Господин Экзамен
Другие калькуляторы
-
Как пользоваться?
- 5*cos(x)*sin(2*x)+5*cos(2*x)*sin(x) если x=-1
- 3*m/m+5-8*m/m2+10*m+25 если m=1/3
- 35*x*23*y если y=4
- sin(270-a) если a=1/2
- Разложить многочлен на множители:
- n^10-n^5
- 8*a^2*b^2-72*a^2*c^2
- 36*m^2-100*n^2
- m^3-27*n^3
- Общий знаменатель:
- (m/m^2-16+64-m+4/m^2-64)/3*m+8/m^2-64
- c*d-d2/c2+d2*(c/c+d+d/c-d)
- (m^2+10*m)/(9-m^2)-(4*m-9)/(9-m^2)
- (2*k-5)/(3-k)+(4+k)/(k-3)
- Умножение столбиком:
- 50*10
- 4039*57
- 25*32
- 30*75
- Деление столбиком:
- 42/5
- 830/4
- 50/2
- 832/7
- Раскрыть скобки в:
- (x-4)*(x-1)
- (x+4)*(x^2-4*x+16)
- 14*a*b^2-17*a*b+5*a^(2*b)+20*a*b-14*a^(2*b)
- 3*(2*x+8)-(5*x+2)
- Разложение числа на простые множители:
- 4356
- 2340
- 2520
- 2280
- График функции y =:
- -1
- Производная:
- -1
- Интеграл d{x}:
-
-1
-
Идентичные выражения
- пять *cos(x)*sin(два *x)+ пять *cos(два *x)*sin(x) если x=- один
- 5 умножить на косинус от (x) умножить на синус от (2 умножить на x) плюс 5 умножить на косинус от (2 умножить на x) умножить на синус от (x) если x равно минус 1
- пять умножить на косинус от (x) умножить на синус от (два умножить на x) плюс пять умножить на косинус от (два умножить на x) умножить на синус от (x) если x равно минус один
- 5cos(x)sin(2x)+5cos(2x)sin(x) если x=-1
- 5cosxsin2x+5cos2xsinx если x=-1
- 5*cos(x)*sin(2*x)+5*cos(2*x)*sin(x) при x=-1
-
Похожие выражения
- 5*cos(x)*sin(2*x)+5*cos(2*x)*sin(x) если x=+1
- 5*cos(x)*sin(2*x)-5*cos(2*x)*sin(x) если x=-1
- 5*cosx*sin(2*x)+5*cos(2*x)*sinx если x=-1
5*cos(x)*sin(2*x)+5*cos(2*x)*sin(x) если x=-1
Решение
Вы ввели
[src]
5*cos(x)*sin(2*x) + 5*cos(2*x)*sin(x)
$$5 \sin{\left(x \right)} \cos{\left(2 x \right)} + 5 \sin{\left(2 x \right)} \cos{\left(x \right)}$$
5*cos(x)*sin(2*x) + 5*cos(2*x)*sin(x)
Подстановка условия
[src]
5*cos(x)*sin(2*x) + 5*cos(2*x)*sin(x) при x = -1
подставляем
5*cos(x)*sin(2*x) + 5*cos(2*x)*sin(x)
$$5 \sin{\left(x \right)} \cos{\left(2 x \right)} + 5 \sin{\left(2 x \right)} \cos{\left(x \right)}$$
5*sin(3*x)
$$5 \sin{\left(3 x \right)}$$
переменные
x = -1
$$x = -1$$
5*sin(3*(-1))
$$5 \sin{\left(3 (-1) \right)}$$
-5*sin(3)
$$- 5 \sin{\left(3 \right)}$$
-5*sin(3)
Объединение рациональных выражений
[src]
5*(cos(x)*sin(2*x) + cos(2*x)*sin(x))
$$5 \left(\sin{\left(x \right)} \cos{\left(2 x \right)} + \sin{\left(2 x \right)} \cos{\left(x \right)}\right)$$
5*(cos(x)*sin(2*x) + cos(2*x)*sin(x))
Тригонометрическая часть
[src]
5*sin(3*x)
$$5 \sin{\left(3 x \right)}$$
5 -------- csc(3*x)
$$\frac{5}{\csc{\left(3 x \right)}}$$
/ pi\
5*cos|3*x - --|
\ 2 /
$$5 \cos{\left(3 x - \frac{\pi}{2} \right)}$$
5 ------------- / pi\ sec|3*x - --| \ 2 /
$$\frac{5}{\sec{\left(3 x - \frac{\pi}{2} \right)}}$$
/3*x\
10*tan|---|
\ 2 /
-------------
2/3*x\
1 + tan |---|
\ 2 /
$$\frac{10 \tan{\left(\frac{3 x}{2} \right)}}{\tan^{2}{\left(\frac{3 x}{2} \right)} + 1}$$
5 5 --------------- + --------------- csc(x)*sec(2*x) csc(2*x)*sec(x)
$$\frac{5}{\csc{\left(2 x \right)} \sec{\left(x \right)}} + \frac{5}{\csc{\left(x \right)} \sec{\left(2 x \right)}}$$
// 0 for 3*x mod pi = 0\ 5*|< | \\sin(3*x) otherwise /
$$5 \left(\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\sin{\left(3 x \right)} & \text{otherwise} \end{cases}\right)$$
