Господин Экзамен
cos(7*x)*cos(4*x)-sin(7*x)*sin(4*x) если x=4
Решение
Вы ввели
[src]
cos(7*x)*cos(4*x) - sin(7*x)*sin(4*x)
$$- \sin{\left(4 x \right)} \sin{\left(7 x \right)} + \cos{\left(4 x \right)} \cos{\left(7 x \right)}$$
cos(7*x)*cos(4*x) - sin(7*x)*sin(4*x)
Подстановка условия
[src]
cos(7*x)*cos(4*x) - sin(7*x)*sin(4*x) при x = 4
подставляем
cos(7*x)*cos(4*x) - sin(7*x)*sin(4*x)
$$- \sin{\left(4 x \right)} \sin{\left(7 x \right)} + \cos{\left(4 x \right)} \cos{\left(7 x \right)}$$
cos(11*x)
$$\cos{\left(11 x \right)}$$
переменные
x = 4
$$x = 4$$
cos(11*(4))
$$\cos{\left(11 (4) \right)}$$
cos(11*4)
$$\cos{\left(11 \cdot 4 \right)}$$
cos(44)
$$\cos{\left(44 \right)}$$
cos(44)
Тригонометрическая часть
[src]
cos(11*x)
$$\cos{\left(11 x \right)}$$
1 --------- sec(11*x)
$$\frac{1}{\sec{\left(11 x \right)}}$$
/pi \ sin|-- + 11*x| \2 /
$$\sin{\left(11 x + \frac{\pi}{2} \right)}$$
1 -------------- /pi \ csc|-- - 11*x| \2 /
$$\frac{1}{\csc{\left(- 11 x + \frac{\pi}{2} \right)}}$$
2/11*x\
1 - tan |----|
\ 2 /
--------------
2/11*x\
1 + tan |----|
\ 2 /
$$\frac{- \tan^{2}{\left(\frac{11 x}{2} \right)} + 1}{\tan^{2}{\left(\frac{11 x}{2} \right)} + 1}$$
/ 1 for 11*x mod 2*pi = 0 < \cos(11*x) otherwise
$$\begin{cases} 1 & \text{for}\: 11 x \bmod 2 \pi = 0 \\\cos{\left(11 x \right)} & \text{otherwise} \end{cases}$$
1 1 ----------------- - ----------------- sec(4*x)*sec(7*x) csc(4*x)*csc(7*x)
$$\frac{1}{\sec{\left(4 x \right)} \sec{\left(7 x \right)}} - \frac{1}{\csc{\left(4 x \right)} \csc{\left(7 x \right)}}$$
/ pi\ / pi\
cos(4*x)*cos(7*x) - cos|4*x - --|*cos|7*x - --|
\ 2 / \ 2 /
$$\cos{\left(4 x \right)} \cos{\left(7 x \right)} - \cos{\left(4 x - \frac{\pi}{2} \right)} \cos{\left(7 x - \frac{\pi}{2} \right)}$$
/pi \ /pi \ sin|-- + 4*x|*sin|-- + 7*x| - sin(4*x)*sin(7*x) \2 / \2 /
$$- \sin{\left(4 x \right)} \sin{\left(7 x \right)} + \sin{\left(4 x + \frac{\pi}{2} \right)} \sin{\left(7 x + \frac{\pi}{2} \right)}$$
1 1
----------------- - ---------------------------
sec(4*x)*sec(7*x) / pi\ / pi\
sec|4*x - --|*sec|7*x - --|
\ 2 / \ 2 /
$$- \frac{1}{\sec{\left(4 x - \frac{\pi}{2} \right)} \sec{\left(7 x - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(4 x \right)} \sec{\left(7 x \right)}}$$
1 1 --------------------------- - ----------------- /pi \ /pi \ csc(4*x)*csc(7*x) csc|-- - 7*x|*csc|-- - 4*x| \2 / \2 /
$$\frac{1}{\csc{\left(- 7 x + \frac{\pi}{2} \right)} \csc{\left(- 4 x + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(4 x \right)} \csc{\left(7 x \right)}}$$
1 1
----------------- - ---------------------------
sec(4*x)*sec(7*x) /pi \ /pi \
sec|-- - 7*x|*sec|-- - 4*x|
\2 / \2 /
$$- \frac{1}{\sec{\left(- 7 x + \frac{\pi}{2} \right)} \sec{\left(- 4 x + \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(4 x \right)} \sec{\left(7 x \right)}}$$
1 1 --------------------------- - --------------------------- /pi \ /pi \ csc(pi - 7*x)*csc(pi - 4*x) csc|-- - 7*x|*csc|-- - 4*x| \2 / \2 /
$$\frac{1}{\csc{\left(- 7 x + \frac{\pi}{2} \right)} \csc{\left(- 4 x + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(- 7 x + \pi \right)} \csc{\left(- 4 x + \pi \right)}}$$
/ 1 for 11*x mod 2*pi = 0 | | 2/11*x\ |-1 + cot |----| < \ 2 / |--------------- otherwise | 2/11*x\ | 1 + cot |----| \ \ 2 /
$$\begin{cases} 1 & \text{for}\: 11 x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{11 x}{2} \right)} - 1}{\cot^{2}{\left(\frac{11 x}{2} \right)} + 1} & \text{otherwise} \end{cases}$$
/7*x\
(-sin(3*x) + 2*sin(4*x) + sin(11*x))*tan|---|
cos(3*x) + cos(11*x) \ 2 /
-------------------- - ---------------------------------------------
2 2
$$- \frac{\left(- \sin{\left(3 x \right)} + 2 \sin{\left(4 x \right)} + \sin{\left(11 x \right)}\right) \tan{\left(\frac{7 x}{2} \right)}}{2} + \frac{\cos{\left(3 x \right)} + \cos{\left(11 x \right)}}{2}$$
2 2/7*x\ -cos(11*x) + cos(3*x) 2 2/7*x\
1 - 2*cos (2*x) - 2*cos |---| - --------------------- + 4*cos (2*x)*cos |---|
\ 2 / 2 \ 2 /
$$4 \cos^{2}{\left(2 x \right)} \cos^{2}{\left(\frac{7 x}{2} \right)} - 2 \cos^{2}{\left(2 x \right)} - 2 \cos^{2}{\left(\frac{7 x}{2} \right)} - \frac{\cos{\left(3 x \right)} - \cos{\left(11 x \right)}}{2} + 1$$
2/ pi\ / 2/ pi\\ / 2/pi 7*x\\
sin |2*x + --|*|1 - cot |2*x + --||*|1 - cot |-- + ---||*(1 + sin(7*x))
cos(3*x) + cos(11*x) \ 4 / \ \ 4 // \ \4 2 //
-------------------- - -----------------------------------------------------------------------
2 2
$$- \frac{\left(- \cot^{2}{\left(2 x + \frac{\pi}{4} \right)} + 1\right) \left(- \cot^{2}{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(7 x \right)} + 1\right) \sin^{2}{\left(2 x + \frac{\pi}{4} \right)}}{2} + \frac{\cos{\left(3 x \right)} + \cos{\left(11 x \right)}}{2}$$
/ 2 \ |/ 2 2 \ 2 2 | / 2 2 \ \\cos (x) - sin (x)/ - 4*cos (x)*sin (x)/*cos(7*x) - 4*\cos (x) - sin (x)/*cos(x)*sin(x)*sin(7*x)
$$- 4 \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \sin{\left(7 x \right)} \cos{\left(x \right)} + \left(- 4 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} + \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)^{2}\right) \cos{\left(7 x \right)}$$
/ 2 \ / 2/7*x\\ /7*x\
\1 - tan (2*x)/*|1 - tan |---|| 4*tan(2*x)*tan|---|
\ \ 2 // \ 2 /
------------------------------- - -------------------------------
