Господин Экзамен
(2*sin(3*a)-3*cos(3*a))^2+2*cos(3*a)+3*sin(3*a) если a=-3/2
Решение
Вы ввели
[src]
2 (2*sin(3*a) - 3*cos(3*a)) + 2*cos(3*a) + 3*sin(3*a)
$$\left(2 \sin{\left(3 a \right)} - 3 \cos{\left(3 a \right)}\right)^{2} + 3 \sin{\left(3 a \right)} + 2 \cos{\left(3 a \right)}$$
(2*sin(3*a) - 3*cos(3*a))^2 + 2*cos(3*a) + 3*sin(3*a)
Подстановка условия
[src]
(2*sin(3*a) - 3*cos(3*a))^2 + 2*cos(3*a) + 3*sin(3*a) при a = -3/2
подставляем
2 (2*sin(3*a) - 3*cos(3*a)) + 2*cos(3*a) + 3*sin(3*a)
$$\left(2 \sin{\left(3 a \right)} - 3 \cos{\left(3 a \right)}\right)^{2} + 3 \sin{\left(3 a \right)} + 2 \cos{\left(3 a \right)}$$
2 (-3*cos(3*a) + 2*sin(3*a)) + 2*cos(3*a) + 3*sin(3*a)
$$\left(2 \sin{\left(3 a \right)} - 3 \cos{\left(3 a \right)}\right)^{2} + 3 \sin{\left(3 a \right)} + 2 \cos{\left(3 a \right)}$$
переменные
a = -3/2
$$a = - \frac{3}{2}$$
2 (-3*cos(3*(-3/2)) + 2*sin(3*(-3/2))) + 2*cos(3*(-3/2)) + 3*sin(3*(-3/2))
$$\left(2 \sin{\left(3 (-3/2) \right)} - 3 \cos{\left(3 (-3/2) \right)}\right)^{2} + 3 \sin{\left(3 (-3/2) \right)} + 2 \cos{\left(3 (-3/2) \right)}$$
2 (-3*cos(9/2) - 2*sin(9/2)) - 3*sin(9/2) + 2*cos(9/2)
$$2 \cos{\left(\frac{9}{2} \right)} - 3 \sin{\left(\frac{9}{2} \right)} + \left(- 3 \cos{\left(\frac{9}{2} \right)} - 2 \sin{\left(\frac{9}{2} \right)}\right)^{2}$$
(-3*cos(9/2) - 2*sin(9/2))^2 - 3*sin(9/2) + 2*cos(9/2)
Численный ответ
[src]
2.0*cos(3*a) + 3.0*sin(3*a) + 9.0*(-cos(3*a) + 0.666666666666667*sin(3*a))^2
2.0*cos(3*a) + 3.0*sin(3*a) + 9.0*(-cos(3*a) + 0.666666666666667*sin(3*a))^2
Степени
[src]
2 / -3*I*a 3*I*a \ / -3*I*a 3*I*a\ | 3*e 3*e / -3*I*a 3*I*a\| 3*I*\- e + e / -3*I*a 3*I*a |- --------- - -------- - I*\- e + e /| - ------------------------ + e + e \ 2 2 / 2
$$e^{3 i a} + \left(- \frac{3 e^{3 i a}}{2} - i \left(e^{3 i a} - e^{- 3 i a}\right) - \frac{3 e^{- 3 i a}}{2}\right)^{2} - \frac{3 i \left(e^{3 i a} - e^{- 3 i a}\right)}{2} + e^{- 3 i a}$$
(-3*exp(-3*i*a)/2 - 3*exp(3*i*a)/2 - i*(-exp(-3*i*a) + exp(3*i*a)))^2 - 3*i*(-exp(-3*i*a) + exp(3*i*a))/2 + exp(-3*i*a) + exp(3*i*a)
Общий знаменатель
[src]
2 2 2*cos(3*a) + 3*sin(3*a) + 4*sin (3*a) + 9*cos (3*a) - 12*cos(3*a)*sin(3*a)
$$4 \sin^{2}{\left(3 a \right)} - 12 \sin{\left(3 a \right)} \cos{\left(3 a \right)} + 9 \cos^{2}{\left(3 a \right)} + 3 \sin{\left(3 a \right)} + 2 \cos{\left(3 a \right)}$$
2*cos(3*a) + 3*sin(3*a) + 4*sin(3*a)^2 + 9*cos(3*a)^2 - 12*cos(3*a)*sin(3*a)
Комбинаторика
[src]
2 2 2*cos(3*a) + 3*sin(3*a) + 4*sin (3*a) + 9*cos (3*a) - 12*cos(3*a)*sin(3*a)
$$4 \sin^{2}{\left(3 a \right)} - 12 \sin{\left(3 a \right)} \cos{\left(3 a \right)} + 9 \cos^{2}{\left(3 a \right)} + 3 \sin{\left(3 a \right)} + 2 \cos{\left(3 a \right)}$$
2*cos(3*a) + 3*sin(3*a) + 4*sin(3*a)^2 + 9*cos(3*a)^2 - 12*cos(3*a)*sin(3*a)
Тригонометрическая часть
[src]
9 2/3*a\ 5*cos(6*a) - - 6*sin(6*a) + 3*sin(3*a) + 4*cos |---| + ---------- 2 \ 2 / 2
$$4 \cos^{2}{\left(\frac{3 a}{2} \right)} + 3 \sin{\left(3 a \right)} - 6 \sin{\left(6 a \right)} + \frac{5 \cos{\left(6 a \right)}}{2} + \frac{9}{2}$$
2 / /pi \ \ /pi \ |- 3*sin|-- + 3*a| + 2*sin(3*a)| + 2*sin|-- + 3*a| + 3*sin(3*a) \ \2 / / \2 /
$$\left(2 \sin{\left(3 a \right)} - 3 \sin{\left(3 a + \frac{\pi}{2} \right)}\right)^{2} + 3 \sin{\left(3 a \right)} + 2 \sin{\left(3 a + \frac{\pi}{2} \right)}$$
2 / 3 2 \ 2 3 |- -------- + --------| + -------- + -------- \ sec(3*a) csc(3*a)/ sec(3*a) csc(3*a)
$$\left(- \frac{3}{\sec{\left(3 a \right)}} + \frac{2}{\csc{\left(3 a \right)}}\right)^{2} + \frac{2}{\sec{\left(3 a \right)}} + \frac{3}{\csc{\left(3 a \right)}}$$
2 / / pi\\ / pi\ |-3*cos(3*a) + 2*cos|3*a - --|| + 2*cos(3*a) + 3*cos|3*a - --| \ \ 2 // \ 2 /
$$\left(- 3 \cos{\left(3 a \right)} + 2 \cos{\left(3 a - \frac{\pi}{2} \right)}\right)^{2} + 2 \cos{\left(3 a \right)} + 3 \cos{\left(3 a - \frac{\pi}{2} \right)}$$
2 / 3 2 \ 2 3 |- -------- + -------------| + -------- + ------------- | sec(3*a) / pi\| sec(3*a) / pi\ | sec|3*a - --|| sec|3*a - --| \ \ 2 // \ 2 /
$$\left(\frac{2}{\sec{\left(3 a - \frac{\pi}{2} \right)}} - \frac{3}{\sec{\left(3 a \right)}}\right)^{2} + \frac{3}{\sec{\left(3 a - \frac{\pi}{2} \right)}} + \frac{2}{\sec{\left(3 a \right)}}$$
2 / 3 2 \ 2 3 |- -------- + -------------| + -------- + ------------- | sec(3*a) /pi \| sec(3*a) /pi \ | sec|-- - 3*a|| sec|-- - 3*a| \ \2 // \2 /
$$\left(\frac{2}{\sec{\left(- 3 a + \frac{\pi}{2} \right)}} - \frac{3}{\sec{\left(3 a \right)}}\right)^{2} + \frac{3}{\sec{\left(- 3 a + \frac{\pi}{2} \right)}} + \frac{2}{\sec{\left(3 a \right)}}$$
2 / 3 2 \ 2 3 |- ------------- + --------| + ------------- + -------- | /pi \ csc(3*a)| /pi \ csc(3*a) | csc|-- - 3*a| | csc|-- - 3*a| \ \2 / / \2 /
$$\left(- \frac{3}{\csc{\left(- 3 a + \frac{\pi}{2} \right)}} + \frac{2}{\csc{\left(3 a \right)}}\right)^{2} + \frac{2}{\csc{\left(- 3 a + \frac{\pi}{2} \right)}} + \frac{3}{\csc{\left(3 a \right)}}$$
2 / 3 2 \ 2 3 |- ------------- + -------------| + ------------- + ------------- | /pi \ csc(pi - 3*a)| /pi \ csc(pi - 3*a) | csc|-- - 3*a| | csc|-- - 3*a| \ \2 / / \2 /
$$\left(- \frac{3}{\csc{\left(- 3 a + \frac{\pi}{2} \right)}} + \frac{2}{\csc{\left(- 3 a + \pi \right)}}\right)^{2} + \frac{2}{\csc{\left(- 3 a + \frac{\pi}{2} \right)}} + \frac{3}{\csc{\left(- 3 a + \pi \right)}}$$
