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