Господин Экзамен
Другие калькуляторы
-
Как пользоваться?
- cos(5*a)*cos(3*a)+sin(5*a)*sin(3*a) если a=-4
- b^(5/6)*3*4*sqrt(b^3) если b=-3/2
- sin(2*x)*cos(2*x) если x=3/2
- x^2+5*x-14 если x=-2
- Разложить многочлен на множители:
- x^2-12*x+20
- 8*x^3+36*x^2*y+54*x*y^2+27*y^3
- x^3+64
- p-p^2
- Общий знаменатель:
- m^2/(m^2-9)-m/(m+3)
- 100*t^2/(100*t^2-4)+40*t/((10*t-2)*(10*t+2))+4/(100*t^2-4)
- 5*b/18*a+a/6*b
- (a+4/a-4-a-4/a+4)/48*a/16-a^2
- Умножение столбиком:
- 25*15
- 14*50
- 48*15
- 40*12
- Деление столбиком:
- 54/8
- 87/9
- 24/6
- 637/5
- Раскрыть скобки в:
- z*(-z)*(-z)^3
- 70*(71^9+71^8+71^7+71^6+71^5+71^4+71^3+71^2+71)+71
- (a+2*b)^10*(a+2*b)
- 5*(8-c)+11*c
- Разложение числа на простые множители:
- 21780
- 10850
- 45630
- 65
- Производная:
- -4
- Интеграл d{x}:
-
-4
- График функции y =:
- -4
-
Идентичные выражения
- cos(пять *a)*cos(три *a)+sin(пять *a)*sin(три *a) если a=- четыре
- косинус от (5 умножить на a) умножить на косинус от (3 умножить на a) плюс синус от (5 умножить на a) умножить на синус от (3 умножить на a) если a равно минус 4
- косинус от (пять умножить на a) умножить на косинус от (три умножить на a) плюс синус от (пять умножить на a) умножить на синус от (три умножить на a) если a равно минус четыре
- cos(5a)cos(3a)+sin(5a)sin(3a) если a=-4
- cos5acos3a+sin5asin3a если a=-4
- cos(5*a)*cos(3*a)+sin(5*a)*sin(3*a) при a=-4
-
Похожие выражения
- cos(5*a)*cos(3*a)+sin(5*a)*sin(3*a) если a=+4
- cos(5*a)*cos(3*a)-sin(5*a)*sin(3*a) если a=-4
cos(5*a)*cos(3*a)+sin(5*a)*sin(3*a) если a=-4
Решение
Вы ввели
[src]
cos(5*a)*cos(3*a) + sin(5*a)*sin(3*a)
$$\sin{\left(3 a \right)} \sin{\left(5 a \right)} + \cos{\left(3 a \right)} \cos{\left(5 a \right)}$$
cos(5*a)*cos(3*a) + sin(5*a)*sin(3*a)
Подстановка условия
[src]
cos(5*a)*cos(3*a) + sin(5*a)*sin(3*a) при a = -4
подставляем
cos(5*a)*cos(3*a) + sin(5*a)*sin(3*a)
$$\sin{\left(3 a \right)} \sin{\left(5 a \right)} + \cos{\left(3 a \right)} \cos{\left(5 a \right)}$$
cos(2*a)
$$\cos{\left(2 a \right)}$$
переменные
a = -4
$$a = -4$$
cos(2*(-4))
$$\cos{\left(2 (-4) \right)}$$
cos(8)
$$\cos{\left(8 \right)}$$
cos(8)
Степени
[src]
/ -5*I*a 5*I*a\ / -3*I*a 3*I*a\ / -5*I*a 5*I*a\ / -3*I*a 3*I*a\ |e e | |e e | \- e + e /*\- e + e / |------- + ------|*|------- + ------| - ----------------------------------------- \ 2 2 / \ 2 2 / 4
$$\left(\frac{e^{3 i a}}{2} + \frac{e^{- 3 i a}}{2}\right) \left(\frac{e^{5 i a}}{2} + \frac{e^{- 5 i a}}{2}\right) - \frac{\left(e^{3 i a} - e^{- 3 i a}\right) \left(e^{5 i a} - e^{- 5 i a}\right)}{4}$$
(exp(-5*i*a)/2 + exp(5*i*a)/2)*(exp(-3*i*a)/2 + exp(3*i*a)/2) - (-exp(-5*i*a) + exp(5*i*a))*(-exp(-3*i*a) + exp(3*i*a))/4
Тригонометрическая часть
[src]
cos(2*a)
$$\cos{\left(2 a \right)}$$
1 -------- sec(2*a)
$$\frac{1}{\sec{\left(2 a \right)}}$$
/pi \ sin|-- + 2*a| \2 /
$$\sin{\left(2 a + \frac{\pi}{2} \right)}$$
1 ------------- /pi \ csc|-- - 2*a| \2 /
$$\frac{1}{\csc{\left(- 2 a + \frac{\pi}{2} \right)}}$$
2
1 - tan (a)
-----------
2
1 + tan (a)
$$\frac{- \tan^{2}{\left(a \right)} + 1}{\tan^{2}{\left(a \right)} + 1}$$
/ 1 for a mod pi = 0 < \cos(2*a) otherwise
$$\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(2 a \right)} & \text{otherwise} \end{cases}$$
1 1 ----------------- + ----------------- csc(3*a)*csc(5*a) sec(3*a)*sec(5*a)
$$\frac{1}{\sec{\left(3 a \right)} \sec{\left(5 a \right)}} + \frac{1}{\csc{\left(3 a \right)} \csc{\left(5 a \right)}}$$
/ pi\ / pi\
cos(3*a)*cos(5*a) + cos|3*a - --|*cos|5*a - --|
\ 2 / \ 2 /
$$\cos{\left(3 a \right)} \cos{\left(5 a \right)} + \cos{\left(3 a - \frac{\pi}{2} \right)} \cos{\left(5 a - \frac{\pi}{2} \right)}$$
/pi \ /pi \
sin(3*a)*sin(5*a) + sin|-- + 3*a|*sin|-- + 5*a|
\2 / \2 /
$$\sin{\left(3 a \right)} \sin{\left(5 a \right)} + \sin{\left(3 a + \frac{\pi}{2} \right)} \sin{\left(5 a + \frac{\pi}{2} \right)}$$
/ 1 for a mod pi = 0 | | 2 <-1 + cot (a) |------------ otherwise | 2 \1 + cot (a)
$$\begin{cases} 1 & \text{for}\: a \bmod \pi = 0 \\\frac{\cot^{2}{\left(a \right)} - 1}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}$$
1 1
----------------- + ---------------------------
sec(3*a)*sec(5*a) / pi\ / pi\
sec|3*a - --|*sec|5*a - --|
\ 2 / \ 2 /
$$\frac{1}{\sec{\left(3 a - \frac{\pi}{2} \right)} \sec{\left(5 a - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(3 a \right)} \sec{\left(5 a \right)}}$$
1 1
----------------- + ---------------------------
csc(3*a)*csc(5*a) /pi \ /pi \
csc|-- - 5*a|*csc|-- - 3*a|
\2 / \2 /
$$\frac{1}{\csc{\left(- 5 a + \frac{\pi}{2} \right)} \csc{\left(- 3 a + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(3 a \right)} \csc{\left(5 a \right)}}$$
1 1
----------------- + ---------------------------
sec(3*a)*sec(5*a) /pi \ /pi \
sec|-- - 5*a|*sec|-- - 3*a|
\2 / \2 /
$$\frac{1}{\sec{\left(- 5 a + \frac{\pi}{2} \right)} \sec{\left(- 3 a + \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(3 a \right)} \sec{\left(5 a \right)}}$$