5*sin(x) 5*sin(3*x) -------- + ---------- + 5*cos(2*x)*sin(x) 2 2
$$5 \sin{\left(x \right)} \cos{\left(2 x \right)} + \frac{5 \sin{\left(x \right)}}{2} + \frac{5 \sin{\left(3 x \right)}}{2}$$
/ pi\ / pi\
5*cos(x)*cos|2*x - --| + 5*cos(2*x)*cos|x - --|
\ 2 / \ 2 /
$$5 \cos{\left(x \right)} \cos{\left(2 x - \frac{\pi}{2} \right)} + 5 \cos{\left(2 x \right)} \cos{\left(x - \frac{\pi}{2} \right)}$$
/pi \ / pi\
5*sin(x)*sin|-- + 2*x| + 5*sin(2*x)*sin|x + --|
\2 / \ 2 /
$$5 \sin{\left(x \right)} \sin{\left(2 x + \frac{\pi}{2} \right)} + 5 \sin{\left(2 x \right)} \sin{\left(x + \frac{\pi}{2} \right)}$$
5*sin(x) 5*sin(3*x) /pi \ -------- + ---------- + 5*sin(x)*sin|-- + 2*x| 2 2 \2 /
$$5 \sin{\left(x \right)} \sin{\left(2 x + \frac{\pi}{2} \right)} + \frac{5 \sin{\left(x \right)}}{2} + \frac{5 \sin{\left(3 x \right)}}{2}$$
/ 2 2 \ 2 5*\cos (x) - sin (x)/*sin(x) + 10*cos (x)*sin(x)
$$10 \sin{\left(x \right)} \cos^{2}{\left(x \right)} + 5 \left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}$$
5 5
-------------------- + --------------------
/ pi\ / pi\
sec(x)*sec|2*x - --| sec(2*x)*sec|x - --|
\ 2 / \ 2 /
$$\frac{5}{\sec{\left(2 x \right)} \sec{\left(x - \frac{\pi}{2} \right)}} + \frac{5}{\sec{\left(x \right)} \sec{\left(2 x - \frac{\pi}{2} \right)}}$$
5 5 5
-------- + ---------- + --------------------
2*csc(x) 2*csc(3*x) /pi \
csc(x)*csc|-- - 2*x|
\2 /
$$\frac{5}{2 \csc{\left(3 x \right)}} + \frac{5}{2 \csc{\left(x \right)}} + \frac{5}{\csc{\left(x \right)} \csc{\left(- 2 x + \frac{\pi}{2} \right)}}$$
5 5
-------------------- + --------------------
/pi \ /pi \
csc(x)*csc|-- - 2*x| csc(2*x)*csc|-- - x|
\2 / \2 /
$$\frac{5}{\csc{\left(2 x \right)} \csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{5}{\csc{\left(x \right)} \csc{\left(- 2 x + \frac{\pi}{2} \right)}}$$
5 5
-------------------- + --------------------
/pi \ /pi \
sec(x)*sec|-- - 2*x| sec(2*x)*sec|-- - x|
\2 / \2 /
$$\frac{5}{\sec{\left(2 x \right)} \sec{\left(- x + \frac{\pi}{2} \right)}} + \frac{5}{\sec{\left(x \right)} \sec{\left(- 2 x + \frac{\pi}{2} \right)}}$$
5 5
------------------------- + -------------------------
/pi \ /pi \
csc(pi - x)*csc|-- - 2*x| csc(pi - 2*x)*csc|-- - x|
\2 / \2 /
$$\frac{5}{\csc{\left(- x + \pi \right)} \csc{\left(- 2 x + \frac{\pi}{2} \right)}} + \frac{5}{\csc{\left(- 2 x + \pi \right)} \csc{\left(- x + \frac{\pi}{2} \right)}}$$
/ pi\ / pi\
5*cos|x - --| 5*cos|3*x - --|
\ 2 / \ 2 / / pi\
------------- + --------------- + 5*cos(2*x)*cos|x - --|
2 2 \ 2 /
$$5 \cos{\left(2 x \right)} \cos{\left(x - \frac{\pi}{2} \right)} + \frac{5 \cos{\left(x - \frac{\pi}{2} \right)}}{2} + \frac{5 \cos{\left(3 x - \frac{\pi}{2} \right)}}{2}$$
5 5 5
------------- + --------------- + --------------------
/ pi\ / pi\ / pi\
2*sec|x - --| 2*sec|3*x - --| sec(2*x)*sec|x - --|
\ 2 / \ 2 / \ 2 /
$$\frac{5}{2 \sec{\left(3 x - \frac{\pi}{2} \right)}} + \frac{5}{2 \sec{\left(x - \frac{\pi}{2} \right)}} + \frac{5}{\sec{\left(2 x \right)} \sec{\left(x - \frac{\pi}{2} \right)}}$$
// 0 for 3*x mod pi = 0\ || | || /3*x\ | || 2*cot|---| | 5*|< \ 2 / | ||------------- otherwise | || 2/3*x\ | ||1 + cot |---| | \\ \ 2 / /
$$5 \left(\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 x}{2} \right)}}{\cot^{2}{\left(\frac{3 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
