^{2}{\left(\frac{7 a}{24} \right)}} + 1} + \frac{- \frac{4 \sin^{4}{\left(\frac{3 a}{16} \right)}}{\sin^{2}{\left(\frac{3 a}{8} \right)}} + 1}{\frac{4 \sin^{4}{\left(\frac{3 a}{16} \right)}}{\sin^{2}{\left(\frac{3 a}{8} \right)}} + 1}$$
// 7*a \ // 3*a \
|| 1 for --- mod 2*pi = 0| || 1 for --- mod 2*pi = 0|
|| 24 | || 8 |
|| | || |
|| 2/7*a\ | || 2/3*a\ |
- |<-1 + cot |---| | + |<-1 + cot |---| |
|| \ 48/ | || \ 16/ |
||-------------- otherwise | ||-------------- otherwise |
|| 2/7*a\ | || 2/3*a\ |
||1 + cot |---| | ||1 + cot |---| |
\\ \ 48/ / \\ \ 16/ /
$$\left(- \begin{cases} 1 & \text{for}\: \frac{7 a}{24} \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{7 a}{48} \right)} - 1}{\cot^{2}{\left(\frac{7 a}{48} \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \frac{3 a}{8} \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{3 a}{16} \right)} - 1}{\cot^{2}{\left(\frac{3 a}{16} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 7*a \ // 3*a \
|| 1 for --- mod 2*pi = 0| || 1 for --- mod 2*pi = 0|
|| 24 | || 8 |
|| | || |
|| 2/7*a\ | || 2/3*a\ |
- |<1 - tan |---| | + |<1 - tan |---| |
|| \ 48/ | || \ 16/ |
||------------- otherwise | ||------------- otherwise |
|| 2/7*a\ | || 2/3*a\ |
||1 + tan |---| | ||1 + tan |---| |
\\ \ 48/ / \\ \ 16/ /
$$\left(- \begin{cases} 1 & \text{for}\: \frac{7 a}{24} \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{7 a}{48} \right)} + 1}{\tan^{2}{\left(\frac{7 a}{48} \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \frac{3 a}{8} \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{3 a}{16} \right)} + 1}{\tan^{2}{\left(\frac{3 a}{16} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
/ /7*a\ /7*a\\ / /3*a\ /3*a\\
2*|-1 - cos|---| + 2*cos|---|| 2*|-1 - cos|---| + 2*cos|---||
\ \ 12/ \ 24// \ \ 4 / \ 8 //
- -------------------------------- + --------------------------------
2 2
/7*a\ / /7*a\\ /3*a\ / /3*a\\
1 - cos|---| + 2*|1 - cos|---|| 1 - cos|---| + 2*|1 - cos|---||
\ 12/ \ \ 24// \ 4 / \ \ 8 //
$$\frac{2 \cdot \left(2 \cos{\left(\frac{3 a}{8} \right)} - \cos{\left(\frac{3 a}{4} \right)} - 1\right)}{2 \left(- \cos{\left(\frac{3 a}{8} \right)} + 1\right)^{2} - \cos{\left(\frac{3 a}{4} \right)} + 1} - \frac{2 \cdot \left(2 \cos{\left(\frac{7 a}{24} \right)} - \cos{\left(\frac{7 a}{12} \right)} - 1\right)}{2 \left(- \cos{\left(\frac{7 a}{24} \right)} + 1\right)^{2} - \cos{\left(\frac{7 a}{12} \right)} + 1}$$
2/ pi 3*a\ 2/ pi 7*a\
cos |- -- + ---| cos |- -- + ---|
\ 2 16/ \ 2 48/
1 - ---------------- 1 - ----------------
2/3*a\ 2/7*a\
cos |---| cos |---|
\ 16/ \ 48/
-------------------- - --------------------
2/ pi 3*a\ 2/ pi 7*a\
cos |- -- + ---| cos |- -- + ---|
\ 2 16/ \ 2 48/
1 + ---------------- 1 + ----------------
2/3*a\ 2/7*a\
cos |---| cos |---|
\ 16/ \ 48/
$$- \frac{1 - \frac{\cos^{2}{\left(\frac{7 a}{48} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{7 a}{48} \right)}}}{1 + \frac{\cos^{2}{\left(\frac{7 a}{48} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{7 a}{48} \right)}}} + \frac{1 - \frac{\cos^{2}{\left(\frac{3 a}{16} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{3 a}{16} \right)}}}{1 + \frac{\cos^{2}{\left(\frac{3 a}{16} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{3 a}{16} \right)}}}$$