/ 2 \ / 2/7*x\\ / 2 \ / 2/7*x\\
\1 + tan (2*x)/*|1 + tan |---|| \1 + tan (2*x)/*|1 + tan |---||
\ \ 2 // \ \ 2 //
$$\frac{\left(- \tan^{2}{\left(2 x \right)} + 1\right) \left(- \tan^{2}{\left(\frac{7 x}{2} \right)} + 1\right)}{\left(\tan^{2}{\left(2 x \right)} + 1\right) \left(\tan^{2}{\left(\frac{7 x}{2} \right)} + 1\right)} - \frac{4 \tan{\left(2 x \right)} \tan{\left(\frac{7 x}{2} \right)}}{\left(\tan^{2}{\left(2 x \right)} + 1\right) \left(\tan^{2}{\left(\frac{7 x}{2} \right)} + 1\right)}$$
/7*x\ / pi\ /pi 7*x\
4*tan(2*x)*tan|---| 4*tan|2*x + --|*tan|-- + ---|
\ 2 / \ 4 / \4 2 /
- ------------------------------- + -----------------------------------------
/ 2 \ / 2/7*x\\ / 2/ pi\\ / 2/pi 7*x\\
\1 + tan (2*x)/*|1 + tan |---|| |1 + tan |2*x + --||*|1 + tan |-- + ---||
\ \ 2 // \ \ 4 // \ \4 2 //
$$\frac{4 \tan{\left(2 x + \frac{\pi}{4} \right)} \tan{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(2 x + \frac{\pi}{4} \right)} + 1\right) \left(\tan^{2}{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)} + 1\right)} - \frac{4 \tan{\left(2 x \right)} \tan{\left(\frac{7 x}{2} \right)}}{\left(\tan^{2}{\left(2 x \right)} + 1\right) \left(\tan^{2}{\left(\frac{7 x}{2} \right)} + 1\right)}$$
/7*x\ / pi\ /pi 7*x\
4*cot(2*x)*cot|---| 4*tan|2*x + --|*tan|-- + ---|
\ 2 / \ 4 / \4 2 /
- ------------------------------- + -----------------------------------------
/ 2 \ / 2/7*x\\ / 2/ pi\\ / 2/pi 7*x\\
\1 + cot (2*x)/*|1 + cot |---|| |1 + tan |2*x + --||*|1 + tan |-- + ---||
\ \ 2 // \ \ 4 // \ \4 2 //
$$- \frac{4 \cot{\left(2 x \right)} \cot{\left(\frac{7 x}{2} \right)}}{\left(\cot^{2}{\left(2 x \right)} + 1\right) \left(\cot^{2}{\left(\frac{7 x}{2} \right)} + 1\right)} + \frac{4 \tan{\left(2 x + \frac{\pi}{4} \right)} \tan{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(2 x + \frac{\pi}{4} \right)} + 1\right) \left(\tan^{2}{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)} + 1\right)}$$
/ 1 \ / 1 \
|1 - ---------|*|1 - ---------|
| 2 | | 2/7*x\|
\ cot (2*x)/ | cot |---||
\ \ 2 // 4
------------------------------- - -------------------------------------------------
/ 1 \ / 1 \ / 1 \ / 1 \ /7*x\
|1 + ---------|*|1 + ---------| |1 + ---------|*|1 + ---------|*cot(2*x)*cot|---|
| 2 | | 2/7*x\| | 2 | | 2/7*x\| \ 2 /
\ cot (2*x)/ | cot |---|| \ cot (2*x)/ | cot |---||
\ \ 2 // \ \ 2 //
$$\frac{\left(1 - \frac{1}{\cot^{2}{\left(2 x \right)}}\right) \left(1 - \frac{1}{\cot^{2}{\left(\frac{7 x}{2} \right)}}\right)}{\left(1 + \frac{1}{\cot^{2}{\left(2 x \right)}}\right) \left(1 + \frac{1}{\cot^{2}{\left(\frac{7 x}{2} \right)}}\right)} - \frac{4}{\left(1 + \frac{1}{\cot^{2}{\left(2 x \right)}}\right) \left(1 + \frac{1}{\cot^{2}{\left(\frac{7 x}{2} \right)}}\right) \cot{\left(2 x \right)} \cot{\left(\frac{7 x}{2} \right)}}$$
/ 2 \ / 2/7*x\\ / 2/ pi\\ / 2/pi 7*x\\
\-1 + cot (2*x)/*|-1 + cot |---|| |-1 + tan |2*x + --||*|-1 + tan |-- + ---||
\ \ 2 // \ \ 4 // \ \4 2 //
--------------------------------- - -------------------------------------------
/ 2 \ / 2/7*x\\ / 2/ pi\\ / 2/pi 7*x\\
\1 + cot (2*x)/*|1 + cot |---|| |1 + tan |2*x + --||*|1 + tan |-- + ---||
\ \ 2 // \ \ 4 // \ \4 2 //
$$- \frac{\left(\tan^{2}{\left(2 x + \frac{\pi}{4} \right)} - 1\right) \left(\tan^{2}{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)} - 1\right)}{\left(\tan^{2}{\left(2 x + \frac{\pi}{4} \right)} + 1\right) \left(\tan^{2}{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)} + 1\right)} + \frac{\left(\cot^{2}{\left(2 x \right)} - 1\right) \left(\cot^{2}{\left(\frac{7 x}{2} \right)} - 1\right)}{\left(\cot^{2}{\left(2 x \right)} + 1\right) \left(\cot^{2}{\left(\frac{7 x}{2} \right)} + 1\right)}$$
/ 2 \ / 2/7*x\\ / 2/ pi\\ / 2/pi 7*x\\
\1 - tan (2*x)/*|1 - tan |---|| |1 - cot |2*x + --||*|1 - cot |-- + ---||
\ \ 2 // \ \ 4 // \ \4 2 //
------------------------------- - -----------------------------------------
/ 2 \ / 2/7*x\\ / 2/ pi\\ / 2/pi 7*x\\
\1 + tan (2*x)/*|1 + tan |---|| |1 + cot |2*x + --||*|1 + cot |-- + ---||
\ \ 2 // \ \ 4 // \ \4 2 //
$$\frac{\left(- \tan^{2}{\left(2 x \right)} + 1\right) \left(- \tan^{2}{\left(\frac{7 x}{2} \right)} + 1\right)}{\left(\tan^{2}{\left(2 x \right)} + 1\right) \left(\tan^{2}{\left(\frac{7 x}{2} \right)} + 1\right)} - \frac{\left(- \cot^{2}{\left(2 x + \frac{\pi}{4} \right)} + 1\right) \left(- \cot^{2}{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)} + 1\right)}{\left(\cot^{2}{\left(2 x + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)} + 1\right)}$$
/ 4/7*x\\
/ 4 \ | 4*sin |---||
| 4*sin (2*x)| | \ 2 /|
|1 - -----------|*|1 - -----------|*(1 + cos(4*x))
| 2 | | 2 |
-cos(10*x) - 2*cos(11*x) + 2*cos(3*x) + cos(18*x) \ sin (4*x) / \ sin (7*x) /
- ------------------------------------------------- + --------------------------------------------------
2 / 4/7*x\\
1 - cos(14*x) + 2*(1 - cos(7*x)) | 4*sin |---||
| \ 2 /|
2*|1 + -----------|
| 2 |
\ sin (7*x) /
$$\frac{\left(- \frac{4 \sin^{4}{\left(2 x \right)}}{\sin^{2}{\left(4 x \right)}} + 1\right) \left(- \frac{4 \sin^{4}{\left(\frac{7 x}{2} \right)}}{\sin^{2}{\left(7 x \right)}} + 1\right) \left(\cos{\left(4 x \right)} + 1\right)}{2 \cdot \left(\frac{4 \sin^{4}{\left(\frac{7 x}{2} \right)}}{\sin^{2}{\left(7 x \right)}} + 1\right)} - \frac{2 \cos{\left(3 x \right)} - \cos{\left(10 x \right)} - 2 \cos{\left(11 x \right)} + \cos{\left(18 x \right)}}{2 \left(- \cos{\left(7 x \right)} + 1\right)^{2} - \cos{\left(14 x \right)} + 1}$$