5*(-sin(x) + sin(3*x)) / 2/x\ sin(2*x)\
---------------------- + 10*|- sin |-|*cos(x) + --------|*sin(x)
2 | \2/ /x\|
| 4*tan|-||
\ \2//
$$10 \left(- \sin^{2}{\left(\frac{x}{2} \right)} \cos{\left(x \right)} + \frac{\sin{\left(2 x \right)}}{4 \tan{\left(\frac{x}{2} \right)}}\right) \sin{\left(x \right)} + \frac{5 \left(- \sin{\left(x \right)} + \sin{\left(3 x \right)}\right)}{2}$$
2 / 2 \ /x\
5*cos (x)*\1 - tan (x)/*(1 + cos(x))*sin|-|
2 \2/
10*cos (x)*sin(x) + -------------------------------------------
/x\
cos|-|
\2/
$$\frac{5 \cdot \left(- \tan^{2}{\left(x \right)} + 1\right) \left(\cos{\left(x \right)} + 1\right) \sin{\left(\frac{x}{2} \right)} \cos^{2}{\left(x \right)}}{\cos{\left(\frac{x}{2} \right)}} + 10 \sin{\left(x \right)} \cos^{2}{\left(x \right)}$$
2 / 2/x pi\\ / 2 \
5*cos (x)*|1 - cot |- + --||*\1 - tan (x)/*(1 + sin(x))
5*(sin(x) + sin(3*x)) \ \2 4 //
--------------------- + -------------------------------------------------------
2 2
$$\frac{5 \cdot \left(- \tan^{2}{\left(x \right)} + 1\right) \left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(x \right)} + 1\right) \cos^{2}{\left(x \right)}}{2} + \frac{5 \left(\sin{\left(x \right)} + \sin{\left(3 x \right)}\right)}{2}$$
/ 2/x\ 2 2 2/x\ 4/x\ \
10*|2*sin |-| + sin (x)*cos(x) - 4*sin (x)*sin |-| - 4*sin |-|*cos(x)|*sin(x)
\ \2/ \2/ \2/ /
-----------------------------------------------------------------------------
2 4/x\
sin (x) + 4*sin |-|
\2/
$$\frac{10 \left(- 4 \sin^{4}{\left(\frac{x}{2} \right)} \cos{\left(x \right)} - 4 \sin^{2}{\left(\frac{x}{2} \right)} \sin^{2}{\left(x \right)} + \sin^{2}{\left(x \right)} \cos{\left(x \right)} + 2 \sin^{2}{\left(\frac{x}{2} \right)}\right) \sin{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}}$$
/x\ /3*x\ / 2 \ /x\
5*tan|-| 5*tan|---| 10*\1 - tan (x)/*tan|-|
\2/ \ 2 / \2/
----------- + ------------- + ---------------------------
2/x\ 2/3*x\ / 2 \ / 2/x\\
1 + tan |-| 1 + tan |---| \1 + tan (x)/*|1 + tan |-||
\2/ \ 2 / \ \2//
$$\frac{10 \cdot \left(- \tan^{2}{\left(x \right)} + 1\right) \tan{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\tan^{2}{\left(x \right)} + 1\right)} + \frac{5 \tan{\left(\frac{3 x}{2} \right)}}{\tan^{2}{\left(\frac{3 x}{2} \right)} + 1} + \frac{5 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}$$
/ 2 \ /x\ / 2/x\\
10*\1 - tan (x)/*tan|-| 10*|1 - tan |-||*tan(x)
\2/ \ \2//
--------------------------- + ---------------------------
/ 2 \ / 2/x\\ / 2 \ / 2/x\\
\1 + tan (x)/*|1 + tan |-|| \1 + tan (x)/*|1 + tan |-||
\ \2// \ \2//
$$\frac{10 \cdot \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \tan{\left(x \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\tan^{2}{\left(x \right)} + 1\right)} + \frac{10 \cdot \left(- \tan^{2}{\left(x \right)} + 1\right) \tan{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\tan^{2}{\left(x \right)} + 1\right)}$$
/x pi\ /x\ / pi\
20*cot(x)*tan|- + --| 20*cot|-|*tan|x + --|
\2 4 / \2/ \ 4 /
-------------------------------- + --------------------------------
/ 2 \ / 2/x pi\\ / 2/x\\ / 2/ pi\\
\1 + cot (x)/*|1 + tan |- + --|| |1 + cot |-||*|1 + tan |x + --||
\ \2 4 // \ \2// \ \ 4 //