1 1
--------------------------- + ---------------------------
csc(pi - 5*a)*csc(pi - 3*a) /pi \ /pi \
csc|-- - 5*a|*csc|-- - 3*a|
\2 / \2 /
$$\frac{1}{\csc{\left(- 5 a + \frac{\pi}{2} \right)} \csc{\left(- 3 a + \frac{\pi}{2} \right)}} + \frac{1}{\csc{\left(- 5 a + \pi \right)} \csc{\left(- 3 a + \pi \right)}}$$
15 3 4 2 cos(8*a) - -- - 16*(1 - cos(2*a)) + 4*(1 - cos(2*a)) + 9*cos(2*a) + 20*(1 - cos(2*a)) - -------- 2 2
$$4 \left(- \cos{\left(2 a \right)} + 1\right)^{4} - 16 \left(- \cos{\left(2 a \right)} + 1\right)^{3} + 20 \left(- \cos{\left(2 a \right)} + 1\right)^{2} + 9 \cos{\left(2 a \right)} - \frac{\cos{\left(8 a \right)}}{2} - \frac{15}{2}$$
15 cos(8*a) 2 4 3 -- + -------- - 20*(1 - cos(2*a)) - 7*cos(2*a) - 4*(1 - cos(2*a)) + 16*(1 - cos(2*a)) 2 2
$$- 4 \left(- \cos{\left(2 a \right)} + 1\right)^{4} + 16 \left(- \cos{\left(2 a \right)} + 1\right)^{3} - 20 \left(- \cos{\left(2 a \right)} + 1\right)^{2} - 7 \cos{\left(2 a \right)} + \frac{\cos{\left(8 a \right)}}{2} + \frac{15}{2}$$
/3*a\ /5*a\
cos(3*a)*cos(5*a) + (1 + cos(3*a)*cos(5*a) + cos(3*a) + cos(5*a))*tan|---|*tan|---|
\ 2 / \ 2 /
$$\left(\cos{\left(3 a \right)} \cos{\left(5 a \right)} + \cos{\left(3 a \right)} + \cos{\left(5 a \right)} + 1\right) \tan{\left(\frac{3 a}{2} \right)} \tan{\left(\frac{5 a}{2} \right)} + \cos{\left(3 a \right)} \cos{\left(5 a \right)}$$
/ 2/pi 3*a\\ / 2/pi 5*a\\
|1 - cot |-- + ---||*|1 - cot |-- + ---||*(1 + sin(3*a))*(1 + sin(5*a))
cos(2*a) + cos(8*a) \ \4 2 // \ \4 2 //
------------------- + -----------------------------------------------------------------------
2 4
$$\frac{\left(- \cot^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(- \cot^{2}{\left(\frac{5 a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(3 a \right)} + 1\right) \left(\sin{\left(5 a \right)} + 1\right)}{4} + \frac{\cos{\left(2 a \right)} + \cos{\left(8 a \right)}}{2}$$
/ 2/3*a\\ / 2/5*a\\ /3*a\ /5*a\ |1 - tan |---||*|1 - tan |---|| 4*tan|---|*tan|---| \ \ 2 // \ \ 2 // \ 2 / \ 2 / ------------------------------- + ------------------------------- / 2/3*a\\ / 2/5*a\\ / 2/3*a\\ / 2/5*a\\ |1 + tan |---||*|1 + tan |---|| |1 + tan |---||*|1 + tan |---|| \ \ 2 // \ \ 2 // \ \ 2 // \ \ 2 //
$$\frac{\left(- \tan^{2}{\left(\frac{3 a}{2} \right)} + 1\right) \left(- \tan^{2}{\left(\frac{5 a}{2} \right)} + 1\right)}{\left(\tan^{2}{\left(\frac{3 a}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{5 a}{2} \right)} + 1\right)} + \frac{4 \tan{\left(\frac{3 a}{2} \right)} \tan{\left(\frac{5 a}{2} \right)}}{\left(\tan^{2}{\left(\frac{3 a}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{5 a}{2} \right)} + 1\right)}$$
/3*a\ /5*a\ /pi 3*a\ /pi 5*a\
4*tan|---|*tan|---| 4*tan|-- + ---|*tan|-- + ---|
\ 2 / \ 2 / \4 2 / \4 2 /
------------------------------- + -----------------------------------------
/ 2/3*a\\ / 2/5*a\\ / 2/pi 3*a\\ / 2/pi 5*a\\
|1 + tan |---||*|1 + tan |---|| |1 + tan |-- + ---||*|1 + tan |-- + ---||
\ \ 2 // \ \ 2 // \ \4 2 // \ \4 2 //
$$\frac{4 \tan{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} \tan{\left(\frac{5 a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\tan^{2}{\left(\frac{5 a}{2} + \frac{\pi}{4} \right)} + 1\right)} + \frac{4 \tan{\left(\frac{3 a}{2} \right)} \tan{\left(\frac{5 a}{2} \right)}}{\left(\tan^{2}{\left(\frac{3 a}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{5 a}{2} \right)} + 1\right)}$$
/3*a\ /5*a\ /pi 3*a\ /pi 5*a\
4*cot|---|*cot|---| 4*tan|-- + ---|*tan|-- + ---|
\ 2 / \ 2 / \4 2 / \4 2 /
------------------------------- + -----------------------------------------
/ 2/3*a\\ / 2/5*a\\ / 2/pi 3*a\\ / 2/pi 5*a\\
|1 + cot |---||*|1 + cot |---|| |1 + tan |-- + ---||*|1 + tan |-- + ---||
\ \ 2 // \ \ 2 // \ \4 2 // \ \4 2 //
$$\frac{4 \cot{\left(\frac{3 a}{2} \right)} \cot{\left(\frac{5 a}{2} \right)}}{\left(\cot^{2}{\left(\frac{3 a}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{5 a}{2} \right)} + 1\right)} + \frac{4 \tan{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} \tan{\left(\frac{5 a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\tan^{2}{\left(\frac{5 a}{2} + \frac{\pi}{4} \right)} + 1\right)}$$
/ 1 \ / 1 \ |1 - ---------|*|1 - ---------| | 2/3*a\| | 2/5*a\| | cot |---|| | cot |---|| \ \ 2 // \ \ 2 // 4 ------------------------------- + ------------------------------------------------- / 1 \ / 1 \ / 1 \ / 1 \ /3*a\ /5*a\ |1 + ---------|*|1 + ---------| |1 + ---------|*|1 + ---------|*cot|---|*cot|---| | 2/3*a\| | 2/5*a\| | 2/3*a\| | 2/5*a\| \ 2 / \ 2 / | cot |---|| | cot |---|| | cot |---|| | cot |---|| \ \ 2 // \ \ 2 // \ \ 2 // \ \ 2 //
$$\frac{\left(1 - \frac{1}{\cot^{2}{\left(\frac{3 a}{2} \right)}}\right) \left(1 - \frac{1}{\cot^{2}{\left(\frac{5 a}{2} \right)}}\right)}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{3 a}{2} \right)}}\right) \left(1 + \frac{1}{\cot^{2}{\left(\frac{5 a}{2} \right)}}\right)} + \frac{4}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{3 a}{2} \right)}}\right) \left(1 + \frac{1}{\cot^{2}{\left(\frac{5 a}{2} \right)}}\right) \cot{\left(\frac{3 a}{2} \right)} \cot{\left(\frac{5 a}{2} \right)}}$$