2 / /3*a\\ /3*a\ |-3*cos(3*a) + 2*(1 + cos(3*a))*tan|---|| + 2*cos(3*a) + 3*(1 + cos(3*a))*tan|---| \ \ 2 // \ 2 /
$$\left(2 \left(\cos{\left(3 a \right)} + 1\right) \tan{\left(\frac{3 a}{2} \right)} - 3 \cos{\left(3 a \right)}\right)^{2} + 3 \left(\cos{\left(3 a \right)} + 1\right) \tan{\left(\frac{3 a}{2} \right)} + 2 \cos{\left(3 a \right)}$$
/ 2/pi 3*a\\
2 3*|1 - cot |-- + ---||*(1 + sin(3*a))
/ / 2/pi 3*a\\ \ \ \4 2 //
|-3*cos(3*a) + |1 - cot |-- + ---||*(1 + sin(3*a))| + 2*cos(3*a) + -------------------------------------
\ \ \4 2 // / 2
$$\frac{3 \cdot \left(- \cot^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(3 a \right)} + 1\right)}{2} + \left(\left(- \cot^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(3 a \right)} + 1\right) - 3 \cos{\left(3 a \right)}\right)^{2} + 2 \cos{\left(3 a \right)}$$
2 / / 2/3*a\\ /3*a\ \ / 2/3*a\\ /3*a\ | 3*|1 - tan |---|| 4*tan|---| | 2*|1 - tan |---|| 6*tan|---| | \ \ 2 // \ 2 / | \ \ 2 // \ 2 / |- ----------------- + -------------| + ----------------- + ------------- | 2/3*a\ 2/3*a\| 2/3*a\ 2/3*a\ | 1 + tan |---| 1 + tan |---|| 1 + tan |---| 1 + tan |---| \ \ 2 / \ 2 // \ 2 / \ 2 /
$$\left(- \frac{3 \cdot \left(- \tan^{2}{\left(\frac{3 a}{2} \right)} + 1\right)}{\tan^{2}{\left(\frac{3 a}{2} \right)} + 1} + \frac{4 \tan{\left(\frac{3 a}{2} \right)}}{\tan^{2}{\left(\frac{3 a}{2} \right)} + 1}\right)^{2} + \frac{2 \cdot \left(- \tan^{2}{\left(\frac{3 a}{2} \right)} + 1\right)}{\tan^{2}{\left(\frac{3 a}{2} \right)} + 1} + \frac{6 \tan{\left(\frac{3 a}{2} \right)}}{\tan^{2}{\left(\frac{3 a}{2} \right)} + 1}$$
2 / /pi 3*a\ /3*a\ \ /pi 3*a\ /3*a\ | 6*tan|-- + ---| 4*cot|---| | 4*tan|-- + ---| 6*cot|---| | \4 2 / \ 2 / | \4 2 / \ 2 / |- ------------------ + -------------| + ------------------ + ------------- | 2/pi 3*a\ 2/3*a\| 2/pi 3*a\ 2/3*a\ | 1 + tan |-- + ---| 1 + cot |---|| 1 + tan |-- + ---| 1 + cot |---| \ \4 2 / \ 2 // \4 2 / \ 2 /
$$\left(\frac{4 \cot{\left(\frac{3 a}{2} \right)}}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} - \frac{6 \tan{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} + 1}\right)^{2} + \frac{6 \cot{\left(\frac{3 a}{2} \right)}}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} + \frac{4 \tan{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} + 1}$$
2 / /pi 3*a\ /3*a\ \ /pi 3*a\ /3*a\ | 6*tan|-- + ---| 4*tan|---| | 4*tan|-- + ---| 6*tan|---| | \4 2 / \ 2 / | \4 2 / \ 2 / |- ------------------ + -------------| + ------------------ + ------------- | 2/pi 3*a\ 2/3*a\| 2/pi 3*a\ 2/3*a\ | 1 + tan |-- + ---| 1 + tan |---|| 1 + tan |-- + ---| 1 + tan |---| \ \4 2 / \ 2 // \4 2 / \ 2 /
$$\left(- \frac{6 \tan{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} + 1} + \frac{4 \tan{\left(\frac{3 a}{2} \right)}}{\tan^{2}{\left(\frac{3 a}{2} \right)} + 1}\right)^{2} + \frac{4 \tan{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} + 1} + \frac{6 \tan{\left(\frac{3 a}{2} \right)}}{\tan^{2}{\left(\frac{3 a}{2} \right)} + 1}$$
2 / / 1 \ \ / 1 \ | 3*|1 - ---------| | 2*|1 - ---------| | | 2/3*a\| | | 2/3*a\| | | cot |---|| | | cot |---|| | \ \ 2 // 4 | \ \ 2 // 6 |- ----------------- + ------------------------| + ----------------- + ------------------------ | 1 / 1 \ /3*a\| 1 / 1 \ /3*a\ | 1 + --------- |1 + ---------|*cot|---|| 1 + --------- |1 + ---------|*cot|---| | 2/3*a\ | 2/3*a\| \ 2 /| 2/3*a\ | 2/3*a\| \ 2 / | cot |---| | cot |---|| | cot |---| | cot |---|| \ \ 2 / \ \ 2 // / \ 2 / \ \ 2 //
$$\left(- \frac{3 \cdot \left(1 - \frac{1}{\cot^{2}{\left(\frac{3 a}{2} \right)}}\right)}{1 + \frac{1}{\cot^{2}{\left(\frac{3 a}{2} \right)}}} + \frac{4}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{3 a}{2} \right)}}\right) \cot{\left(\frac{3 a}{2} \right)}}\right)^{2} + \frac{2 \cdot \left(1 - \frac{1}{\cot^{2}{\left(\frac{3 a}{2} \right)}}\right)}{1 + \frac{1}{\cot^{2}{\left(\frac{3 a}{2} \right)}}} + \frac{6}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{3 a}{2} \right)}}\right) \cot{\left(\frac{3 a}{2} \right)}}$$
2 / / 2/3*a\\ / 2/pi 3*a\\\ / 2/3*a\\ / 2/pi 3*a\\ | 3*|-1 + cot |---|| 2*|-1 + tan |-- + ---||| 2*|-1 + cot |---|| 3*|-1 + tan |-- + ---|| | \ \ 2 // \ \4 2 //| \ \ 2 // \ \4 2 // |- ------------------ + -----------------------| + ------------------ + ----------------------- | 2/3*a\ 2/pi 3*a\ | 2/3*a\ 2/pi 3*a\ | 1 + cot |---| 1 + tan |-- + ---| | 1 + cot |---| 1 + tan |-- + ---| \ \ 2 / \4 2 / / \ 2 / \4 2 /
$$\left(\frac{2 \left(\tan^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} - 1\right)}{\tan^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} + 1} - \frac{3 \left(\cot^{2}{\left(\frac{3 a}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1}\right)^{2} + \frac{3 \left(\tan^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} - 1\right)}{\tan^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} + 1} + \frac{2 \left(\cot^{2}{\left(\frac{3 a}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1}$$