$$\frac{20 \tan{\left(x + \frac{\pi}{4} \right)} \cot{\left(\frac{x}{2} \right)}}{\left(\tan^{2}{\left(x + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)} + \frac{20 \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)} \cot{\left(x \right)}}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)}$$
/x pi\ /x\ / pi\
20*tan(x)*tan|- + --| 20*tan|-|*tan|x + --|
\2 4 / \2/ \ 4 /
-------------------------------- + --------------------------------
/ 2 \ / 2/x pi\\ / 2/x\\ / 2/ pi\\
\1 + tan (x)/*|1 + tan |- + --|| |1 + tan |-||*|1 + tan |x + --||
\ \2 4 // \ \2// \ \ 4 //
$$\frac{20 \tan{\left(x \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(x \right)} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)} + \frac{20 \tan{\left(\frac{x}{2} \right)} \tan{\left(x + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\tan^{2}{\left(x + \frac{\pi}{4} \right)} + 1\right)}$$
/ 1 \
10*|1 - -------| / 1 \
| 2/x\| 10*|1 - -------|
| cot |-|| | 2 |
\ \2// \ cot (x)/
---------------------------------- + ----------------------------------
/ 1 \ / 1 \ / 1 \ / 1 \ /x\
|1 + -------|*|1 + -------|*cot(x) |1 + -------|*|1 + -------|*cot|-|
| 2 | | 2/x\| | 2 | | 2/x\| \2/
\ cot (x)/ | cot |-|| \ cot (x)/ | cot |-||
\ \2// \ \2//
$$\frac{10 \cdot \left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right) \left(1 + \frac{1}{\cot^{2}{\left(x \right)}}\right) \cot{\left(x \right)}} + \frac{10 \cdot \left(1 - \frac{1}{\cot^{2}{\left(x \right)}}\right)}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right) \left(1 + \frac{1}{\cot^{2}{\left(x \right)}}\right) \cot{\left(\frac{x}{2} \right)}}$$
/ 2 \ / 2/x pi\\ / 2/x\\ / 2/ pi\\
5*\-1 + cot (x)/*|-1 + tan |- + --|| 5*|-1 + cot |-||*|-1 + tan |x + --||
\ \2 4 // \ \2// \ \ 4 //
------------------------------------ + ------------------------------------
/ 2 \ / 2/x pi\\ / 2/x\\ / 2/ pi\\
\1 + cot (x)/*|1 + tan |- + --|| |1 + cot |-||*|1 + tan |x + --||
\ \2 4 // \ \2// \ \ 4 //
$$\frac{5 \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right) \left(\cot^{2}{\left(x \right)} - 1\right)}{\left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(x \right)} + 1\right)} + \frac{5 \left(\tan^{2}{\left(x + \frac{\pi}{4} \right)} - 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{\left(\tan^{2}{\left(x + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
/ 2/ pi\\ / 2/x\\ / 2/x pi\\ / 2 \ 5*|1 - cot |x + --||*|1 - tan |-|| 5*|1 - cot |- + --||*\1 - tan (x)/ \ \ 4 // \ \2// \ \2 4 // ---------------------------------- + ---------------------------------- / 2/ pi\\ / 2/x\\ / 2/x pi\\ / 2 \ |1 + cot |x + --||*|1 + tan |-|| |1 + cot |- + --||*\1 + tan (x)/ \ \ 4 // \ \2// \ \2 4 //
$$\frac{5 \cdot \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(- \cot^{2}{\left(x + \frac{\pi}{4} \right)} + 1\right)}{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(\cot^{2}{\left(x + \frac{\pi}{4} \right)} + 1\right)} + \frac{5 \cdot \left(- \tan^{2}{\left(x \right)} + 1\right) \left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)}{\left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)}$$