// 0 for 3*a mod pi = 0\ // 0 for 5*a mod pi = 0\ // 1 for 3*a mod 2*pi = 0\ // 1 for 5*a mod 2*pi = 0\ |< |*|< | + |< |*|< | \\sin(3*a) otherwise / \\sin(5*a) otherwise / \\cos(3*a) otherwise / \\cos(5*a) otherwise /
$$\left(\left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\sin{\left(3 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 5 a \bmod \pi = 0 \\\sin{\left(5 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\cos{\left(3 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 a \bmod 2 \pi = 0 \\\cos{\left(5 a \right)} & \text{otherwise} \end{cases}\right)\right)$$
/ 2/3*a\\ / 2/5*a\\ / 2/pi 3*a\\ / 2/pi 5*a\\ |-1 + cot |---||*|-1 + cot |---|| |-1 + tan |-- + ---||*|-1 + tan |-- + ---|| \ \ 2 // \ \ 2 // \ \4 2 // \ \4 2 // --------------------------------- + ------------------------------------------- / 2/3*a\\ / 2/5*a\\ / 2/pi 3*a\\ / 2/pi 5*a\\ |1 + cot |---||*|1 + cot |---|| |1 + tan |-- + ---||*|1 + tan |-- + ---|| \ \ 2 // \ \ 2 // \ \4 2 // \ \4 2 //
$$\frac{\left(\tan^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} - 1\right) \left(\tan^{2}{\left(\frac{5 a}{2} + \frac{\pi}{4} \right)} - 1\right)}{\left(\tan^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\tan^{2}{\left(\frac{5 a}{2} + \frac{\pi}{4} \right)} + 1\right)} + \frac{\left(\cot^{2}{\left(\frac{3 a}{2} \right)} - 1\right) \left(\cot^{2}{\left(\frac{5 a}{2} \right)} - 1\right)}{\left(\cot^{2}{\left(\frac{3 a}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{5 a}{2} \right)} + 1\right)}$$
/ 2/pi 3*a\\ / 2/pi 5*a\\ / 2/3*a\\ / 2/5*a\\ |1 - cot |-- + ---||*|1 - cot |-- + ---|| |1 - tan |---||*|1 - tan |---|| \ \4 2 // \ \4 2 // \ \ 2 // \ \ 2 // ----------------------------------------- + ------------------------------- / 2/pi 3*a\\ / 2/pi 5*a\\ / 2/3*a\\ / 2/5*a\\ |1 + cot |-- + ---||*|1 + cot |-- + ---|| |1 + tan |---||*|1 + tan |---|| \ \4 2 // \ \4 2 // \ \ 2 // \ \ 2 //
$$\frac{\left(- \tan^{2}{\left(\frac{3 a}{2} \right)} + 1\right) \left(- \tan^{2}{\left(\frac{5 a}{2} \right)} + 1\right)}{\left(\tan^{2}{\left(\frac{3 a}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{5 a}{2} \right)} + 1\right)} + \frac{\left(- \cot^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(- \cot^{2}{\left(\frac{5 a}{2} + \frac{\pi}{4} \right)} + 1\right)}{\left(\cot^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{5 a}{2} + \frac{\pi}{4} \right)} + 1\right)}$$
// 0 for 3*a mod pi = 0\ // 0 for 5*a mod pi = 0\ || | || | // 1 for 3*a mod 2*pi = 0\ // 1 for 5*a mod 2*pi = 0\ |< / pi\ |*|< / pi\ | + |< |*|< | ||cos|3*a - --| otherwise | ||cos|5*a - --| otherwise | \\cos(3*a) otherwise / \\cos(5*a) otherwise / \\ \ 2 / / \\ \ 2 / /
$$\left(\left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\cos{\left(3 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 5 a \bmod \pi = 0 \\\cos{\left(5 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\cos{\left(3 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 a \bmod 2 \pi = 0 \\\cos{\left(5 a \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 3*a mod 2*pi = 0\ // 1 for 5*a mod 2*pi = 0\
// 0 for 3*a mod pi = 0\ // 0 for 5*a mod pi = 0\ || | || |
|< |*|< | + |< /pi \ |*|< /pi \ |
\\sin(3*a) otherwise / \\sin(5*a) otherwise / ||sin|-- + 3*a| otherwise | ||sin|-- + 5*a| otherwise |
\\ \2 / / \\ \2 / /
$$\left(\left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\sin{\left(3 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 5 a \bmod \pi = 0 \\\sin{\left(5 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\sin{\left(3 a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 a \bmod 2 \pi = 0 \\\sin{\left(5 a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right)$$
// / 3*pi\ \ // / 3*pi\ \
// 1 for 3*a mod 2*pi = 0\ // 1 for 5*a mod 2*pi = 0\ || 1 for |3*a + ----| mod 2*pi = 0| || 1 for |5*a + ----| mod 2*pi = 0|
|< |*|< | + |< \ 2 / |*|< \ 2 / |
\\cos(3*a) otherwise / \\cos(5*a) otherwise / || | || |
\\sin(3*a) otherwise / \\sin(5*a) otherwise /
$$\left(\left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\cos{\left(3 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 a \bmod 2 \pi = 0 \\\cos{\left(5 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: \left(3 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(3 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(5 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(5 a \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for 3*a mod pi = 0\ // 0 for 5*a mod pi = 0\ || | || | // 1 for 3*a mod 2*pi = 0\ // 1 for 5*a mod 2*pi = 0\ || 1 | || 1 | || | || | |<------------- otherwise |*|<------------- otherwise | + |< 1 |*|< 1 | || / pi\ | || / pi\ | ||-------- otherwise | ||-------- otherwise | ||sec|3*a - --| | ||sec|5*a - --| | \\sec(3*a) / \\sec(5*a) / \\ \ 2 / / \\ \ 2 / /