// 1 for 2*x mod pi = 0\ || | // 1 for 7*x mod 2*pi = 0\ // 0 for 4*x mod pi = 0\ // 0 for 7*x mod pi = 0\ |< 2 / 2 \ |*|< | - |< |*|< | ||sin (2*x)*\-1 + cot (2*x)/ otherwise | \\cos(7*x) otherwise / \\sin(4*x) otherwise / \\sin(7*x) otherwise / \\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 x \bmod \pi = 0 \\\sin{\left(4 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 7 x \bmod \pi = 0 \\\sin{\left(7 x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 x \bmod \pi = 0 \\\left(\cot^{2}{\left(2 x \right)} - 1\right) \sin^{2}{\left(2 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 7 x \bmod 2 \pi = 0 \\\cos{\left(7 x \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 2*x mod pi = 0\ || | || 1 | ||-1 + --------- | // 1 for 7*x mod 2*pi = 0\ // 0 for 4*x mod pi = 0\ // 0 for 7*x mod pi = 0\ |< 2 |*|< | - |< |*|< | || tan (2*x) | \\cos(7*x) otherwise / \\sin(4*x) otherwise / \\sin(7*x) otherwise / ||-------------- otherwise | || 2 | \\ csc (2*x) /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 x \bmod \pi = 0 \\\sin{\left(4 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 7 x \bmod \pi = 0 \\\sin{\left(7 x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(2 x \right)}}}{\csc^{2}{\left(2 x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 7 x \bmod 2 \pi = 0 \\\cos{\left(7 x \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 2*x mod pi = 0\ || | || 2 / 1 \ | // 1 for 7*x mod 2*pi = 0\ // 0 for 4*x mod pi = 0\ // 0 for 7*x mod pi = 0\ |
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 x \bmod \pi = 0 \\\sin{\left(4 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 7 x \bmod \pi = 0 \\\sin{\left(7 x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 x \bmod \pi = 0 \\\left(-1 + \frac{1}{\tan^{2}{\left(2 x \right)}}\right) \sin^{2}{\left(2 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 7 x \bmod 2 \pi = 0 \\\cos{\left(7 x \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 2*x mod pi = 0\ || | || 4 2 / 1 \ | // 1 for 7*x mod 2*pi = 0\ // 0 for 4*x mod pi = 0\ // 0 for 7*x mod pi = 0\ |<4*cos (x)*tan (x)*|-1 + ---------| otherwise |*|< | - |< |*|< | || | 2 | | \\cos(7*x) otherwise / \\sin(4*x) otherwise / \\sin(7*x) otherwise / || \ tan (2*x)/ | \\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 x \bmod \pi = 0 \\\sin{\left(4 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 7 x \bmod \pi = 0 \\\sin{\left(7 x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 x \bmod \pi = 0 \\4 \left(-1 + \frac{1}{\tan^{2}{\left(2 x \right)}}\right) \cos^{4}{\left(x \right)} \tan^{2}{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 7 x \bmod 2 \pi = 0 \\\cos{\left(7 x \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 2*x mod pi = 0\ || | || / 2 \ | // 1 for 7*x mod 2*pi = 0\ // 0 for 4*x mod pi = 0\ // 0 for 7*x mod pi = 0\ |< 2 | sin (4*x) | |*|< | - |< |*|< | ||sin (2*x)*|-1 + -----------| otherwise | \\cos(7*x) otherwise / \\sin(4*x) otherwise / \\sin(7*x) otherwise / || | 4 | | \\ \ 4*sin (2*x)/ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 x \bmod \pi = 0 \\\sin{\left(4 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 7 x \bmod \pi = 0 \\\sin{\left(7 x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 x \bmod \pi = 0 \\\left(-1 + \frac{\sin^{2}{\left(4 x \right)}}{4 \sin^{4}{\left(2 x \right)}}\right) \sin^{2}{\left(2 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 7 x \bmod 2 \pi = 0 \\\cos{\left(7 x \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 2*x mod pi = 0\ || | // 1 for 7*x mod 2*pi = 0\ || / 2 \ | || | // 0 for 4*x mod pi = 0\ // 0 for 7*x mod pi = 0\ |< 2 | sin (4*x) | |*|< /pi \ | - |< |*|< | ||sin (2*x)*|-1 + -----------| otherwise | ||sin|-- + 7*x| otherwise | \\sin(4*x) otherwise / \\sin(7*x) otherwise / || | 4 | | \\ \2 / / \\ \ 4*sin (2*x)/ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 x \bmod \pi = 0 \\\sin{\left(4 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 7 x \bmod \pi = 0 \\\sin{\left(7 x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 x \bmod \pi = 0 \\\left(-1 + \frac{\sin^{2}{\left(4 x \right)}}{4 \sin^{4}{\left(2 x \right)}}\right) \sin^{2}{\left(2 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 7 x \bmod 2 \pi = 0 \\\sin{\left(7 x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 2*x mod pi = 0\ // 0 for 7*x mod pi = 0\ || | || | || 2 / 1 \ | // 1 for 7*x mod 2*pi = 0\ // 0 for 4*x mod pi = 0\ ||1 - cos(7*x) | |
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 x \bmod \pi = 0 \\\sin{\left(4 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 7 x \bmod \pi = 0 \\\frac{- \cos{\left(7 x \right)} + 1}{\tan{\left(\frac{7 x}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 x \bmod \pi = 0 \\\left(-1 + \frac{1}{\tan^{2}{\left(2 x \right)}}\right) \sin^{2}{\left(2 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 7 x \bmod 2 \pi = 0 \\\cos{\left(7 x \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 2*x mod pi = 0\ // / 3*pi\ \ // / 3*pi\ \ || | // 1 for 7*x mod 2*pi = 0\ || 1 for |4*x + ----| mod 2*pi = 0| || 1 for |7*x + ----| mod 2*pi = 0| |< 2 / 2 \ |*|< | - |< \ 2 / |*|< \ 2 / | ||sin (2*x)*\-1 + cot (2*x)/ otherwise | \\cos(7*x) otherwise / || | || | \\ / \\sin(4*x) otherwise / \\sin(7*x) otherwise /