// 0 for x mod pi = 0\ // 1 for x mod pi = 0\ // 0 for 2*x mod pi = 0\ // 1 for x mod 2*pi = 0\ 5*|< |*|< | + 5*|< |*|< | \\sin(x) otherwise / \\cos(2*x) otherwise / \\sin(2*x) otherwise / \\cos(x) otherwise /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for x mod pi = 0\ // 0 for 2*x mod pi = 0\ || | // 1 for x mod pi = 0\ || | // 1 for x mod 2*pi = 0\ 5*|< / pi\ |*|< | + 5*|< / pi\ |*|< | ||cos|x - --| otherwise | \\cos(2*x) otherwise / ||cos|2*x - --| otherwise | \\cos(x) otherwise / \\ \ 2 / / \\ \ 2 / /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\cos{\left(2 x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
// 0 for x mod pi = 0\ || | // 0 for 2*x mod pi = 0\ || |
5*|< |*|< /pi \ | + 5*|< |*|< / pi\ |
\\sin(x) otherwise / ||sin|-- + 2*x| otherwise | \\sin(2*x) otherwise / ||sin|x + --| otherwise |
\\ \2 / / \\ \ 2 / /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(2 x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\sin{\left(x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for x mod pi = 0\ // 0 for 3*x mod pi = 0\
5*|< | 5*|< |
\\sin(x) otherwise / \\sin(3*x) otherwise / // 0 for x mod pi = 0\ // 1 for x mod pi = 0\
----------------------------- + --------------------------------- + 5*|< |*|< |
2 2 \\sin(x) otherwise / \\cos(2*x) otherwise /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(\frac{5 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right)}{2}\right) + \left(\frac{5 \left(\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\sin{\left(3 x \right)} & \text{otherwise} \end{cases}\right)}{2}\right)$$
// 0 for x mod pi = 0\ || | ||1 - cos(x) | // 1 for x mod pi = 0\ // 0 for 2*x mod pi = 0\ // 1 for x mod 2*pi = 0\ 5*|<---------- otherwise |*|< | + 5*|< |*|< | || /x\ | \\cos(2*x) otherwise / \\sin(2*x) otherwise / \\cos(x) otherwise / || tan|-| | \\ \2/ /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{- \cos{\left(x \right)} + 1}{\tan{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right)\right)$$
// / 3*pi\ \ // / 3*pi\ \
// 1 for x mod pi = 0\ || 1 for |x + ----| mod 2*pi = 0| // 1 for x mod 2*pi = 0\ || 1 for |2*x + ----| mod 2*pi = 0|
5*|< |*|< \ 2 / | + 5*|< |*|< \ 2 / |
\\cos(2*x) otherwise / || | \\cos(x) otherwise / || |
\\sin(x) otherwise / \\sin(2*x) otherwise /
$$\left(5 \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(2 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for x mod pi = 0\ // 0 for 2*x mod pi = 0\ || | // 1 for x mod pi = 0\ || | // 1 for x mod 2*pi = 0\ || 1 | || | || 1 | || | 5*|<----------- otherwise |*|< 1 | + 5*|<------------- otherwise |*|< 1 | || / pi\ | ||-------- otherwise | || / pi\ | ||------ otherwise | ||sec|x - --| | \\sec(2*x) / ||sec|2*x - --| | \\sec(x) / \\ \ 2 / / \\ \ 2 / /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\sec{\left(2 x \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{1}{\sec{\left(2 x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
// 0 for x mod pi = 0\ || | // 0 for 2*x mod pi = 0\ || |
|| | || 1 | || | || 1 |
5*|< 1 |*|<------------- otherwise | + 5*|< 1 |*|<----------- otherwise |
||------ otherwise | || /pi \ | ||-------- otherwise | || /pi \ |
\\csc(x) / ||csc|-- - 2*x| | \\csc(2*x) / ||csc|-- - x| |
\\ \2 / / \\ \2 / /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\csc{\left(x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{1}{\csc{\left(2 x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right)$$