2 / / 2/3*a\\ / 2/pi 3*a\\\ / 2/3*a\\ / 2/pi 3*a\\ | 3*|1 - tan |---|| 2*|1 - cot |-- + ---||| 2*|1 - tan |---|| 3*|1 - cot |-- + ---|| | \ \ 2 // \ \4 2 //| \ \ 2 // \ \4 2 // |- ----------------- + ----------------------| + ----------------- + ---------------------- | 2/3*a\ 2/pi 3*a\ | 2/3*a\ 2/pi 3*a\ | 1 + tan |---| 1 + cot |-- + ---| | 1 + tan |---| 1 + cot |-- + ---| \ \ 2 / \4 2 / / \ 2 / \4 2 /
$$\left(- \frac{3 \cdot \left(- \tan^{2}{\left(\frac{3 a}{2} \right)} + 1\right)}{\tan^{2}{\left(\frac{3 a}{2} \right)} + 1} + \frac{2 \cdot \left(- \cot^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} + 1\right)}{\cot^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} + 1}\right)^{2} + \frac{2 \cdot \left(- \tan^{2}{\left(\frac{3 a}{2} \right)} + 1\right)}{\tan^{2}{\left(\frac{3 a}{2} \right)} + 1} + \frac{3 \cdot \left(- \cot^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} + 1\right)}{\cot^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} + 1}$$
2 / // 1 for 3*a mod 2*pi = 0\ // 0 for 3*a mod pi = 0\\ // 1 for 3*a mod 2*pi = 0\ // 0 for 3*a mod pi = 0\ |- 3*|< | + 2*|< || + 2*|< | + 3*|< | \ \\cos(3*a) otherwise / \\sin(3*a) otherwise // \\cos(3*a) otherwise / \\sin(3*a) otherwise /
$$\left(\left(\left(2 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\sin{\left(3 a \right)} & \text{otherwise} \end{cases}\right)\right) - \left(3 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\cos{\left(3 a \right)} & \text{otherwise} \end{cases}\right)\right)\right)^{2}\right) + \left(3 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\sin{\left(3 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\cos{\left(3 a \right)} & \text{otherwise} \end{cases}\right)\right)$$
2 / // 0 for 3*a mod pi = 0\\ // 0 for 3*a mod pi = 0\ | // 1 for 3*a mod 2*pi = 0\ || || // 1 for 3*a mod 2*pi = 0\ || | |- 3*|< | + 2*|< / pi\ || + 2*|< | + 3*|< / pi\ | | \\cos(3*a) otherwise / ||cos|3*a - --| otherwise || \\cos(3*a) otherwise / ||cos|3*a - --| otherwise | \ \\ \ 2 / // \\ \ 2 / /
$$\left(\left(\left(2 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\cos{\left(3 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) - \left(3 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\cos{\left(3 a \right)} & \text{otherwise} \end{cases}\right)\right)\right)^{2}\right) + \left(3 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\cos{\left(3 a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\cos{\left(3 a \right)} & \text{otherwise} \end{cases}\right)\right)$$
2 / // 1 for 3*a mod 2*pi = 0\ \ // 1 for 3*a mod 2*pi = 0\ | || | // 0 for 3*a mod pi = 0\| || | // 0 for 3*a mod pi = 0\ |- 3*|< /pi \ | + 2*|< || + 2*|< /pi \ | + 3*|< | | ||sin|-- + 3*a| otherwise | \\sin(3*a) otherwise /| ||sin|-- + 3*a| otherwise | \\sin(3*a) otherwise / \ \\ \2 / / / \\ \2 / /
$$\left(\left(\left(2 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\sin{\left(3 a \right)} & \text{otherwise} \end{cases}\right)\right) - \left(3 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\sin{\left(3 a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right)\right)^{2}\right) + \left(3 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\sin{\left(3 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\sin{\left(3 a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right)$$
2 / 2/3*a\ \ 2/3*a\ | 8*sin |---|*sin(3*a) | 12*sin |---|*sin(3*a) | 6*(-1 - cos(6*a) + 2*cos(3*a)) \ 2 / | 4*(-1 - cos(6*a) + 2*cos(3*a)) \ 2 / |- -------------------------------- + -----------------------| + -------------------------------- + ----------------------- | 2 2 4/3*a\| 2 2 4/3*a\ | 1 - cos(6*a) + 2*(1 - cos(3*a)) sin (3*a) + 4*sin |---|| 1 - cos(6*a) + 2*(1 - cos(3*a)) sin (3*a) + 4*sin |---| \ \ 2 // \ 2 /
$$\left(\frac{8 \sin^{2}{\left(\frac{3 a}{2} \right)} \sin{\left(3 a \right)}}{4 \sin^{4}{\left(\frac{3 a}{2} \right)} + \sin^{2}{\left(3 a \right)}} - \frac{6 \cdot \left(2 \cos{\left(3 a \right)} - \cos{\left(6 a \right)} - 1\right)}{2 \left(- \cos{\left(3 a \right)} + 1\right)^{2} - \cos{\left(6 a \right)} + 1}\right)^{2} + \frac{12 \sin^{2}{\left(\frac{3 a}{2} \right)} \sin{\left(3 a \right)}}{4 \sin^{4}{\left(\frac{3 a}{2} \right)} + \sin^{2}{\left(3 a \right)}} + \frac{4 \cdot \left(2 \cos{\left(3 a \right)} - \cos{\left(6 a \right)} - 1\right)}{2 \left(- \cos{\left(3 a \right)} + 1\right)^{2} - \cos{\left(6 a \right)} + 1}$$
2 / // / 3*pi\ \\ // / 3*pi\ \ | // 1 for 3*a mod 2*pi = 0\ || 1 for |3*a + ----| mod 2*pi = 0|| // 1 for 3*a mod 2*pi = 0\ || 1 for |3*a + ----| mod 2*pi = 0| |- 3*|< | + 2*|< \ 2 / || + 2*|< | + 3*|< \ 2 / | | \\cos(3*a) otherwise / || || \\cos(3*a) otherwise / || | \ \\sin(3*a) otherwise // \\sin(3*a) otherwise /
$$\left(\left(\left(- 3 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\cos{\left(3 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: \left(3 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(3 a \right)} & \text{otherwise} \end{cases}\right)\right)\right)^{2}\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\cos{\left(3 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 1 & \text{for}\: \left(3 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(3 a \right)} & \text{otherwise} \end{cases}\right)\right)$$