$$\left(\left(\begin{cases} 1 & \text{for}\: 2 x \bmod \pi = 0 \\\left(\cot^{2}{\left(2 x \right)} - 1\right) \sin^{2}{\left(2 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 7 x \bmod 2 \pi = 0 \\\cos{\left(7 x \right)} & \text{otherwise} \end{cases}\right)\right) - \left(\left(\begin{cases} 1 & \text{for}\: \left(4 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(4 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(7 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(7 x \right)} & \text{otherwise} \end{cases}\right)\right)$$
// /pi \ \
// /pi \ \ || 0 for |-- + 7*x| mod pi = 0|
|| 0 for |-- + 4*x| mod pi = 0| || \2 / | // 0 for 4*x mod pi = 0\ // 0 for 7*x mod pi = 0\
|< \2 / |*|< | - |< |*|< |
|| | || /pi 7*x\ | \\sin(4*x) otherwise / \\sin(7*x) otherwise /
\\cos(4*x) otherwise / ||(1 + sin(7*x))*cot|-- + ---| otherwise |
\\ \4 2 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 x \bmod \pi = 0 \\\sin{\left(4 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 7 x \bmod \pi = 0 \\\sin{\left(7 x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: \left(4 x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\cos{\left(4 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(7 x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(7 x \right)} + 1\right) \cot{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 2*x mod pi = 0\ || | || 2 | // 1 for 7*x mod 2*pi = 0\ || csc (2*x) | || | // 0 for 4*x mod pi = 0\ // 0 for 7*x mod pi = 0\ ||-1 + -------------- | || 1 | || | || | |< 2/pi \ |*|<------------- otherwise | - |< 1 |*|< 1 | || csc |-- - 2*x| | || /pi \ | ||-------- otherwise | ||-------- otherwise | || \2 / | ||csc|-- - 7*x| | \\csc(4*x) / \\csc(7*x) / ||------------------- otherwise | \\ \2 / / || 2 | \\ csc (2*x) /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 x \bmod \pi = 0 \\\frac{1}{\csc{\left(4 x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 7 x \bmod \pi = 0 \\\frac{1}{\csc{\left(7 x \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{\frac{\csc^{2}{\left(2 x \right)}}{\csc^{2}{\left(- 2 x + \frac{\pi}{2} \right)}} - 1}{\csc^{2}{\left(2 x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 7 x \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- 7 x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 2*x mod pi = 0\ || | || / 2 \ | // 0 for 4*x mod pi = 0\ // 0 for 7*x mod pi = 0\ || 2/ pi\ | cos (2*x) | | // 1 for 7*x mod 2*pi = 0\ || | || | |
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 x \bmod \pi = 0 \\\cos{\left(4 x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 7 x \bmod \pi = 0 \\\cos{\left(7 x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 x \bmod \pi = 0 \\\left(\frac{\cos^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x - \frac{\pi}{2} \right)}} - 1\right) \cos^{2}{\left(2 x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 7 x \bmod 2 \pi = 0 \\\cos{\left(7 x \right)} & \text{otherwise} \end{cases}\right)\right)$$
/ 4/7*x\\
/ 4 \ | 4*sin |---||
| 4*sin (2*x)| | \ 2 /|
|1 - -----------|*|1 - -----------| 2 2/7*x\
| 2 | | 2 | 16*sin (2*x)*sin |---|
\ sin (4*x) / \ sin (7*x) / \ 2 /
----------------------------------- - -----------------------------------------------------
/ 4/7*x\\ / 4/7*x\\
/ 4 \ | 4*sin |---|| / 4 \ | 4*sin |---||
| 4*sin (2*x)| | \ 2 /| | 4*sin (2*x)| | \ 2 /|
|1 + -----------|*|1 + -----------| |1 + -----------|*|1 + -----------|*sin(4*x)*sin(7*x)
| 2 | | 2 | | 2 | | 2 |
\ sin (4*x) / \ sin (7*x) / \ sin (4*x) / \ sin (7*x) /
$$\frac{\left(- \frac{4 \sin^{4}{\left(2 x \right)}}{\sin^{2}{\left(4 x \right)}} + 1\right) \left(- \frac{4 \sin^{4}{\left(\frac{7 x}{2} \right)}}{\sin^{2}{\left(7 x \right)}} + 1\right)}{\left(\frac{4 \sin^{4}{\left(2 x \right)}}{\sin^{2}{\left(4 x \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{7 x}{2} \right)}}{\sin^{2}{\left(7 x \right)}} + 1\right)} - \frac{16 \sin^{2}{\left(2 x \right)} \sin^{2}{\left(\frac{7 x}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(2 x \right)}}{\sin^{2}{\left(4 x \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{7 x}{2} \right)}}{\sin^{2}{\left(7 x \right)}} + 1\right) \sin{\left(4 x \right)} \sin{\left(7 x \right)}}$$
// 1 for 2*x mod pi = 0\ || | || 2/ pi\ | || sec |2*x - --| | // 0 for 4*x mod pi = 0\ // 0 for 7*x mod pi = 0\ || \ 2 / | // 1 for 7*x mod 2*pi = 0\ || | || | ||-1 + -------------- | || | || 1 | || 1 | |< 2 |*|< 1 | - |<------------- otherwise |*|<------------- otherwise | || sec (2*x) | ||-------- otherwise | || / pi\ | || / pi\ | ||------------------- otherwise | \\sec(7*x) / ||sec|4*x - --| | ||sec|7*x - --| | || 2/ pi\ | \\ \ 2 / / \\ \ 2 / / || sec |2*x - --| | || \ 2 / | \\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 x \bmod \pi = 0 \\\frac{1}{\sec{\left(4 x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 7 x \bmod \pi = 0 \\\frac{1}{\sec{\left(7 x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(2 x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(2 x \right)}}}{\sec^{2}{\left(2 x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 7 x \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(7 x \right)}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 7*x mod 2*pi = 0\ // 0 for 7*x mod pi = 0\