// / pi\ \
// /pi \ \ || 0 for |x + --| mod pi = 0|
// 0 for x mod pi = 0\ || 0 for |-- + 2*x| mod pi = 0| // 0 for 2*x mod pi = 0\ || \ 2 / |
5*|< |*|< \2 / | + 5*|< |*|< |
\\sin(x) otherwise / || | \\sin(2*x) otherwise / || /x pi\ |
\\cos(2*x) otherwise / ||(1 + sin(x))*cot|- + --| otherwise |
\\ \2 4 / /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(2 x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(x \right)} + 1\right) \cot{\left(\frac{x}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right)$$
/ 4/x\\
/ 4 \ | 4*sin |-||
2/x\ | 4*sin (x)| 2 | \2/|
20*sin |-|*|1 - ---------| 20*sin (x)*|1 - ---------|
\2/ | 2 | | 2 |
\ sin (2*x)/ \ sin (x) /
-------------------------------------- + ----------------------------------------
/ 4/x\\ / 4/x\\
| 4*sin |-|| / 4 \ | 4*sin |-|| / 4 \
| \2/| | 4*sin (x)| | \2/| | 4*sin (x)|
|1 + ---------|*|1 + ---------|*sin(x) |1 + ---------|*|1 + ---------|*sin(2*x)
| 2 | | 2 | | 2 | | 2 |
\ sin (x) / \ sin (2*x)/ \ sin (x) / \ sin (2*x)/
$$\frac{20 \left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \sin^{2}{\left(x \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1\right) \sin{\left(2 x \right)}} + \frac{20 \left(- \frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1\right) \sin^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1\right) \sin{\left(x \right)}}$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || | // 1 for x mod pi = 0\ // 0 for 2*x mod pi = 0\ || | || /x\ | || | || | || 2/x\ | || 2*cot|-| | || 2 | || 2*cot(x) | ||-1 + cot |-| | 5*|< \2/ |*|<-1 + cot (x) | + 5*|<----------- otherwise |*|< \2/ | ||----------- otherwise | ||------------ otherwise | || 2 | ||------------ otherwise | || 2/x\ | || 2 | ||1 + cot (x) | || 2/x\ | ||1 + cot |-| | \\1 + cot (x) / \\ / ||1 + cot |-| | \\ \2/ / \\ \2/ /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || | // 1 for x mod pi = 0\ // 0 for 2*x mod pi = 0\ || | || /x\ | || | || | || 2/x\ | || 2*tan|-| | || 2 | || 2*tan(x) | ||1 - tan |-| | 5*|< \2/ |*|<1 - tan (x) | + 5*|<----------- otherwise |*|< \2/ | ||----------- otherwise | ||----------- otherwise | || 2 | ||----------- otherwise | || 2/x\ | || 2 | ||1 + tan (x) | || 2/x\ | ||1 + tan |-| | \\1 + tan (x) / \\ / ||1 + tan |-| | \\ \2/ / \\ \2/ /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{- \tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{x}{2} \right)} + 1}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for x mod pi = 0\ || | || 2*sin(x) | // 1 for x mod 2*pi = 0\ ||---------------------------- otherwise | || | || / 2 \ | // 1 for x mod pi = 0\ // 0 for 2*x mod pi = 0\ || 2 | 5*|< | sin (x) | |*|< | + 5*|< |*|< -4 + 4*sin (x) + 4*cos(x) | ||(1 - cos(x))*|1 + ---------| | \\cos(2*x) otherwise / \\sin(2*x) otherwise / ||--------------------------- otherwise | || | 4/x\| | || 2 2 | || | 4*sin |-|| | \\2*(1 - cos(x)) + 2*sin (x) / || \ \2// | \\ /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \sin{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \left(- \cos{\left(x \right)} + 1\right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{4 \sin^{2}{\left(x \right)} + 4 \cos{\left(x \right)} - 4}{2 \left(- \cos{\left(x \right)} + 1\right)^{2} + 2 \sin^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for x mod pi = 0\ // 0 for 3*x mod pi = 0\