2 / // 0 for 3*a mod pi = 0\\ // 0 for 3*a mod pi = 0\ | // 1 for 3*a mod 2*pi = 0\ || || // 1 for 3*a mod 2*pi = 0\ || | | || | || 1 || || | || 1 | |- 3*|< 1 | + 2*|<------------- otherwise || + 2*|< 1 | + 3*|<------------- otherwise | | ||-------- otherwise | || / pi\ || ||-------- otherwise | || / pi\ | | \\sec(3*a) / ||sec|3*a - --| || \\sec(3*a) / ||sec|3*a - --| | \ \\ \ 2 / // \\ \ 2 / /
$$\left(\left(\left(2 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{1}{\sec{\left(3 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(3 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(3 a \right)}} & \text{otherwise} \end{cases}\right)\right)\right)^{2}\right) + \left(3 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{1}{\sec{\left(3 a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(3 a \right)}} & \text{otherwise} \end{cases}\right)\right)$$
2 / // 1 for 3*a mod 2*pi = 0\ \ // 1 for 3*a mod 2*pi = 0\ | || | // 0 for 3*a mod pi = 0\| || | // 0 for 3*a mod pi = 0\ | || 1 | || || || 1 | || | |- 3*|<------------- otherwise | + 2*|< 1 || + 2*|<------------- otherwise | + 3*|< 1 | | || /pi \ | ||-------- otherwise || || /pi \ | ||-------- otherwise | | ||csc|-- - 3*a| | \\csc(3*a) /| ||csc|-- - 3*a| | \\csc(3*a) / \ \\ \2 / / / \\ \2 / /
$$\left(\left(\left(2 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{1}{\csc{\left(3 a \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(3 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- 3 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right)\right)^{2}\right) + \left(3 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{1}{\csc{\left(3 a \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- 3 a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right)$$
2 / // 0 for 3*a mod pi = 0\\ // 0 for 3*a mod pi = 0\ | || || || | | // 1 for 3*a mod 2*pi = 0\ ||1 - cos(3*a) || // 1 for 3*a mod 2*pi = 0\ ||1 - cos(3*a) | |- 3*|< | + 2*|<------------ otherwise || + 2*|< | + 3*|<------------ otherwise | | \\cos(3*a) otherwise / || /3*a\ || \\cos(3*a) otherwise / || /3*a\ | | || tan|---| || || tan|---| | \ \\ \ 2 / // \\ \ 2 / /
$$\left(\left(\left(2 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{- \cos{\left(3 a \right)} + 1}{\tan{\left(\frac{3 a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(3 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\cos{\left(3 a \right)} & \text{otherwise} \end{cases}\right)\right)\right)^{2}\right) + \left(3 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{- \cos{\left(3 a \right)} + 1}{\tan{\left(\frac{3 a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\cos{\left(3 a \right)} & \text{otherwise} \end{cases}\right)\right)$$
2 / / 4/3*a\\ \ / 4/3*a\\ | | 4*sin |---|| | | 4*sin |---|| | | \ 2 /| | | \ 2 /| | 3*|1 - -----------| 2/3*a\ | 2*|1 - -----------| 2/3*a\ | | 2 | 8*sin |---| | | 2 | 12*sin |---| | \ sin (3*a) / \ 2 / | \ sin (3*a) / \ 2 / |- ------------------- + --------------------------| + ------------------- + -------------------------- | 4/3*a\ / 4/3*a\\ | 4/3*a\ / 4/3*a\\ | 4*sin |---| | 4*sin |---|| | 4*sin |---| | 4*sin |---|| | \ 2 / | \ 2 /| | \ 2 / | \ 2 /| | 1 + ----------- |1 + -----------|*sin(3*a)| 1 + ----------- |1 + -----------|*sin(3*a) | 2 | 2 | | 2 | 2 | \ sin (3*a) \ sin (3*a) / / sin (3*a) \ sin (3*a) /
$$\left(- \frac{3 \left(- \frac{4 \sin^{4}{\left(\frac{3 a}{2} \right)}}{\sin^{2}{\left(3 a \right)}} + 1\right)}{\frac{4 \sin^{4}{\left(\frac{3 a}{2} \right)}}{\sin^{2}{\left(3 a \right)}} + 1} + \frac{8 \sin^{2}{\left(\frac{3 a}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{3 a}{2} \right)}}{\sin^{2}{\left(3 a \right)}} + 1\right) \sin{\left(3 a \right)}}\right)^{2} + \frac{2 \left(- \frac{4 \sin^{4}{\left(\frac{3 a}{2} \right)}}{\sin^{2}{\left(3 a \right)}} + 1\right)}{\frac{4 \sin^{4}{\left(\frac{3 a}{2} \right)}}{\sin^{2}{\left(3 a \right)}} + 1} + \frac{12 \sin^{2}{\left(\frac{3 a}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{3 a}{2} \right)}}{\sin^{2}{\left(3 a \right)}} + 1\right) \sin{\left(3 a \right)}}$$
2 / // /pi \ \ \ // /pi \ \ | || 0 for |-- + 3*a| mod pi = 0| | || 0 for |-- + 3*a| mod pi = 0| | || \2 / | // 0 for 3*a mod pi = 0\| || \2 / | // 0 for 3*a mod pi = 0\ |- 3*|< | + 2*|< || + 2*|< | + 3*|< | | || /pi 3*a\ | \\sin(3*a) otherwise /| || /pi 3*a\ | \\sin(3*a) otherwise / | ||(1 + sin(3*a))*cot|-- + ---| otherwise | | ||(1 + sin(3*a))*cot|-- + ---| otherwise | \ \\ \4 2 / / / \\ \4 2 / /
$$\left(\left(\left(2 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\sin{\left(3 a \right)} & \text{otherwise} \end{cases}\right)\right) - \left(3 \left(\begin{cases} 0 & \text{for}\: \left(3 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(3 a \right)} + 1\right) \cot{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right)\right)^{2}\right) + \left(3 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\sin{\left(3 a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: \left(3 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(3 a \right)} + 1\right) \cot{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right)$$