2/3*a\ 2/7*a\
sec |---| sec |---|
\ 16/ \ 48/
1 - ---------------- 1 - ----------------
2/ pi 3*a\ 2/ pi 7*a\
sec |- -- + ---| sec |- -- + ---|
\ 2 16/ \ 2 48/
-------------------- - --------------------
2/3*a\ 2/7*a\
sec |---| sec |---|
\ 16/ \ 48/
1 + ---------------- 1 + ----------------
2/ pi 3*a\ 2/ pi 7*a\
sec |- -- + ---| sec |- -- + ---|
\ 2 16/ \ 2 48/
$$- \frac{- \frac{\sec^{2}{\left(\frac{7 a}{48} \right)}}{\sec^{2}{\left(\frac{7 a}{48} - \frac{\pi}{2} \right)}} + 1}{\frac{\sec^{2}{\left(\frac{7 a}{48} \right)}}{\sec^{2}{\left(\frac{7 a}{48} - \frac{\pi}{2} \right)}} + 1} + \frac{- \frac{\sec^{2}{\left(\frac{3 a}{16} \right)}}{\sec^{2}{\left(\frac{3 a}{16} - \frac{\pi}{2} \right)}} + 1}{\frac{\sec^{2}{\left(\frac{3 a}{16} \right)}}{\sec^{2}{\left(\frac{3 a}{16} - \frac{\pi}{2} \right)}} + 1}$$
// 7*a \ // 3*a \
|| 1 for --- mod 2*pi = 0| || 1 for --- mod 2*pi = 0|
|| 24 | || 8 |
|| | || |
|| 1 | || 1 |
||-1 + --------- | ||-1 + --------- |
|| 2/7*a\ | || 2/3*a\ |
- |< tan |---| | + |< tan |---| |
|| \ 48/ | || \ 16/ |
||-------------- otherwise | ||-------------- otherwise |
|| 1 | || 1 |
||1 + --------- | ||1 + --------- |
|| 2/7*a\ | || 2/3*a\ |
|| tan |---| | || tan |---| |
\\ \ 48/ / \\ \ 16/ /
$$\left(- \begin{cases} 1 & \text{for}\: \frac{7 a}{24} \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{7 a}{48} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{7 a}{48} \right)}}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \frac{3 a}{8} \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{3 a}{16} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{3 a}{16} \right)}}} & \text{otherwise} \end{cases}\right)$$
2/pi 3*a\ 2/pi 7*a\
csc |-- - ---| csc |-- - ---|
\2 16/ \2 48/
1 - -------------- 1 - --------------
2/3*a\ 2/7*a\
csc |---| csc |---|
\ 16/ \ 48/
------------------ - ------------------
2/pi 3*a\ 2/pi 7*a\
csc |-- - ---| csc |-- - ---|
\2 16/ \2 48/
1 + -------------- 1 + --------------
2/3*a\ 2/7*a\
csc |---| csc |---|
\ 16/ \ 48/
$$- \frac{1 - \frac{\csc^{2}{\left(- \frac{7 a}{48} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{7 a}{48} \right)}}}{1 + \frac{\csc^{2}{\left(- \frac{7 a}{48} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{7 a}{48} \right)}}} + \frac{1 - \frac{\csc^{2}{\left(- \frac{3 a}{16} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{3 a}{16} \right)}}}{1 + \frac{\csc^{2}{\left(- \frac{3 a}{16} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{3 a}{16} \right)}}}$$
// 7*a \ // 3*a \
|| 1 for --- mod 2*pi = 0| || 1 for --- mod 2*pi = 0|
|| 24 | || 8 |
|| | || |
||/ 7*a | ||/ 3*a |
- |<| 1 for --- mod 2*pi = 0 | + |<| 1 for --- mod 2*pi = 0 |
||| 24 | ||| 8 |
||< otherwise | ||< otherwise |
||| /7*a\ | ||| /3*a\ |
|||cos|---| otherwise | |||cos|---| otherwise |
\\\ \ 24/ / \\\ \ 8 / /