$$\left(\left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{1}{\sec{\left(3 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 5 a \bmod \pi = 0 \\\frac{1}{\sec{\left(5 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(3 a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 a \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(5 a \right)}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 3*a mod 2*pi = 0\ // 1 for 5*a mod 2*pi = 0\
// 0 for 3*a mod pi = 0\ // 0 for 5*a mod pi = 0\ || | || |
|| | || | || 1 | || 1 |
|< 1 |*|< 1 | + |<------------- otherwise |*|<------------- otherwise |
||-------- otherwise | ||-------- otherwise | || /pi \ | || /pi \ |
\\csc(3*a) / \\csc(5*a) / ||csc|-- - 3*a| | ||csc|-- - 5*a| |
\\ \2 / / \\ \2 / /
$$\left(\left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{1}{\csc{\left(3 a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 5 a \bmod \pi = 0 \\\frac{1}{\csc{\left(5 a \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- 3 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 a \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- 5 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for 3*a mod pi = 0\ // 0 for 5*a mod pi = 0\ || | || | ||1 - cos(3*a) | ||1 - cos(5*a) | // 1 for 3*a mod 2*pi = 0\ // 1 for 5*a mod 2*pi = 0\ |<------------ otherwise |*|<------------ otherwise | + |< |*|< | || /3*a\ | || /5*a\ | \\cos(3*a) otherwise / \\cos(5*a) otherwise / || tan|---| | || tan|---| | \\ \ 2 / / \\ \ 2 / /
$$\left(\left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{- \cos{\left(3 a \right)} + 1}{\tan{\left(\frac{3 a}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 5 a \bmod \pi = 0 \\\frac{- \cos{\left(5 a \right)} + 1}{\tan{\left(\frac{5 a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\cos{\left(3 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 a \bmod 2 \pi = 0 \\\cos{\left(5 a \right)} & \text{otherwise} \end{cases}\right)\right)$$
// /pi \ \ // /pi \ \
|| 0 for |-- + 3*a| mod pi = 0| || 0 for |-- + 5*a| mod pi = 0|
// 0 for 3*a mod pi = 0\ // 0 for 5*a mod pi = 0\ || \2 / | || \2 / |
|< |*|< | + |< |*|< |
\\sin(3*a) otherwise / \\sin(5*a) otherwise / || /pi 3*a\ | || /pi 5*a\ |
||(1 + sin(3*a))*cot|-- + ---| otherwise | ||(1 + sin(5*a))*cot|-- + ---| otherwise |
\\ \4 2 / / \\ \4 2 / /
$$\left(\left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\sin{\left(3 a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 5 a \bmod \pi = 0 \\\sin{\left(5 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: \left(3 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(3 a \right)} + 1\right) \cot{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(5 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(5 a \right)} + 1\right) \cot{\left(\frac{5 a}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right)$$
/ 4/3*a\\ / 4/5*a\\ | 4*sin |---|| | 4*sin |---|| | \ 2 /| | \ 2 /| |1 - -----------|*|1 - -----------| 2/3*a\ 2/5*a\ | 2 | | 2 | 16*sin |---|*sin |---| \ sin (3*a) / \ sin (5*a) / \ 2 / \ 2 / ----------------------------------- + ----------------------------------------------------- / 4/3*a\\ / 4/5*a\\ / 4/3*a\\ / 4/5*a\\ | 4*sin |---|| | 4*sin |---|| | 4*sin |---|| | 4*sin |---|| | \ 2 /| | \ 2 /| | \ 2 /| | \ 2 /| |1 + -----------|*|1 + -----------| |1 + -----------|*|1 + -----------|*sin(3*a)*sin(5*a) | 2 | | 2 | | 2 | | 2 | \ sin (3*a) / \ sin (5*a) / \ sin (3*a) / \ sin (5*a) /
$$\frac{\left(- \frac{4 \sin^{4}{\left(\frac{3 a}{2} \right)}}{\sin^{2}{\left(3 a \right)}} + 1\right) \left(- \frac{4 \sin^{4}{\left(\frac{5 a}{2} \right)}}{\sin^{2}{\left(5 a \right)}} + 1\right)}{\left(\frac{4 \sin^{4}{\left(\frac{3 a}{2} \right)}}{\sin^{2}{\left(3 a \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{5 a}{2} \right)}}{\sin^{2}{\left(5 a \right)}} + 1\right)} + \frac{16 \sin^{2}{\left(\frac{3 a}{2} \right)} \sin^{2}{\left(\frac{5 a}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{3 a}{2} \right)}}{\sin^{2}{\left(3 a \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{5 a}{2} \right)}}{\sin^{2}{\left(5 a \right)}} + 1\right) \sin{\left(3 a \right)} \sin{\left(5 a \right)}}$$
// 0 for 3*a mod pi = 0\ // 0 for 5*a mod pi = 0\ // 1 for 3*a mod 2*pi = 0\ // 1 for 5*a mod 2*pi = 0\ || | || | || | || | || /3*a\ | || /5*a\ | || 2/3*a\ | || 2/5*a\ | || 2*cot|---| | || 2*cot|---| | ||-1 + cot |---| | ||-1 + cot |---| | |< \ 2 / |*|< \ 2 / | + |< \ 2 / |*|< \ 2 / | ||------------- otherwise | ||------------- otherwise | ||-------------- otherwise | ||-------------- otherwise | || 2/3*a\ | || 2/5*a\ | || 2/3*a\ | || 2/5*a\ | ||1 + cot |---| | ||1 + cot |---| | ||1 + cot |---| | ||1 + cot |---| | \\ \ 2 / / \\ \ 2 / / \\ \ 2 / / \\ \ 2 / /