2 / // 1 for 3*a mod 2*pi = 0\ // 0 for 3*a mod pi = 0\\ // 1 for 3*a mod 2*pi = 0\ // 0 for 3*a mod pi = 0\ | || | || || || | || | | || 2/3*a\ | || /3*a\ || || 2/3*a\ | || /3*a\ | | ||-1 + cot |---| | || 2*cot|---| || ||-1 + cot |---| | || 2*cot|---| | |- 3*|< \ 2 / | + 2*|< \ 2 / || + 2*|< \ 2 / | + 3*|< \ 2 / | | ||-------------- otherwise | ||------------- otherwise || ||-------------- otherwise | ||------------- otherwise | | || 2/3*a\ | || 2/3*a\ || || 2/3*a\ | || 2/3*a\ | | ||1 + cot |---| | ||1 + cot |---| || ||1 + cot |---| | ||1 + cot |---| | \ \\ \ 2 / / \\ \ 2 / // \\ \ 2 / / \\ \ 2 / /
$$\left(\left(\left(2 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 a}{2} \right)}}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(3 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{3 a}{2} \right)} - 1}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)\right)^{2}\right) + \left(3 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 a}{2} \right)}}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{3 a}{2} \right)} - 1}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
2 / // 1 for 3*a mod 2*pi = 0\ // 0 for 3*a mod pi = 0\\ // 1 for 3*a mod 2*pi = 0\ // 0 for 3*a mod pi = 0\ | || | || || || | || | | || 2/3*a\ | || /3*a\ || || 2/3*a\ | || /3*a\ | | ||1 - tan |---| | || 2*tan|---| || ||1 - tan |---| | || 2*tan|---| | |- 3*|< \ 2 / | + 2*|< \ 2 / || + 2*|< \ 2 / | + 3*|< \ 2 / | | ||------------- otherwise | ||------------- otherwise || ||------------- otherwise | ||------------- otherwise | | || 2/3*a\ | || 2/3*a\ || || 2/3*a\ | || 2/3*a\ | | ||1 + tan |---| | ||1 + tan |---| || ||1 + tan |---| | ||1 + tan |---| | \ \\ \ 2 / / \\ \ 2 / // \\ \ 2 / / \\ \ 2 / /
$$\left(\left(\left(2 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{3 a}{2} \right)}}{\tan^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(3 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{3 a}{2} \right)} + 1}{\tan^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)\right)^{2}\right) + \left(3 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{3 a}{2} \right)}}{\tan^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{3 a}{2} \right)} + 1}{\tan^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
2
/ // 1 for 3*a mod 2*pi = 0\ // 0 for 3*a mod pi = 0\\ // 1 for 3*a mod 2*pi = 0\ // 0 for 3*a mod pi = 0\
| || | || || || | || |
|- 3*|
$$\left(\left(\left(2 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\sin{\left(3 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) - \left(3 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\cos{\left(3 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)\right)^{2}\right) + \left(3 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\sin{\left(3 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\cos{\left(3 a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
2 / / 2/3*a\ \ \ / 2/3*a\ \ | | sec |---| | | | sec |---| | | | \ 2 / | | | \ 2 / | | 3*|1 - ----------------| | 2*|1 - ----------------| | | 2/ pi 3*a\| /3*a\ | | 2/ pi 3*a\| /3*a\ | | sec |- -- + ---|| 4*sec|---| | | sec |- -- + ---|| 6*sec|---| | \ \ 2 2 // \ 2 / | \ \ 2 2 // \ 2 / |- ------------------------ + --------------------------------------| + ------------------------ + -------------------------------------- | 2/3*a\ / 2/3*a\ \ | 2/3*a\ / 2/3*a\ \ | sec |---| | sec |---| | | sec |---| | sec |---| | | \ 2 / | \ 2 / | / pi 3*a\| \ 2 / | \ 2 / | / pi 3*a\ | 1 + ---------------- |1 + ----------------|*sec|- -- + ---|| 1 + ---------------- |1 + ----------------|*sec|- -- + ---| | 2/ pi 3*a\ | 2/ pi 3*a\| \ 2 2 /| 2/ pi 3*a\ | 2/ pi 3*a\| \ 2 2 / | sec |- -- + ---| | sec |- -- + ---|| | sec |- -- + ---| | sec |- -- + ---|| \ \ 2 2 / \ \ 2 2 // / \ 2 2 / \ \ 2 2 //
$$\left(- \frac{3 \left(- \frac{\sec^{2}{\left(\frac{3 a}{2} \right)}}{\sec^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}} + 1\right)}{\frac{\sec^{2}{\left(\frac{3 a}{2} \right)}}{\sec^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}} + 1} + \frac{4 \sec{\left(\frac{3 a}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{3 a}{2} \right)}}{\sec^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}}\right)^{2} + \frac{2 \left(- \frac{\sec^{2}{\left(\frac{3 a}{2} \right)}}{\sec^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}} + 1\right)}{\frac{\sec^{2}{\left(\frac{3 a}{2} \right)}}{\sec^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}} + 1} + \frac{6 \sec{\left(\frac{3 a}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{3 a}{2} \right)}}{\sec^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}}$$
2 / / 2/ pi 3*a\\ \ / 2/ pi 3*a\\ | | cos |- -- + ---|| | | cos |- -- + ---|| | | \ 2 2 /| | | \ 2 2 /| | 3*|1 - ----------------| | 2*|1 - ----------------| | | 2/3*a\ | / pi 3*a\ | | 2/3*a\ | / pi 3*a\ | | cos |---| | 4*cos|- -- + ---| | | cos |---| | 6*cos|- -- + ---| | \ \ 2 / / \ 2 2 / | \ \ 2 / / \ 2 2 / |- ------------------------ + -------------------------------| + ------------------------ + ------------------------------- | 2/ pi 3*a\ / 2/ pi 3*a\\ | 2/ pi 3*a\ / 2/ pi 3*a\\ | cos |- -- + ---| | cos |- -- + ---|| | cos |- -- + ---| | cos |- -- + ---|| | \ 2 2 / | \ 2 2 /| /3*a\| \ 2 2 / | \ 2 2 /| /3*a\ | 1 + ---------------- |1 + ----------------|*cos|---|| 1 + ---------------- |1 + ----------------|*cos|---| | 2/3*a\ | 2/3*a\ | \ 2 /| 2/3*a\ | 2/3*a\ | \ 2 / | cos |---| | cos |---| | | cos |---| | cos |---| | \ \ 2 / \ \ 2 / / / \ 2 / \ \ 2 / /