// 1 for 2*x mod pi = 0\ || | // 0 for 4*x mod pi = 0\ || |
|| | || 2/7*x\ | || | || /7*x\ |
|| 2 | ||-1 + cot |---| | || 2*cot(2*x) | || 2*cot|---| |
|<-1 + cot (2*x) |*|< \ 2 / | - |<------------- otherwise |*|< \ 2 / |
||-------------- otherwise | ||-------------- otherwise | || 2 | ||------------- otherwise |
|| 2 | || 2/7*x\ | ||1 + cot (2*x) | || 2/7*x\ |
\\1 + cot (2*x) / ||1 + cot |---| | \\ / ||1 + cot |---| |
\\ \ 2 / / \\ \ 2 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 x \bmod \pi = 0 \\\frac{2 \cot{\left(2 x \right)}}{\cot^{2}{\left(2 x \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 7 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{7 x}{2} \right)}}{\cot^{2}{\left(\frac{7 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{\cot^{2}{\left(2 x \right)} - 1}{\cot^{2}{\left(2 x \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 7 x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{7 x}{2} \right)} - 1}{\cot^{2}{\left(\frac{7 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 2*x mod pi = 0\ || | // 1 for 7*x mod 2*pi = 0\ // 0 for 7*x mod pi = 0\ || 2 / 1 \ | || | // 0 for 4*x mod pi = 0\ || | ||4*tan (x)*|-1 + ---------| | || 2/7*x\ | || | || /7*x\ | || | 2 | | ||1 - tan |---| | || 2*tan(2*x) | || 2*tan|---| | |< \ tan (2*x)/ |*|< \ 2 / | - |<------------- otherwise |*|< \ 2 / | ||-------------------------- otherwise | ||------------- otherwise | || 2 | ||------------- otherwise | || 2 | || 2/7*x\ | ||1 + tan (2*x) | || 2/7*x\ | || / 2 \ | ||1 + tan |---| | \\ / ||1 + tan |---| | || \1 + tan (x)/ | \\ \ 2 / / \\ \ 2 / / \\ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 x \bmod \pi = 0 \\\frac{2 \tan{\left(2 x \right)}}{\tan^{2}{\left(2 x \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 7 x \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{7 x}{2} \right)}}{\tan^{2}{\left(\frac{7 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{4 \left(-1 + \frac{1}{\tan^{2}{\left(2 x \right)}}\right) \tan^{2}{\left(x \right)}}{\left(\tan^{2}{\left(x \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 7 x \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{7 x}{2} \right)} + 1}{\tan^{2}{\left(\frac{7 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
/ 2/ pi\\ / 2/ pi 7*x\\
| cos |2*x - --|| | cos |- -- + ---||
| \ 2 /| | \ 2 2 /|
|1 - --------------|*|1 - ----------------|
| 2 | | 2/7*x\ | / pi\ / pi 7*x\
\ cos (2*x) / | cos |---| | 4*cos|2*x - --|*cos|- -- + ---|
\ \ 2 / / \ 2 / \ 2 2 /
------------------------------------------- - -------------------------------------------------------------
/ 2/ pi\\ / 2/ pi 7*x\\ / 2/ pi\\ / 2/ pi 7*x\\
| cos |2*x - --|| | cos |- -- + ---|| | cos |2*x - --|| | cos |- -- + ---||
| \ 2 /| | \ 2 2 /| | \ 2 /| | \ 2 2 /| /7*x\
|1 + --------------|*|1 + ----------------| |1 + --------------|*|1 + ----------------|*cos(2*x)*cos|---|
| 2 | | 2/7*x\ | | 2 | | 2/7*x\ | \ 2 /
\ cos (2*x) / | cos |---| | \ cos (2*x) / | cos |---| |
\ \ 2 / / \ \ 2 / /
$$\frac{\left(1 - \frac{\cos^{2}{\left(2 x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(2 x \right)}}\right) \left(1 - \frac{\cos^{2}{\left(\frac{7 x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{7 x}{2} \right)}}\right)}{\left(1 + \frac{\cos^{2}{\left(2 x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(2 x \right)}}\right) \left(1 + \frac{\cos^{2}{\left(\frac{7 x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{7 x}{2} \right)}}\right)} - \frac{4 \cos{\left(2 x - \frac{\pi}{2} \right)} \cos{\left(\frac{7 x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(2 x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(2 x \right)}}\right) \left(1 + \frac{\cos^{2}{\left(\frac{7 x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{7 x}{2} \right)}}\right) \cos{\left(2 x \right)} \cos{\left(\frac{7 x}{2} \right)}}$$
/ 2/7*x\ \
/ 2 \ | sec |---| |
| sec (2*x) | | \ 2 / |
|1 - --------------|*|1 - ----------------|
| 2/ pi\| | 2/ pi 7*x\| /7*x\
| sec |2*x - --|| | sec |- -- + ---|| 4*sec(2*x)*sec|---|
\ \ 2 // \ \ 2 2 // \ 2 /
------------------------------------------- - -------------------------------------------------------------------------
/ 2/7*x\ \ / 2/7*x\ \
/ 2 \ | sec |---| | / 2 \ | sec |---| |
| sec (2*x) | | \ 2 / | | sec (2*x) | | \ 2 / | / pi\ / pi 7*x\
|1 + --------------|*|1 + ----------------| |1 + --------------|*|1 + ----------------|*sec|2*x - --|*sec|- -- + ---|
| 2/ pi\| | 2/ pi 7*x\| | 2/ pi\| | 2/ pi 7*x\| \ 2 / \ 2 2 /
| sec |2*x - --|| | sec |- -- + ---|| | sec |2*x - --|| | sec |- -- + ---||
\ \ 2 // \ \ 2 2 // \ \ 2 // \ \ 2 2 //