$$\left(\left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 a}{2} \right)}}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 5 a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{5 a}{2} \right)}}{\cot^{2}{\left(\frac{5 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{3 a}{2} \right)} - 1}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{5 a}{2} \right)} - 1}{\cot^{2}{\left(\frac{5 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for 3*a mod pi = 0\ // 0 for 5*a mod pi = 0\ // 1 for 3*a mod 2*pi = 0\ // 1 for 5*a mod 2*pi = 0\ || | || | || | || | || /3*a\ | || /5*a\ | || 2/3*a\ | || 2/5*a\ | || 2*tan|---| | || 2*tan|---| | ||1 - tan |---| | ||1 - tan |---| | |< \ 2 / |*|< \ 2 / | + |< \ 2 / |*|< \ 2 / | ||------------- otherwise | ||------------- otherwise | ||------------- otherwise | ||------------- otherwise | || 2/3*a\ | || 2/5*a\ | || 2/3*a\ | || 2/5*a\ | ||1 + tan |---| | ||1 + tan |---| | ||1 + tan |---| | ||1 + tan |---| | \\ \ 2 / / \\ \ 2 / / \\ \ 2 / / \\ \ 2 / /
$$\left(\left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{3 a}{2} \right)}}{\tan^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 5 a \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{5 a}{2} \right)}}{\tan^{2}{\left(\frac{5 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{3 a}{2} \right)} + 1}{\tan^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 a \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{5 a}{2} \right)} + 1}{\tan^{2}{\left(\frac{5 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for 3*a mod pi = 0\ // 0 for 5*a mod pi = 0\ // 1 for 3*a mod 2*pi = 0\ // 1 for 5*a mod 2*pi = 0\
|| | || | || | || |
|
$$\left(\left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\sin{\left(3 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 5 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 5 a \bmod \pi = 0 \\\sin{\left(5 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\cos{\left(3 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: 5 a \bmod 2 \pi = 0 \\\cos{\left(5 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 3*a mod 2*pi = 0\ // 1 for 5*a mod 2*pi = 0\
|| | || |
// 0 for 3*a mod pi = 0\ // 0 for 5*a mod pi = 0\ || 1 | || 1 |
|| | || | ||-1 + --------- | ||-1 + --------- |
|| 2 | || 2 | || 2/3*a\ | || 2/5*a\ |
||------------------------ otherwise | ||------------------------ otherwise | || tan |---| | || tan |---| |
|
$$\left(\left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{3 a}{2} \right)}}\right) \tan{\left(\frac{3 a}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 5 a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{5 a}{2} \right)}}\right) \tan{\left(\frac{5 a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{3 a}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{3 a}{2} \right)}}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 a \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{5 a}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{5 a}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)$$
// /pi \ \ // /pi \ \
// 0 for 3*a mod pi = 0\ // 0 for 5*a mod pi = 0\ || 0 for |-- + 3*a| mod pi = 0| || 0 for |-- + 5*a| mod pi = 0|
|| | || | || \2 / | || \2 / |
|| /3*a\ | || /5*a\ | || | || |
|| 2*cot|---| | || 2*cot|---| | || /pi 3*a\ | || /pi 5*a\ |
|< \ 2 / |*|< \ 2 / | + |< 2*cot|-- + ---| |*|< 2*cot|-- + ---| |
||------------- otherwise | ||------------- otherwise | || \4 2 / | || \4 2 / |
|| 2/3*a\ | || 2/5*a\ | ||------------------ otherwise | ||------------------ otherwise |
||1 + cot |---| | ||1 + cot |---| | || 2/pi 3*a\ | || 2/pi 5*a\ |
\\ \ 2 / / \\ \ 2 / / ||1 + cot |-- + ---| | ||1 + cot |-- + ---| |
\\ \4 2 / / \\ \4 2 / /
$$\left(\left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 a}{2} \right)}}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 5 a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{5 a}{2} \right)}}{\cot^{2}{\left(\frac{5 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: \left(3 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(5 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{5 a}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{5 a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
/ 2/ pi 3*a\\ / 2/ pi 5*a\\ | cos |- -- + ---|| | cos |- -- + ---|| | \ 2 2 /| | \ 2 2 /| |1 - ----------------|*|1 - ----------------| | 2/3*a\ | | 2/5*a\ | / pi 3*a\ / pi 5*a\ | cos |---| | | cos |---| | 4*cos|- -- + ---|*cos|- -- + ---| \ \ 2 / / \ \ 2 / / \ 2 2 / \ 2 2 / --------------------------------------------- + --------------------------------------------------------------- / 2/ pi 3*a\\ / 2/ pi 5*a\\ / 2/ pi 3*a\\ / 2/ pi 5*a\\ | cos |- -- + ---|| | cos |- -- + ---|| | cos |- -- + ---|| | cos |- -- + ---|| | \ 2 2 /| | \ 2 2 /| | \ 2 2 /| | \ 2 2 /| /3*a\ /5*a\ |1 + ----------------|*|1 + ----------------| |1 + ----------------|*|1 + ----------------|*cos|---|*cos|---| | 2/3*a\ | | 2/5*a\ | | 2/3*a\ | | 2/5*a\ | \ 2 / \ 2 / | cos |---| | | cos |---| | | cos |---| | | cos |---| | \ \ 2 / / \ \ 2 / / \ \ 2 / / \ \ 2 / /