$$\left(- \frac{3 \cdot \left(1 - \frac{\cos^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{3 a}{2} \right)}}\right)}{1 + \frac{\cos^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{3 a}{2} \right)}}} + \frac{4 \cos{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{3 a}{2} \right)}}\right) \cos{\left(\frac{3 a}{2} \right)}}\right)^{2} + \frac{2 \cdot \left(1 - \frac{\cos^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{3 a}{2} \right)}}\right)}{1 + \frac{\cos^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{3 a}{2} \right)}}} + \frac{6 \cos{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{3 a}{2} \right)}}\right) \cos{\left(\frac{3 a}{2} \right)}}$$
2
/ // 1 for 3*a mod 2*pi = 0\ \ // 1 for 3*a mod 2*pi = 0\
| || | | || |
| || 1 | // 0 for 3*a mod pi = 0\| || 1 | // 0 for 3*a mod pi = 0\
| ||-1 + --------- | || || ||-1 + --------- | || |
| || 2/3*a\ | || 2 || || 2/3*a\ | || 2 |
| || tan |---| | ||------------------------ otherwise || || tan |---| | ||------------------------ otherwise |
|- 3*|< \ 2 / | + 2*|
$$\left(\left(\left(2 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{3 a}{2} \right)}}\right) \tan{\left(\frac{3 a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(3 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{3 a}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{3 a}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)\right)^{2}\right) + \left(3 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{3 a}{2} \right)}}\right) \tan{\left(\frac{3 a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{3 a}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{3 a}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)$$
2 / / 2/pi 3*a\\ \ / 2/pi 3*a\\ | | csc |-- - ---|| | | csc |-- - ---|| | | \2 2 /| | | \2 2 /| | 3*|1 - --------------| | 2*|1 - --------------| | | 2/3*a\ | /pi 3*a\ | | 2/3*a\ | /pi 3*a\ | | csc |---| | 4*csc|-- - ---| | | csc |---| | 6*csc|-- - ---| | \ \ 2 / / \2 2 / | \ \ 2 / / \2 2 / |- ---------------------- + -----------------------------| + ---------------------- + ----------------------------- | 2/pi 3*a\ / 2/pi 3*a\\ | 2/pi 3*a\ / 2/pi 3*a\\ | csc |-- - ---| | csc |-- - ---|| | csc |-- - ---| | csc |-- - ---|| | \2 2 / | \2 2 /| /3*a\| \2 2 / | \2 2 /| /3*a\ | 1 + -------------- |1 + --------------|*csc|---|| 1 + -------------- |1 + --------------|*csc|---| | 2/3*a\ | 2/3*a\ | \ 2 /| 2/3*a\ | 2/3*a\ | \ 2 / | csc |---| | csc |---| | | csc |---| | csc |---| | \ \ 2 / \ \ 2 / / / \ 2 / \ \ 2 / /
$$\left(- \frac{3 \cdot \left(1 - \frac{\csc^{2}{\left(- \frac{3 a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{3 a}{2} \right)}}\right)}{1 + \frac{\csc^{2}{\left(- \frac{3 a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{3 a}{2} \right)}}} + \frac{4 \csc{\left(- \frac{3 a}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{3 a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{3 a}{2} \right)}}\right) \csc{\left(\frac{3 a}{2} \right)}}\right)^{2} + \frac{2 \cdot \left(1 - \frac{\csc^{2}{\left(- \frac{3 a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{3 a}{2} \right)}}\right)}{1 + \frac{\csc^{2}{\left(- \frac{3 a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{3 a}{2} \right)}}} + \frac{6 \csc{\left(- \frac{3 a}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{3 a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{3 a}{2} \right)}}\right) \csc{\left(\frac{3 a}{2} \right)}}$$
2 / // /pi \ \ \ // /pi \ \ | || 0 for |-- + 3*a| mod pi = 0| // 0 for 3*a mod pi = 0\| || 0 for |-- + 3*a| mod pi = 0| // 0 for 3*a mod pi = 0\ | || \2 / | || || || \2 / | || | | || | || /3*a\ || || | || /3*a\ | | || /pi 3*a\ | || 2*cot|---| || || /pi 3*a\ | || 2*cot|---| | |- 3*|< 2*cot|-- + ---| | + 2*|< \ 2 / || + 2*|< 2*cot|-- + ---| | + 3*|< \ 2 / | | || \4 2 / | ||------------- otherwise || || \4 2 / | ||------------- otherwise | | ||------------------ otherwise | || 2/3*a\ || ||------------------ otherwise | || 2/3*a\ | | || 2/pi 3*a\ | ||1 + cot |---| || || 2/pi 3*a\ | ||1 + cot |---| | | ||1 + cot |-- + ---| | \\ \ 2 / /| ||1 + cot |-- + ---| | \\ \ 2 / / \ \\ \4 2 / / / \\ \4 2 / /
$$\left(\left(\left(2 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 a}{2} \right)}}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(3 \left(\begin{cases} 0 & \text{for}\: \left(3 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)\right)^{2}\right) + \left(3 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 a}{2} \right)}}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 0 & \text{for}\: \left(3 a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
2 / // / 3*pi\ \\ // / 3*pi\ \ | // 1 for 3*a mod 2*pi = 0\ || 1 for |3*a + ----| mod 2*pi = 0|| // 1 for 3*a mod 2*pi = 0\ || 1 for |3*a + ----| mod 2*pi = 0| | || | || \ 2 / || || | || \ 2 / | | || 2/3*a\ | || || || 2/3*a\ | || | | ||-1 + cot |---| | || 2/pi 3*a\ || ||-1 + cot |---| | || 2/pi 3*a\ | |- 3*|< \ 2 / | + 2*|<-1 + tan |-- + ---| || + 2*|< \ 2 / | + 3*|<-1 + tan |-- + ---| | | ||-------------- otherwise | || \4 2 / || ||-------------- otherwise | || \4 2 / | | || 2/3*a\ | ||------------------- otherwise || || 2/3*a\ | ||------------------- otherwise | | ||1 + cot |---| | || 2/pi 3*a\ || ||1 + cot |---| | || 2/pi 3*a\ | | \\ \ 2 / / || 1 + tan |-- + ---| || \\ \ 2 / / || 1 + tan |-- + ---| | \ \\ \4 2 / // \\ \4 2 / /