$$\left(- \begin{cases} 1 & \text{for}\: \frac{7 a}{24} \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \frac{7 a}{24} \bmod 2 \pi = 0 \\\cos{\left(\frac{7 a}{24} \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \frac{3 a}{8} \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \frac{3 a}{8} \bmod 2 \pi = 0 \\\cos{\left(\frac{3 a}{8} \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// /pi 7*a\ \ // /pi 3*a\ \
|| 0 for |-- + ---| mod pi = 0| || 0 for |-- + ---| mod pi = 0|
|| \2 24/ | || \2 8 / |
- |< | + |< |
||/ /7*a\\ /pi 7*a\ | ||/ /3*a\\ /pi 3*a\ |
|||1 + sin|---||*cot|-- + ---| otherwise | |||1 + sin|---||*cot|-- + ---| otherwise |
\\\ \ 24// \4 48/ / \\\ \ 8 // \4 16/ /
$$\left(- \begin{cases} 0 & \text{for}\: \left(\frac{7 a}{24} + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(\frac{7 a}{24} \right)} + 1\right) \cot{\left(\frac{7 a}{48} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(\frac{3 a}{8} + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(\frac{3 a}{8} \right)} + 1\right) \cot{\left(\frac{3 a}{16} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)$$
// /pi 7*a\ \ // /pi 3*a\ \
|| 0 for |-- + ---| mod pi = 0| || 0 for |-- + ---| mod pi = 0|
|| \2 24/ | || \2 8 / |
|| | || |
|| /pi 7*a\ | || /pi 3*a\ |
- |< 2*cot|-- + ---| | + |< 2*cot|-- + ---| |
|| \4 48/ | || \4 16/ |
||------------------ otherwise | ||------------------ otherwise |
|| 2/pi 7*a\ | || 2/pi 3*a\ |
||1 + cot |-- + ---| | ||1 + cot |-- + ---| |
\\ \4 48/ / \\ \4 16/ /
$$\left(- \begin{cases} 0 & \text{for}\: \left(\frac{7 a}{24} + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{7 a}{48} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{7 a}{48} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 0 & \text{for}\: \left(\frac{3 a}{8} + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 a}{16} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{3 a}{16} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 7*a \ // 3*a \
|| 1 for --- mod 2*pi = 0| || 1 for --- mod 2*pi = 0|
|| 24 | || 8 |
|| | || |
|| /7*a\ /7*a\ | || /3*a\ /3*a\ |
|| -2 - 2*cos|---| + 4*cos|---| | || -2 - 2*cos|---| + 4*cos|---| |
- |< \ 12/ \ 24/ | + |< \ 4 / \ 8 / |
||-------------------------------- otherwise | ||-------------------------------- otherwise |
|| 2 | || 2 |
|| /7*a\ / /7*a\\ | || /3*a\ / /3*a\\ |
||1 - cos|---| + 2*|1 - cos|---|| | ||1 - cos|---| + 2*|1 - cos|---|| |
|| \ 12/ \ \ 24// | || \ 4 / \ \ 8 // |
\\ / \\ /
$$\left(- \begin{cases} 1 & \text{for}\: \frac{7 a}{24} \bmod 2 \pi = 0 \\\frac{4 \cos{\left(\frac{7 a}{24} \right)} - 2 \cos{\left(\frac{7 a}{12} \right)} - 2}{2 \left(- \cos{\left(\frac{7 a}{24} \right)} + 1\right)^{2} - \cos{\left(\frac{7 a}{12} \right)} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \frac{3 a}{8} \bmod 2 \pi = 0 \\\frac{4 \cos{\left(\frac{3 a}{8} \right)} - 2 \cos{\left(\frac{3 a}{4} \right)} - 2}{2 \left(- \cos{\left(\frac{3 a}{8} \right)} + 1\right)^{2} - \cos{\left(\frac{3 a}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 7*a \ // 3*a \
|| 1 for --- mod 2*pi = 0| || 1 for --- mod 2*pi = 0|