$$\frac{\left(1 - \frac{\cos^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{3 a}{2} \right)}}\right) \left(1 - \frac{\cos^{2}{\left(\frac{5 a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{5 a}{2} \right)}}\right)}{\left(1 + \frac{\cos^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{3 a}{2} \right)}}\right) \left(1 + \frac{\cos^{2}{\left(\frac{5 a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{5 a}{2} \right)}}\right)} + \frac{4 \cos{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)} \cos{\left(\frac{5 a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{3 a}{2} \right)}}\right) \left(1 + \frac{\cos^{2}{\left(\frac{5 a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{5 a}{2} \right)}}\right) \cos{\left(\frac{3 a}{2} \right)} \cos{\left(\frac{5 a}{2} \right)}}$$
/ 2/3*a\ \ / 2/5*a\ \ | sec |---| | | sec |---| | | \ 2 / | | \ 2 / | |1 - ----------------|*|1 - ----------------| | 2/ pi 3*a\| | 2/ pi 5*a\| /3*a\ /5*a\ | sec |- -- + ---|| | sec |- -- + ---|| 4*sec|---|*sec|---| \ \ 2 2 // \ \ 2 2 // \ 2 / \ 2 / --------------------------------------------- + ----------------------------------------------------------------------------- / 2/3*a\ \ / 2/5*a\ \ / 2/3*a\ \ / 2/5*a\ \ | sec |---| | | sec |---| | | sec |---| | | sec |---| | | \ 2 / | | \ 2 / | | \ 2 / | | \ 2 / | / pi 3*a\ / pi 5*a\ |1 + ----------------|*|1 + ----------------| |1 + ----------------|*|1 + ----------------|*sec|- -- + ---|*sec|- -- + ---| | 2/ pi 3*a\| | 2/ pi 5*a\| | 2/ pi 3*a\| | 2/ pi 5*a\| \ 2 2 / \ 2 2 / | sec |- -- + ---|| | sec |- -- + ---|| | sec |- -- + ---|| | sec |- -- + ---|| \ \ 2 2 // \ \ 2 2 // \ \ 2 2 // \ \ 2 2 //
$$\frac{\left(- \frac{\sec^{2}{\left(\frac{3 a}{2} \right)}}{\sec^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}} + 1\right) \left(- \frac{\sec^{2}{\left(\frac{5 a}{2} \right)}}{\sec^{2}{\left(\frac{5 a}{2} - \frac{\pi}{2} \right)}} + 1\right)}{\left(\frac{\sec^{2}{\left(\frac{3 a}{2} \right)}}{\sec^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}} + 1\right) \left(\frac{\sec^{2}{\left(\frac{5 a}{2} \right)}}{\sec^{2}{\left(\frac{5 a}{2} - \frac{\pi}{2} \right)}} + 1\right)} + \frac{4 \sec{\left(\frac{3 a}{2} \right)} \sec{\left(\frac{5 a}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{3 a}{2} \right)}}{\sec^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}} + 1\right) \left(\frac{\sec^{2}{\left(\frac{5 a}{2} \right)}}{\sec^{2}{\left(\frac{5 a}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)} \sec{\left(\frac{5 a}{2} - \frac{\pi}{2} \right)}}$$
/ 2/pi 3*a\\ / 2/pi 5*a\\ | csc |-- - ---|| | csc |-- - ---|| | \2 2 /| | \2 2 /| |1 - --------------|*|1 - --------------| | 2/3*a\ | | 2/5*a\ | /pi 5*a\ /pi 3*a\ | csc |---| | | csc |---| | 4*csc|-- - ---|*csc|-- - ---| \ \ 2 / / \ \ 2 / / \2 2 / \2 2 / ----------------------------------------- + ----------------------------------------------------------- / 2/pi 3*a\\ / 2/pi 5*a\\ / 2/pi 3*a\\ / 2/pi 5*a\\ | csc |-- - ---|| | csc |-- - ---|| | csc |-- - ---|| | csc |-- - ---|| | \2 2 /| | \2 2 /| | \2 2 /| | \2 2 /| /3*a\ /5*a\ |1 + --------------|*|1 + --------------| |1 + --------------|*|1 + --------------|*csc|---|*csc|---| | 2/3*a\ | | 2/5*a\ | | 2/3*a\ | | 2/5*a\ | \ 2 / \ 2 / | csc |---| | | csc |---| | | csc |---| | | csc |---| | \ \ 2 / / \ \ 2 / / \ \ 2 / / \ \ 2 / /
$$\frac{\left(1 - \frac{\csc^{2}{\left(- \frac{3 a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{3 a}{2} \right)}}\right) \left(1 - \frac{\csc^{2}{\left(- \frac{5 a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{5 a}{2} \right)}}\right)}{\left(1 + \frac{\csc^{2}{\left(- \frac{3 a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{3 a}{2} \right)}}\right) \left(1 + \frac{\csc^{2}{\left(- \frac{5 a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{5 a}{2} \right)}}\right)} + \frac{4 \csc{\left(- \frac{5 a}{2} + \frac{\pi}{2} \right)} \csc{\left(- \frac{3 a}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{3 a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{3 a}{2} \right)}}\right) \left(1 + \frac{\csc^{2}{\left(- \frac{5 a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{5 a}{2} \right)}}\right) \csc{\left(\frac{3 a}{2} \right)} \csc{\left(\frac{5 a}{2} \right)}}$$
// / 3*pi\ \ // / 3*pi\ \
// 1 for 3*a mod 2*pi = 0\ // 1 for 5*a mod 2*pi = 0\ || 1 for |3*a + ----| mod 2*pi = 0| || 1 for |5*a + ----| mod 2*pi = 0|
|| | || | || \ 2 / | || \ 2 / |
|| 2/3*a\ | || 2/5*a\ | || | || |
||-1 + cot |---| | ||-1 + cot |---| | || 2/pi 3*a\ | || 2/pi 5*a\ |
|< \ 2 / |*|< \ 2 / | + |<-1 + tan |-- + ---| |*|<-1 + tan |-- + ---| |
||-------------- otherwise | ||-------------- otherwise | || \4 2 / | || \4 2 / |