$$\left(\left(\left(- 3 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{3 a}{2} \right)} - 1}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: \left(3 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)\right)^{2}\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{3 a}{2} \right)} - 1}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(3 \left(\begin{cases} 1 & \text{for}\: \left(3 a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{3 a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
2 / // 1 for 3*a mod 2*pi = 0\ // 0 for 3*a mod pi = 0\\ // 1 for 3*a mod 2*pi = 0\ // 0 for 3*a mod pi = 0\ | || | || || || | || | | || -2 - 2*cos(6*a) + 4*cos(3*a) | || -2*sin(6*a) + 4*sin(3*a) || || -2 - 2*cos(6*a) + 4*cos(3*a) | || -2*sin(6*a) + 4*sin(3*a) | |- 3*|<-------------------------------- otherwise | + 2*|<-------------------------------- otherwise || + 2*|<-------------------------------- otherwise | + 3*|<-------------------------------- otherwise | | || 2 | || 2 || || 2 | || 2 | | ||1 - cos(6*a) + 2*(1 - cos(3*a)) | ||1 - cos(6*a) + 2*(1 - cos(3*a)) || ||1 - cos(6*a) + 2*(1 - cos(3*a)) | ||1 - cos(6*a) + 2*(1 - cos(3*a)) | \ \\ / \\ // \\ / \\ /
$$\left(\left(\left(2 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{4 \sin{\left(3 a \right)} - 2 \sin{\left(6 a \right)}}{2 \left(- \cos{\left(3 a \right)} + 1\right)^{2} - \cos{\left(6 a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(3 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{4 \cos{\left(3 a \right)} - 2 \cos{\left(6 a \right)} - 2}{2 \left(- \cos{\left(3 a \right)} + 1\right)^{2} - \cos{\left(6 a \right)} + 1} & \text{otherwise} \end{cases}\right)\right)\right)^{2}\right) + \left(3 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{4 \sin{\left(3 a \right)} - 2 \sin{\left(6 a \right)}}{2 \left(- \cos{\left(3 a \right)} + 1\right)^{2} - \cos{\left(6 a \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{4 \cos{\left(3 a \right)} - 2 \cos{\left(6 a \right)} - 2}{2 \left(- \cos{\left(3 a \right)} + 1\right)^{2} - \cos{\left(6 a \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
2 / // 1 for 3*a mod 2*pi = 0\ \ // 1 for 3*a mod 2*pi = 0\ | || | | || | | || 2 | // 0 for 3*a mod pi = 0\| || 2 | // 0 for 3*a mod pi = 0\ | || sin (3*a) | || || || sin (3*a) | || | | ||-1 + ----------- | || sin(3*a) || ||-1 + ----------- | || sin(3*a) | | || 4/3*a\ | ||--------------------------- otherwise || || 4/3*a\ | ||--------------------------- otherwise | | || 4*sin |---| | ||/ 2 \ || || 4*sin |---| | ||/ 2 \ | |- 3*|< \ 2 / | + 2*|<| sin (3*a) | 2/3*a\ || + 2*|< \ 2 / | + 3*|<| sin (3*a) | 2/3*a\ | | ||---------------- otherwise | |||1 + -----------|*sin |---| || ||---------------- otherwise | |||1 + -----------|*sin |---| | | || 2 | ||| 4/3*a\| \ 2 / || || 2 | ||| 4/3*a\| \ 2 / | | || sin (3*a) | ||| 4*sin |---|| || || sin (3*a) | ||| 4*sin |---|| | | ||1 + ----------- | ||\ \ 2 // || ||1 + ----------- | ||\ \ 2 // | | || 4/3*a\ | \\ /| || 4/3*a\ | \\ / | || 4*sin |---| | | || 4*sin |---| | \ \\ \ 2 / / / \\ \ 2 / /
$$\left(\left(\left(2 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{\sin{\left(3 a \right)}}{\left(1 + \frac{\sin^{2}{\left(3 a \right)}}{4 \sin^{4}{\left(\frac{3 a}{2} \right)}}\right) \sin^{2}{\left(\frac{3 a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(3 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(3 a \right)}}{4 \sin^{4}{\left(\frac{3 a}{2} \right)}}}{1 + \frac{\sin^{2}{\left(3 a \right)}}{4 \sin^{4}{\left(\frac{3 a}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)\right)^{2}\right) + \left(3 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{\sin{\left(3 a \right)}}{\left(1 + \frac{\sin^{2}{\left(3 a \right)}}{4 \sin^{4}{\left(\frac{3 a}{2} \right)}}\right) \sin^{2}{\left(\frac{3 a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(3 a \right)}}{4 \sin^{4}{\left(\frac{3 a}{2} \right)}}}{1 + \frac{\sin^{2}{\left(3 a \right)}}{4 \sin^{4}{\left(\frac{3 a}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)$$
2 / // 1 for 3*a mod 2*pi = 0\ // 0 for 3*a mod pi = 0\\ // 1 for 3*a mod 2*pi = 0\ // 0 for 3*a mod pi = 0\ | || | || || || | || | | ||/ 1 for 3*a mod 2*pi = 0 | ||/ 0 for 3*a mod pi = 0 || ||/ 1 for 3*a mod 2*pi = 0 | ||/ 0 for 3*a mod pi = 0 | | ||| | ||| || ||| | ||| | | ||| 2/3*a\ | ||| /3*a\ || ||| 2/3*a\ | ||| /3*a\ | |- 3*|<|-1 + cot |---| | + 2*|<| 2*cot|---| || + 2*|<|-1 + cot |---| | + 3*|<| 2*cot|---| | | ||< \ 2 / otherwise | ||< \ 2 / otherwise || ||< \ 2 / otherwise | ||< \ 2 / otherwise | | |||-------------- otherwise | |||------------- otherwise || |||-------------- otherwise | |||------------- otherwise | | ||| 2/3*a\ | ||| 2/3*a\ || ||| 2/3*a\ | ||| 2/3*a\ | | |||1 + cot |---| | |||1 + cot |---| || |||1 + cot |---| | |||1 + cot |---| | \ \\\ \ 2 / / \\\ \ 2 / // \\\ \ 2 / / \\\ \ 2 / /