$$\frac{\left(- \frac{\sec^{2}{\left(2 x \right)}}{\sec^{2}{\left(2 x - \frac{\pi}{2} \right)}} + 1\right) \left(- \frac{\sec^{2}{\left(\frac{7 x}{2} \right)}}{\sec^{2}{\left(\frac{7 x}{2} - \frac{\pi}{2} \right)}} + 1\right)}{\left(\frac{\sec^{2}{\left(2 x \right)}}{\sec^{2}{\left(2 x - \frac{\pi}{2} \right)}} + 1\right) \left(\frac{\sec^{2}{\left(\frac{7 x}{2} \right)}}{\sec^{2}{\left(\frac{7 x}{2} - \frac{\pi}{2} \right)}} + 1\right)} - \frac{4 \sec{\left(2 x \right)} \sec{\left(\frac{7 x}{2} \right)}}{\left(\frac{\sec^{2}{\left(2 x \right)}}{\sec^{2}{\left(2 x - \frac{\pi}{2} \right)}} + 1\right) \left(\frac{\sec^{2}{\left(\frac{7 x}{2} \right)}}{\sec^{2}{\left(\frac{7 x}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(2 x - \frac{\pi}{2} \right)} \sec{\left(\frac{7 x}{2} - \frac{\pi}{2} \right)}}$$
// 1 for 7*x mod 2*pi = 0\
// 1 for 2*x mod pi = 0\ || |
|| | || 1 | // 0 for 4*x mod pi = 0\ // 0 for 7*x mod pi = 0\
|| 1 | ||-1 + --------- | || | || |
||-1 + --------- | || 2/7*x\ | || 2 | || 2 |
|| 2 | || tan |---| | ||------------------------ otherwise | ||------------------------ otherwise |
|< tan (2*x) |*|< \ 2 / | - |
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 x \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(2 x \right)}}\right) \tan{\left(2 x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 7 x \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{7 x}{2} \right)}}\right) \tan{\left(\frac{7 x}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(2 x \right)}}}{1 + \frac{1}{\tan^{2}{\left(2 x \right)}}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 7 x \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{7 x}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{7 x}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)$$
/ 2/pi \\ / 2/pi 7*x\\
| csc |-- - 2*x|| | csc |-- - ---||
| \2 /| | \2 2 /|
|1 - --------------|*|1 - --------------|
| 2 | | 2/7*x\ | /pi \ /pi 7*x\
\ csc (2*x) / | csc |---| | 4*csc|-- - 2*x|*csc|-- - ---|
\ \ 2 / / \2 / \2 2 /
----------------------------------------- - -----------------------------------------------------------
/ 2/pi \\ / 2/pi 7*x\\ / 2/pi \\ / 2/pi 7*x\\
| csc |-- - 2*x|| | csc |-- - ---|| | csc |-- - 2*x|| | csc |-- - ---||
| \2 /| | \2 2 /| | \2 /| | \2 2 /| /7*x\
|1 + --------------|*|1 + --------------| |1 + --------------|*|1 + --------------|*csc(2*x)*csc|---|
| 2 | | 2/7*x\ | | 2 | | 2/7*x\ | \ 2 /
\ csc (2*x) / | csc |---| | \ csc (2*x) / | csc |---| |
\ \ 2 / / \ \ 2 / /
$$\frac{\left(1 - \frac{\csc^{2}{\left(- 2 x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(2 x \right)}}\right) \left(1 - \frac{\csc^{2}{\left(- \frac{7 x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{7 x}{2} \right)}}\right)}{\left(1 + \frac{\csc^{2}{\left(- 2 x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(2 x \right)}}\right) \left(1 + \frac{\csc^{2}{\left(- \frac{7 x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{7 x}{2} \right)}}\right)} - \frac{4 \csc{\left(- \frac{7 x}{2} + \frac{\pi}{2} \right)} \csc{\left(- 2 x + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- 2 x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(2 x \right)}}\right) \left(1 + \frac{\csc^{2}{\left(- \frac{7 x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{7 x}{2} \right)}}\right) \csc{\left(2 x \right)} \csc{\left(\frac{7 x}{2} \right)}}$$
// /pi \ \ // /pi \ \ || 0 for |-- + 4*x| mod pi = 0| || 0 for |-- + 7*x| mod pi = 0| // 0 for 7*x mod pi = 0\ || \2 / | || \2 / | // 0 for 4*x mod pi = 0\ || | || | || | || | || /7*x\ | || / pi\ | || /pi 7*x\ | || 2*cot(2*x) | || 2*cot|---| | |< 2*cot|2*x + --| |*|< 2*cot|-- + ---| | - |<------------- otherwise |*|< \ 2 / | || \ 4 / | || \4 2 / | || 2 | ||------------- otherwise | ||------------------ otherwise | ||------------------ otherwise | ||1 + cot (2*x) | || 2/7*x\ | || 2/ pi\ | || 2/pi 7*x\ | \\ / ||1 + cot |---| | ||1 + cot |2*x + --| | ||1 + cot |-- + ---| | \\ \ 2 / / \\ \ 4 / / \\ \4 2 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 x \bmod \pi = 0 \\\frac{2 \cot{\left(2 x \right)}}{\cot^{2}{\left(2 x \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 7 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{7 x}{2} \right)}}{\cot^{2}{\left(\frac{7 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 0 & \text{for}\: \left(4 x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(2 x + \frac{\pi}{4} \right)}}{\cot^{2}{\left(2 x + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(7 x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 2*x mod pi = 0\ || | // 1 for 7*x mod 2*pi = 0\ // 0 for 7*x mod pi = 0\ || / 2 \ | || | || | || | sin (4*x) | | || -2 - 2*cos(14*x) + 4*cos(7*x) | // 0 for 4*x mod pi = 0\ || -2*sin(14*x) + 4*sin(7*x) | |<(1 - cos(4*x))*|-1 + -----------| |*|<--------------------------------- otherwise | - |< |*|<--------------------------------- otherwise | || | 4 | | || 2 | \\sin(4*x) otherwise / || 2 | || \ 4*sin (2*x)/ | ||1 - cos(14*x) + 2*(1 - cos(7*x)) | ||1 - cos(14*x) + 2*(1 - cos(7*x)) | ||--------------------------------- otherwise | \\ / \\ / \\ 2 /