|| 2/3*a\ | || 2/5*a\ | ||------------------- otherwise | ||------------------- otherwise |
||1 + cot |---| | ||1 + cot |---| | || 2/pi 3*a\ | || 2/pi 5*a\ |
\\ \ 2 / / \\ \ 2 / / || 1 + tan |-- + ---| | || 1 + tan |-- + ---| |
\\ \4 2 / / \\ \4 2 / /
$$\left(\left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{3 a}{2} \right)} - 1}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{5 a}{2} \right)} - 1}{\cot^{2}{\left(\frac{5 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: \left(3 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(5 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{5 a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{5 a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for 3*a mod pi = 0\ // 0 for 5*a mod pi = 0\ // 1 for 3*a mod 2*pi = 0\ // 1 for 5*a mod 2*pi = 0\ || | || | || | || | || -2*sin(6*a) + 4*sin(3*a) | || -2*sin(10*a) + 4*sin(5*a) | || -2 - 2*cos(6*a) + 4*cos(3*a) | || -2 - 2*cos(10*a) + 4*cos(5*a) | |<-------------------------------- otherwise |*|<--------------------------------- otherwise | + |<-------------------------------- otherwise |*|<--------------------------------- otherwise | || 2 | || 2 | || 2 | || 2 | ||1 - cos(6*a) + 2*(1 - cos(3*a)) | ||1 - cos(10*a) + 2*(1 - cos(5*a)) | ||1 - cos(6*a) + 2*(1 - cos(3*a)) | ||1 - cos(10*a) + 2*(1 - cos(5*a)) | \\ / \\ / \\ / \\ /
$$\left(\left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{4 \sin{\left(3 a \right)} - 2 \sin{\left(6 a \right)}}{2 \left(- \cos{\left(3 a \right)} + 1\right)^{2} - \cos{\left(6 a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 5 a \bmod \pi = 0 \\\frac{4 \sin{\left(5 a \right)} - 2 \sin{\left(10 a \right)}}{2 \left(- \cos{\left(5 a \right)} + 1\right)^{2} - \cos{\left(10 a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{4 \cos{\left(3 a \right)} - 2 \cos{\left(6 a \right)} - 2}{2 \left(- \cos{\left(3 a \right)} + 1\right)^{2} - \cos{\left(6 a \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 a \bmod 2 \pi = 0 \\\frac{4 \cos{\left(5 a \right)} - 2 \cos{\left(10 a \right)} - 2}{2 \left(- \cos{\left(5 a \right)} + 1\right)^{2} - \cos{\left(10 a \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 3*a mod 2*pi = 0\ // 1 for 5*a mod 2*pi = 0\
|| | || |
// 0 for 3*a mod pi = 0\ // 0 for 5*a mod pi = 0\ || 2 | || 2 |
|| | || | || sin (3*a) | || sin (5*a) |
|| sin(3*a) | || sin(5*a) | ||-1 + ----------- | ||-1 + ----------- |
||--------------------------- otherwise | ||--------------------------- otherwise | || 4/3*a\ | || 4/5*a\ |
||/ 2 \ | ||/ 2 \ | || 4*sin |---| | || 4*sin |---| |
|<| sin (3*a) | 2/3*a\ |*|<| sin (5*a) | 2/5*a\ | + |< \ 2 / |*|< \ 2 / |
|||1 + -----------|*sin |---| | |||1 + -----------|*sin |---| | ||---------------- otherwise | ||---------------- otherwise |
||| 4/3*a\| \ 2 / | ||| 4/5*a\| \ 2 / | || 2 | || 2 |
||| 4*sin |---|| | ||| 4*sin |---|| | || sin (3*a) | || sin (5*a) |
||\ \ 2 // | ||\ \ 2 // | ||1 + ----------- | ||1 + ----------- |
\\ / \\ / || 4/3*a\ | || 4/5*a\ |
|| 4*sin |---| | || 4*sin |---| |
\\ \ 2 / / \\ \ 2 / /
$$\left(\left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{\sin{\left(3 a \right)}}{\left(1 + \frac{\sin^{2}{\left(3 a \right)}}{4 \sin^{4}{\left(\frac{3 a}{2} \right)}}\right) \sin^{2}{\left(\frac{3 a}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 5 a \bmod \pi = 0 \\\frac{\sin{\left(5 a \right)}}{\left(1 + \frac{\sin^{2}{\left(5 a \right)}}{4 \sin^{4}{\left(\frac{5 a}{2} \right)}}\right) \sin^{2}{\left(\frac{5 a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(3 a \right)}}{4 \sin^{4}{\left(\frac{3 a}{2} \right)}}}{1 + \frac{\sin^{2}{\left(3 a \right)}}{4 \sin^{4}{\left(\frac{3 a}{2} \right)}}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(5 a \right)}}{4 \sin^{4}{\left(\frac{5 a}{2} \right)}}}{1 + \frac{\sin^{2}{\left(5 a \right)}}{4 \sin^{4}{\left(\frac{5 a}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for 3*a mod pi = 0\ // 0 for 5*a mod pi = 0\ // 1 for 3*a mod 2*pi = 0\ // 1 for 5*a mod 2*pi = 0\ || | || | || | || | ||/ 0 for 3*a mod pi = 0 | ||/ 0 for 5*a mod pi = 0 | ||/ 1 for 3*a mod 2*pi = 0 | ||/ 1 for 5*a mod 2*pi = 0 | ||| | ||| | ||| | ||| | ||| /3*a\ | ||| /5*a\ | ||| 2/3*a\ | ||| 2/5*a\ | |<| 2*cot|---| |*|<| 2*cot|---| | + |<|-1 + cot |---| |*|<|-1 + cot |---| | ||< \ 2 / otherwise | ||< \ 2 / otherwise | ||< \ 2 / otherwise | ||< \ 2 / otherwise | |||------------- otherwise | |||------------- otherwise | |||-------------- otherwise | |||-------------- otherwise | ||| 2/3*a\ | ||| 2/5*a\ | ||| 2/3*a\ | ||| 2/5*a\ | |||1 + cot |---| | |||1 + cot |---| | |||1 + cot |---| | |||1 + cot |---| | \\\ \ 2 / / \\\ \ 2 / / \\\ \ 2 / / \\\ \ 2 / /