$$\left(\left(\left(2 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 a}{2} \right)}}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) - \left(3 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{3 a}{2} \right)} - 1}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)\right)^{2}\right) + \left(3 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 a}{2} \right)}}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{3 a}{2} \right)} - 1}{\cot^{2}{\left(\frac{3 a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
2
/ // 1 for 3*a mod 2*pi = 0\ \ // 1 for 3*a mod 2*pi = 0\
| || | | || |
| || 2/3*a\ | // 0 for 3*a mod pi = 0\| || 2/3*a\ | // 0 for 3*a mod pi = 0\
| || cos |---| | || || || cos |---| | || |
| || \ 2 / | || /3*a\ || || \ 2 / | || /3*a\ |
| ||-1 + ---------------- | || 2*cos|---| || ||-1 + ---------------- | || 2*cos|---| |
| || 2/ pi 3*a\ | || \ 2 / || || 2/ pi 3*a\ | || \ 2 / |
| || cos |- -- + ---| | ||-------------------------------------- otherwise || || cos |- -- + ---| | ||-------------------------------------- otherwise |
|- 3*|< \ 2 2 / | + 2*|
$$\left(\left(\left(2 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{3 a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{3 a}{2} \right)}}{\cos^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(3 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{3 a}{2} \right)}}{\cos^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{3 a}{2} \right)}}{\cos^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right)\right)^{2}\right) + \left(3 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{3 a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{3 a}{2} \right)}}{\cos^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{3 a}{2} \right)}}{\cos^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{3 a}{2} \right)}}{\cos^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right)$$
2
/ // 1 for 3*a mod 2*pi = 0\ \ // 1 for 3*a mod 2*pi = 0\
| || | | || |
| || 2/ pi 3*a\ | // 0 for 3*a mod pi = 0\| || 2/ pi 3*a\ | // 0 for 3*a mod pi = 0\
| || sec |- -- + ---| | || || || sec |- -- + ---| | || |
| || \ 2 2 / | || / pi 3*a\ || || \ 2 2 / | || / pi 3*a\ |
| ||-1 + ---------------- | || 2*sec|- -- + ---| || ||-1 + ---------------- | || 2*sec|- -- + ---| |
| || 2/3*a\ | || \ 2 2 / || || 2/3*a\ | || \ 2 2 / |
| || sec |---| | ||------------------------------- otherwise || || sec |---| | ||------------------------------- otherwise |
|- 3*|< \ 2 / | + 2*|
$$\left(\left(\left(2 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{3 a}{2} \right)}}\right) \sec{\left(\frac{3 a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(3 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{3 a}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{3 a}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)\right)^{2}\right) + \left(3 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{3 a}{2} \right)}}\right) \sec{\left(\frac{3 a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{3 a}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{3 a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{3 a}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)$$
2
/ // 1 for 3*a mod 2*pi = 0\ \ // 1 for 3*a mod 2*pi = 0\
| || | | || |
| || 2/3*a\ | // 0 for 3*a mod pi = 0\| || 2/3*a\ | // 0 for 3*a mod pi = 0\
| || csc |---| | || || || csc |---| | || |
| || \ 2 / | || /3*a\ || || \ 2 / | || /3*a\ |
| ||-1 + -------------- | || 2*csc|---| || ||-1 + -------------- | || 2*csc|---| |
| || 2/pi 3*a\ | || \ 2 / || || 2/pi 3*a\ | || \ 2 / |
| || csc |-- - ---| | ||---------------------------------- otherwise || || csc |-- - ---| | ||---------------------------------- otherwise |
|- 3*|< \2 2 / | + 2*|
$$\left(\left(\left(2 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{3 a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{3 a}{2} \right)}}{\csc^{2}{\left(- \frac{3 a}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{3 a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(3 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{3 a}{2} \right)}}{\csc^{2}{\left(- \frac{3 a}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{3 a}{2} \right)}}{\csc^{2}{\left(- \frac{3 a}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right)\right)^{2}\right) + \left(3 \left(\begin{cases} 0 & \text{for}\: 3 a \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{3 a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{3 a}{2} \right)}}{\csc^{2}{\left(- \frac{3 a}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{3 a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(2 \left(\begin{cases} 1 & \text{for}\: 3 a \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{3 a}{2} \right)}}{\csc^{2}{\left(- \frac{3 a}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{3 a}{2} \right)}}{\csc^{2}{\left(- \frac{3 a}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right)$$
(-3*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)) + 2*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)))^2 + 2*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)) + 3*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))