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 x \bmod \pi = 0 \\\sin{\left(4 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 7 x \bmod \pi = 0 \\\frac{4 \sin{\left(7 x \right)} - 2 \sin{\left(14 x \right)}}{2 \left(- \cos{\left(7 x \right)} + 1\right)^{2} - \cos{\left(14 x \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(4 x \right)}}{4 \sin^{4}{\left(2 x \right)}}\right) \left(- \cos{\left(4 x \right)} + 1\right)}{2} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 7 x \bmod 2 \pi = 0 \\\frac{4 \cos{\left(7 x \right)} - 2 \cos{\left(14 x \right)} - 2}{2 \left(- \cos{\left(7 x \right)} + 1\right)^{2} - \cos{\left(14 x \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// / 3*pi\ \ // / 3*pi\ \
// 1 for 7*x mod 2*pi = 0\ || 1 for |4*x + ----| mod 2*pi = 0| || 1 for |7*x + ----| mod 2*pi = 0|
// 1 for 2*x mod pi = 0\ || | || \ 2 / | || \ 2 / |
|| | || 2/7*x\ | || | || |
|| 2 | ||-1 + cot |---| | || 2/ pi\ | || 2/pi 7*x\ |
|<-1 + cot (2*x) |*|< \ 2 / | - |<-1 + tan |2*x + --| |*|<-1 + tan |-- + ---| |
||-------------- otherwise | ||-------------- otherwise | || \ 4 / | || \4 2 / |
|| 2 | || 2/7*x\ | ||------------------- otherwise | ||------------------- otherwise |
\\1 + cot (2*x) / ||1 + cot |---| | || 2/ pi\ | || 2/pi 7*x\ |
\\ \ 2 / / || 1 + tan |2*x + --| | || 1 + tan |-- + ---| |
\\ \ 4 / / \\ \4 2 / /
$$\left(\left(\begin{cases} 1 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{\cot^{2}{\left(2 x \right)} - 1}{\cot^{2}{\left(2 x \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 7 x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{7 x}{2} \right)} - 1}{\cot^{2}{\left(\frac{7 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(\left(\begin{cases} 1 & \text{for}\: \left(4 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(2 x + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(2 x + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(7 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{7 x}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 2*x mod pi = 0\
|| | // 1 for 7*x mod 2*pi = 0\ // 0 for 4*x mod pi = 0\ // 0 for 7*x mod pi = 0\
|| // 0 for 2*x mod pi = 0\ | || | || | || |
|
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 4 x \bmod \pi = 0 \\\sin{\left(4 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 7 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 7 x \bmod \pi = 0 \\\sin{\left(7 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 x \bmod \pi = 0 \\\left(\cot^{2}{\left(2 x \right)} - 1\right) \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{- \cos{\left(4 x \right)} + 1}{2} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 7 x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: 7 x \bmod 2 \pi = 0 \\\cos{\left(7 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 7*x mod 2*pi = 0\
// 1 for 2*x mod pi = 0\ || |
|| | || 2 | // 0 for 7*x mod pi = 0\
|| 2 | || sin (7*x) | // 0 for 4*x mod pi = 0\ || |
|| sin (4*x) | ||-1 + ----------- | || | || sin(7*x) |
||-1 + ----------- | || 4/7*x\ | || sin(4*x) | ||--------------------------- otherwise |
|| 4 | || 4*sin |---| | ||--------------------------- otherwise | ||/ 2 \ |
|< 4*sin (2*x) |*|< \ 2 / | - |
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 x \bmod \pi = 0 \\\frac{\sin{\left(4 x \right)}}{\left(1 + \frac{\sin^{2}{\left(4 x \right)}}{4 \sin^{4}{\left(2 x \right)}}\right) \sin^{2}{\left(2 x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 7 x \bmod \pi = 0 \\\frac{\sin{\left(7 x \right)}}{\left(1 + \frac{\sin^{2}{\left(7 x \right)}}{4 \sin^{4}{\left(\frac{7 x}{2} \right)}}\right) \sin^{2}{\left(\frac{7 x}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(4 x \right)}}{4 \sin^{4}{\left(2 x \right)}}}{1 + \frac{\sin^{2}{\left(4 x \right)}}{4 \sin^{4}{\left(2 x \right)}}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 7 x \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(7 x \right)}}{4 \sin^{4}{\left(\frac{7 x}{2} \right)}}}{1 + \frac{\sin^{2}{\left(7 x \right)}}{4 \sin^{4}{\left(\frac{7 x}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 2*x mod pi = 0\ // 1 for 7*x mod 2*pi = 0\ // 0 for 7*x mod pi = 0\
|| | || | // 0 for 4*x mod pi = 0\ || |
|| // 0 for 2*x mod pi = 0\ | ||/ 1 for 7*x mod 2*pi = 0 | || | ||/ 0 for 7*x mod pi = 0 |
|| || | | ||| | ||/ 0 for 4*x mod pi = 0 | ||| |
|| || 2 | | ||| 2/7*x\ | ||| | ||| /7*x\ |
|
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 4 x \bmod \pi = 0 \\\frac{2 \cot{\left(2 x \right)}}{\cot^{2}{\left(2 x \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 7 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 7 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{7 x}{2} \right)}}{\cot^{2}{\left(\frac{7 x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 x \bmod \pi = 0 \\\left(\cot^{2}{\left(2 x \right)} - 1\right) \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(x \right)}}{\left(\cot^{2}{\left(x \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right) & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 7 x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: 7 x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{7 x}{2} \right)} - 1}{\cot^{2}{\left(\frac{7 x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 7*x mod 2*pi = 0\
// 1 for 2*x mod pi = 0\ || |
|| | || 2/7*x\ | // 0 for 7*x mod pi = 0\
|| 2 | || cos |---| | // 0 for 4*x mod pi = 0\ || |
|| cos (2*x) | || \ 2 / | || | || /7*x\ |
||-1 + -------------- | ||-1 + ---------------- | || 2*cos(2*x) | || 2*cos|---| |