|| | || |
|| /x\ | || /3*x\ |
|| 2*cot|-| | || 2*cot|---| |
5*|< \2/ | 5*|< \ 2 / |
||----------- otherwise | ||------------- otherwise | // 0 for x mod pi = 0\
|| 2/x\ | || 2/3*x\ | || | // 1 for x mod pi = 0\
||1 + cot |-| | ||1 + cot |---| | || /x\ | || |
\\ \2/ / \\ \ 2 / / || 2*cot|-| | || 2 |
---------------------------------- + -------------------------------------- + 5*|< \2/ |*|<-1 + cot (x) |
2 2 ||----------- otherwise | ||------------ otherwise |
|| 2/x\ | || 2 |
||1 + cot |-| | \\1 + cot (x) /
\\ \2/ /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(\frac{5 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)}{2}\right) + \left(\frac{5 \left(\begin{cases} 0 & \text{for}\: 3 x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{3 x}{2} \right)}}{\cot^{2}{\left(\frac{3 x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)}{2}\right)$$
/ 2/x\ \
| sec |-| | / 2 \
| \2/ | | sec (x) | /x\
10*|1 - ------------|*sec(x) 10*|1 - ------------|*sec|-|
| 2/x pi\| | 2/ pi\| \2/
| sec |- - --|| | sec |x - --||
\ \2 2 // \ \ 2 //
------------------------------------------------- + -------------------------------------------------
/ 2/x\ \ / 2/x\ \
/ 2 \ | sec |-| | / 2 \ | sec |-| |
| sec (x) | | \2/ | / pi\ | sec (x) | | \2/ | /x pi\
|1 + ------------|*|1 + ------------|*sec|x - --| |1 + ------------|*|1 + ------------|*sec|- - --|
| 2/ pi\| | 2/x pi\| \ 2 / | 2/ pi\| | 2/x pi\| \2 2 /
| sec |x - --|| | sec |- - --|| | sec |x - --|| | sec |- - --||
\ \ 2 // \ \2 2 // \ \ 2 // \ \2 2 //
$$\frac{10 \left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(x \right)}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \left(\frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(x - \frac{\pi}{2} \right)}} + \frac{10 \left(- \frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{x}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \left(\frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$
/ 2/x pi\\
| cos |- - --|| / 2/ pi\\
| \2 2 /| / pi\ | cos |x - --||
10*|1 - ------------|*cos|x - --| | \ 2 /| /x pi\
| 2/x\ | \ 2 / 10*|1 - ------------|*cos|- - --|
| cos |-| | | 2 | \2 2 /
\ \2/ / \ cos (x) /
-------------------------------------------- + --------------------------------------------
/ 2/ pi\\ / 2/x pi\\ / 2/ pi\\ / 2/x pi\\
| cos |x - --|| | cos |- - --|| | cos |x - --|| | cos |- - --||
| \ 2 /| | \2 2 /| | \ 2 /| | \2 2 /| /x\
|1 + ------------|*|1 + ------------|*cos(x) |1 + ------------|*|1 + ------------|*cos|-|
| 2 | | 2/x\ | | 2 | | 2/x\ | \2/
\ cos (x) / | cos |-| | \ cos (x) / | cos |-| |
\ \2/ / \ \2/ /
$$\frac{10 \cdot \left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right) \cos{\left(x - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right) \left(1 + \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right) \cos{\left(x \right)}} + \frac{10 \cdot \left(1 - \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right) \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right) \left(1 + \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right) \cos{\left(\frac{x}{2} \right)}}$$
// 1 for x mod 2*pi = 0\
// 1 for x mod pi = 0\ || |