$$\left(\left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 a}{2} \right)}}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 5 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 5 a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{5 a}{2} \right)}}{\cot^{2}{\left(\frac{5 a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{3 a}{2} \right)} - 1}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: 5 a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{5 a}{2} \right)} - 1}{\cot^{2}{\left(\frac{5 a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 3*a mod 2*pi = 0\ // 1 for 5*a mod 2*pi = 0\
|| | || |
// 0 for 3*a mod pi = 0\ // 0 for 5*a mod pi = 0\ || 2/3*a\ | || 2/5*a\ |
|| | || | || cos |---| | || cos |---| |
|| /3*a\ | || /5*a\ | || \ 2 / | || \ 2 / |
|| 2*cos|---| | || 2*cos|---| | ||-1 + ---------------- | ||-1 + ---------------- |
|| \ 2 / | || \ 2 / | || 2/ pi 3*a\ | || 2/ pi 5*a\ |
||-------------------------------------- otherwise | ||-------------------------------------- otherwise | || cos |- -- + ---| | || cos |- -- + ---| |
|
$$\left(\left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{3 a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{3 a}{2} \right)}}{\cos^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 5 a \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{5 a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{5 a}{2} \right)}}{\cos^{2}{\left(\frac{5 a}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{5 a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{3 a}{2} \right)}}{\cos^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{3 a}{2} \right)}}{\cos^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 a \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{5 a}{2} \right)}}{\cos^{2}{\left(\frac{5 a}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{5 a}{2} \right)}}{\cos^{2}{\left(\frac{5 a}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 3*a mod 2*pi = 0\ // 1 for 5*a mod 2*pi = 0\
|| | || |
// 0 for 3*a mod pi = 0\ // 0 for 5*a mod pi = 0\ || 2/ pi 3*a\ | || 2/ pi 5*a\ |
|| | || | || sec |- -- + ---| | || sec |- -- + ---| |
|| / pi 3*a\ | || / pi 5*a\ | || \ 2 2 / | || \ 2 2 / |
|| 2*sec|- -- + ---| | || 2*sec|- -- + ---| | ||-1 + ---------------- | ||-1 + ---------------- |
|| \ 2 2 / | || \ 2 2 / | || 2/3*a\ | || 2/5*a\ |
||------------------------------- otherwise | ||------------------------------- otherwise | || sec |---| | || sec |---| |
|
$$\left(\left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{3 a}{2} \right)}}\right) \sec{\left(\frac{3 a}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 5 a \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{5 a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{5 a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{5 a}{2} \right)}}\right) \sec{\left(\frac{5 a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{3 a}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{3 a}{2} \right)}}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{5 a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{5 a}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{5 a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{5 a}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 3*a mod 2*pi = 0\ // 1 for 5*a mod 2*pi = 0\
|| | || |
// 0 for 3*a mod pi = 0\ // 0 for 5*a mod pi = 0\ || 2/3*a\ | || 2/5*a\ |
|| | || | || csc |---| | || csc |---| |
|| /3*a\ | || /5*a\ | || \ 2 / | || \ 2 / |
|| 2*csc|---| | || 2*csc|---| | ||-1 + -------------- | ||-1 + -------------- |
|| \ 2 / | || \ 2 / | || 2/pi 3*a\ | || 2/pi 5*a\ |
||---------------------------------- otherwise | ||---------------------------------- otherwise | || csc |-- - ---| | || csc |-- - ---| |
|
$$\left(\left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{3 a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{3 a}{2} \right)}}{\csc^{2}{\left(- \frac{3 a}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{3 a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 5 a \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{5 a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{5 a}{2} \right)}}{\csc^{2}{\left(- \frac{5 a}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{5 a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{3 a}{2} \right)}}{\csc^{2}{\left(- \frac{3 a}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{3 a}{2} \right)}}{\csc^{2}{\left(- \frac{3 a}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 5 a \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{5 a}{2} \right)}}{\csc^{2}{\left(- \frac{5 a}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{5 a}{2} \right)}}{\csc^{2}{\left(- \frac{5 a}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right)$$
Piecewise((0, Mod(3*a = pi, 0)), (2*csc(3*a/2)/((1 + csc(3*a/2)^2/csc(pi/2 - 3*a/2)^2)*csc(pi/2 - 3*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(3*a = 2*pi, 0)), ((-1 + csc(3*a/2)^2/csc(pi/2 - 3*a/2)^2)/(1 + csc(3*a/2)^2/csc(pi/2 - 3*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))