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