|| 24 | || 8 |
|| | || |
||/ 7*a | ||/ 3*a |
||| 1 for --- mod 2*pi = 0 | ||| 1 for --- mod 2*pi = 0 |
||| 24 | ||| 8 |
- |<| | + |<| |
||| 2/7*a\ | ||| 2/3*a\ |
||<-1 + cot |---| otherwise | ||<-1 + cot |---| otherwise |
||| \ 48/ | ||| \ 16/ |
|||-------------- otherwise | |||-------------- otherwise |
||| 2/7*a\ | ||| 2/3*a\ |
|||1 + cot |---| | |||1 + cot |---| |
\\\ \ 48/ / \\\ \ 16/ /
$$\left(- \begin{cases} 1 & \text{for}\: \frac{7 a}{24} \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \frac{7 a}{24} \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{7 a}{48} \right)} - 1}{\cot^{2}{\left(\frac{7 a}{48} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \frac{3 a}{8} \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: \frac{3 a}{8} \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{3 a}{16} \right)} - 1}{\cot^{2}{\left(\frac{3 a}{16} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 7*a \ // 3*a \
|| 1 for --- mod 2*pi = 0| || 1 for --- mod 2*pi = 0|
|| 24 | || 8 |
|| | || |
|| 2/7*a\ | || 2/3*a\ |
|| sin |---| | || sin |---| |
|| \ 24/ | || \ 8 / |
||-1 + ----------- | ||-1 + ----------- |
|| 4/7*a\ | || 4/3*a\ |
- |< 4*sin |---| | + |< 4*sin |---| |
|| \ 48/ | || \ 16/ |
||---------------- otherwise | ||---------------- otherwise |
|| 2/7*a\ | || 2/3*a\ |
|| sin |---| | || sin |---| |
|| \ 24/ | || \ 8 / |
||1 + ----------- | ||1 + ----------- |
|| 4/7*a\ | || 4/3*a\ |
|| 4*sin |---| | || 4*sin |---| |
\\ \ 48/ / \\ \ 16/ /
$$\left(- \begin{cases} 1 & \text{for}\: \frac{7 a}{24} \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(\frac{7 a}{24} \right)}}{4 \sin^{4}{\left(\frac{7 a}{48} \right)}}}{1 + \frac{\sin^{2}{\left(\frac{7 a}{24} \right)}}{4 \sin^{4}{\left(\frac{7 a}{48} \right)}}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \frac{3 a}{8} \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(\frac{3 a}{8} \right)}}{4 \sin^{4}{\left(\frac{3 a}{16} \right)}}}{1 + \frac{\sin^{2}{\left(\frac{3 a}{8} \right)}}{4 \sin^{4}{\left(\frac{3 a}{16} \right)}}} & \text{otherwise} \end{cases}\right)$$
// 7*a \ // 3*a \
|| 1 for --- mod 2*pi = 0| || 1 for --- mod 2*pi = 0|
|| 24 | || 8 |
|| | || |
|| 2/7*a\ | || 2/3*a\ |
|| cos |---| | || cos |---| |
|| \ 48/ | || \ 16/ |
||-1 + ---------------- | ||-1 + ---------------- |
|| 2/ pi 7*a\ | || 2/ pi 3*a\ |
- |< cos |- -- + ---| | + |< cos |- -- + ---| |
|| \ 2 48/ | || \ 2 16/ |
||--------------------- otherwise | ||--------------------- otherwise |
|| 2/7*a\ | || 2/3*a\ |
|| cos |---| | || cos |---| |
|| \ 48/ | || \ 16/ |
|| 1 + ---------------- | || 1 + ---------------- |
|| 2/ pi 7*a\ | || 2/ pi 3*a\ |
|| cos |- -- + ---| | || cos |- -- + ---| |
\\ \ 2 48/ / \\ \ 2 16/ /
$$\left(- \begin{cases} 1 & \text{for}\: \frac{7 a}{24} \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{7 a}{48} \right)}}{\cos^{2}{\left(\frac{7 a}{48} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{7 a}{48} \right)}}{\cos^{2}{\left(\frac{7 a}{48} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \frac{3 a}{8} \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{3 a}{16} \right)}}{\cos^{2}{\left(\frac{3 a}{16} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{3 a}{16} \right)}}{\cos^{2}{\left(\frac{3 a}{16} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)$$