|| 2/ pi\ | || 2/ pi 7*x\ | ||---------------------------------- otherwise | || \ 2 / |
|| cos |2*x - --| | || cos |- -- + ---| | ||/ 2 \ | ||-------------------------------------- otherwise |
|< \ 2 / |*|< \ 2 2 / | - |<| cos (2*x) | / pi\ |*|
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 x \bmod \pi = 0 \\\frac{2 \cos{\left(2 x \right)}}{\left(\frac{\cos^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(2 x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 7 x \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{7 x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{7 x}{2} \right)}}{\cos^{2}{\left(\frac{7 x}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{7 x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{\frac{\cos^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 7 x \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{7 x}{2} \right)}}{\cos^{2}{\left(\frac{7 x}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{7 x}{2} \right)}}{\cos^{2}{\left(\frac{7 x}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 7*x mod 2*pi = 0\
// 1 for 2*x mod pi = 0\ || |
|| | || 2/ pi 7*x\ | // 0 for 4*x mod pi = 0\ // 0 for 7*x mod pi = 0\
|| 2/ pi\ | || sec |- -- + ---| | || | || |
|| sec |2*x - --| | || \ 2 2 / | || / pi\ | || / pi 7*x\ |
|| \ 2 / | ||-1 + ---------------- | || 2*sec|2*x - --| | || 2*sec|- -- + ---| |
||-1 + -------------- | || 2/7*x\ | || \ 2 / | || \ 2 2 / |
|| 2 | || sec |---| | ||----------------------------- otherwise | ||------------------------------- otherwise |
|< sec (2*x) |*|< \ 2 / | - |
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 x \bmod \pi = 0 \\\frac{2 \sec{\left(2 x - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(2 x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(2 x \right)}}\right) \sec{\left(2 x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 7 x \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{7 x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{7 x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{7 x}{2} \right)}}\right) \sec{\left(\frac{7 x}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(2 x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(2 x \right)}}}{1 + \frac{\sec^{2}{\left(2 x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(2 x \right)}}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 7 x \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{7 x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{7 x}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{7 x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{7 x}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for 7*x mod 2*pi = 0\
// 1 for 2*x mod pi = 0\ || |
|| | || 2/7*x\ | // 0 for 7*x mod pi = 0\
|| 2 | || csc |---| | // 0 for 4*x mod pi = 0\ || |
|| csc (2*x) | || \ 2 / | || | || /7*x\ |
||-1 + -------------- | ||-1 + -------------- | || 2*csc(2*x) | || 2*csc|---| |
|| 2/pi \ | || 2/pi 7*x\ | ||---------------------------------- otherwise | || \ 2 / |
|| csc |-- - 2*x| | || csc |-- - ---| | ||/ 2 \ | ||---------------------------------- otherwise |
|< \2 / |*|< \2 2 / | - |<| csc (2*x) | /pi \ |*|
$$\left(- \left(\begin{cases} 0 & \text{for}\: 4 x \bmod \pi = 0 \\\frac{2 \csc{\left(2 x \right)}}{\left(\frac{\csc^{2}{\left(2 x \right)}}{\csc^{2}{\left(- 2 x + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- 2 x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: 7 x \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{7 x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{7 x}{2} \right)}}{\csc^{2}{\left(- \frac{7 x}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{7 x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(\left(\begin{cases} 1 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{\frac{\csc^{2}{\left(2 x \right)}}{\csc^{2}{\left(- 2 x + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(2 x \right)}}{\csc^{2}{\left(- 2 x + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: 7 x \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{7 x}{2} \right)}}{\csc^{2}{\left(- \frac{7 x}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{7 x}{2} \right)}}{\csc^{2}{\left(- \frac{7 x}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right)$$
Piecewise((1, Mod(2*x = pi, 0)), ((-1 + csc(2*x)^2/csc(pi/2 - 2*x)^2)/(1 + csc(2*x)^2/csc(pi/2 - 2*x)^2), True))*Piecewise((1, Mod(7*x = 2*pi, 0)), ((-1 + csc(7*x/2)^2/csc(pi/2 - 7*x/2)^2)/(1 + csc(7*x/2)^2/csc(pi/2 - 7*x/2)^2), True)) - Piecewise((0, Mod(4*x = pi, 0)), (2*csc(2*x)/((1 + csc(2*x)^2/csc(pi/2 - 2*x)^2)*csc(pi/2 - 2*x)), True))*Piecewise((0, Mod(7*x = pi, 0)), (2*csc(7*x/2)/((1 + csc(7*x/2)^2/csc(pi/2 - 7*x/2)^2)*csc(pi/2 - 7*x/2)), True))
Степени
[src]
/ -7*I*x 7*I*x\ / -4*I*x 4*I*x\ / -7*I*x 7*I*x\ / -4*I*x 4*I*x\ |e e | |e e | \- e + e /*\- e + e / |------- + ------|*|------- + ------| + ----------------------------------------- \ 2 2 / \ 2 2 / 4
$$\left(\frac{e^{4 i x}}{2} + \frac{e^{- 4 i x}}{2}\right) \left(\frac{e^{7 i x}}{2} + \frac{e^{- 7 i x}}{2}\right) + \frac{\left(e^{4 i x} - e^{- 4 i x}\right) \left(e^{7 i x} - e^{- 7 i x}\right)}{4}$$
(exp(-7*i*x)/2 + exp(7*i*x)/2)*(exp(-4*i*x)/2 + exp(4*i*x)/2) + (-exp(-7*i*x) + exp(7*i*x))*(-exp(-4*i*x) + exp(4*i*x))/4