// 7*a \ // 3*a \
|| 1 for --- mod 2*pi = 0| || 1 for --- mod 2*pi = 0|
|| 24 | || 8 |
|| | || |
|| 2/ pi 7*a\ | || 2/ pi 3*a\ |
|| sec |- -- + ---| | || sec |- -- + ---| |
|| \ 2 48/ | || \ 2 16/ |
||-1 + ---------------- | ||-1 + ---------------- |
|| 2/7*a\ | || 2/3*a\ |
- |< sec |---| | + |< sec |---| |
|| \ 48/ | || \ 16/ |
||--------------------- otherwise | ||--------------------- otherwise |
|| 2/ pi 7*a\ | || 2/ pi 3*a\ |
|| sec |- -- + ---| | || sec |- -- + ---| |
|| \ 2 48/ | || \ 2 16/ |
|| 1 + ---------------- | || 1 + ---------------- |
|| 2/7*a\ | || 2/3*a\ |
|| sec |---| | || sec |---| |
\\ \ 48/ / \\ \ 16/ /
$$\left(- \begin{cases} 1 & \text{for}\: \frac{7 a}{24} \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{7 a}{48} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{7 a}{48} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{7 a}{48} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{7 a}{48} \right)}}} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \frac{3 a}{8} \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{3 a}{16} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{3 a}{16} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{3 a}{16} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{3 a}{16} \right)}}} & \text{otherwise} \end{cases}\right)$$
// 7*a \ // 3*a \
|| 1 for --- mod 2*pi = 0| || 1 for --- mod 2*pi = 0|
|| 24 | || 8 |
|| | || |
|| 2/7*a\ | || 2/3*a\ |
|| csc |---| | || csc |---| |
|| \ 48/ | || \ 16/ |
||-1 + -------------- | ||-1 + -------------- |
|| 2/pi 7*a\ | || 2/pi 3*a\ |
- |< csc |-- - ---| | + |< csc |-- - ---| |
|| \2 48/ | || \2 16/ |
||------------------- otherwise | ||------------------- otherwise |
|| 2/7*a\ | || 2/3*a\ |
|| csc |---| | || csc |---| |
|| \ 48/ | || \ 16/ |
|| 1 + -------------- | || 1 + -------------- |
|| 2/pi 7*a\ | || 2/pi 3*a\ |
|| csc |-- - ---| | || csc |-- - ---| |
\\ \2 48/ / \\ \2 16/ /
$$\left(- \begin{cases} 1 & \text{for}\: \frac{7 a}{24} \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{7 a}{48} \right)}}{\csc^{2}{\left(- \frac{7 a}{48} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{7 a}{48} \right)}}{\csc^{2}{\left(- \frac{7 a}{48} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} 1 & \text{for}\: \frac{3 a}{8} \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{3 a}{16} \right)}}{\csc^{2}{\left(- \frac{3 a}{16} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{3 a}{16} \right)}}{\csc^{2}{\left(- \frac{3 a}{16} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)$$
-Piecewise((1, Mod(7*a/24 = 2*pi, 0)), ((-1 + csc(7*a/48)^2/csc(pi/2 - 7*a/48)^2)/(1 + csc(7*a/48)^2/csc(pi/2 - 7*a/48)^2), True)) + Piecewise((1, Mod(3*a/8 = 2*pi, 0)), ((-1 + csc(3*a/16)^2/csc(pi/2 - 3*a/16)^2)/(1 + csc(3*a/16)^2/csc(pi/2 - 3*a/16)^2), True))