// 0 for x mod pi = 0\ || | // 0 for 2*x mod pi = 0\ || 1 |
|| | || 1 | || | ||-1 + ------- |
|| 2 | ||-1 + ------- | || 2 | || 2/x\ |
||-------------------- otherwise | || 2 | ||-------------------- otherwise | || tan |-| |
5*|
$$\left(5 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \tan{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(x \right)}}}{1 + \frac{1}{\tan^{2}{\left(x \right)}}} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(x \right)}}\right) \tan{\left(x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)$$
/ 2/pi x\\
| csc |-- - -|| / 2/pi \\
| \2 2/| /pi \ | csc |-- - x||
10*|1 - ------------|*csc|-- - x| | \2 /| /pi x\
| 2/x\ | \2 / 10*|1 - ------------|*csc|-- - -|
| csc |-| | | 2 | \2 2/
\ \2/ / \ csc (x) /
-------------------------------------------- + --------------------------------------------
/ 2/pi \\ / 2/pi x\\ / 2/pi \\ / 2/pi x\\
| csc |-- - x|| | csc |-- - -|| | csc |-- - x|| | csc |-- - -||
| \2 /| | \2 2/| | \2 /| | \2 2/| /x\
|1 + ------------|*|1 + ------------|*csc(x) |1 + ------------|*|1 + ------------|*csc|-|
| 2 | | 2/x\ | | 2 | | 2/x\ | \2/
\ csc (x) / | csc |-| | \ csc (x) / | csc |-| |
\ \2/ / \ \2/ /
$$\frac{10 \cdot \left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \csc{\left(- x + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \left(1 + \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}\right) \csc{\left(x \right)}} + \frac{10 \cdot \left(1 - \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}\right) \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \left(1 + \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}\right) \csc{\left(\frac{x}{2} \right)}}$$
// /pi \ \ // / pi\ \
// 0 for x mod pi = 0\ || 0 for |-- + 2*x| mod pi = 0| || 0 for |x + --| mod pi = 0|
|| | || \2 / | // 0 for 2*x mod pi = 0\ || \ 2 / |
|| /x\ | || | || | || |
|| 2*cot|-| | || / pi\ | || 2*cot(x) | || /x pi\ |
5*|< \2/ |*|< 2*cot|x + --| | + 5*|<----------- otherwise |*|< 2*cot|- + --| |
||----------- otherwise | || \ 4 / | || 2 | || \2 4 / |
|| 2/x\ | ||---------------- otherwise | ||1 + cot (x) | ||---------------- otherwise |
||1 + cot |-| | || 2/ pi\ | \\ / || 2/x pi\ |
\\ \2/ / ||1 + cot |x + --| | ||1 + cot |- + --| |
\\ \ 4 / / \\ \2 4 / /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(2 x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(x + \frac{\pi}{4} \right)}}{\cot^{2}{\left(x + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for x mod pi = 0\ // 1 for x mod pi = 0\ // 0 for 2*x mod pi = 0\ // 1 for x mod 2*pi = 0\
|| | || | || | || |
5*|
$$\left(5 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
// / 3*pi\ \ // / 3*pi\ \
|| 1 for |x + ----| mod 2*pi = 0| // 1 for x mod 2*pi = 0\ || 1 for |2*x + ----| mod 2*pi = 0|
// 1 for x mod pi = 0\ || \ 2 / | || | || \ 2 / |
|| | || | || 2/x\ | || |
|| 2 | || 2/x pi\ | ||-1 + cot |-| | || 2/ pi\ |
5*|<-1 + cot (x) |*|<-1 + tan |- + --| | + 5*|< \2/ |*|<-1 + tan |x + --| |
||------------ otherwise | || \2 4 / | ||------------ otherwise | || \ 4 / |
|| 2 | ||----------------- otherwise | || 2/x\ | ||----------------- otherwise |
\\1 + cot (x) / || 2/x pi\ | ||1 + cot |-| | || 2/ pi\ |
|| 1 + tan |- + --| | \\ \2/ / || 1 + tan |x + --| |
\\ \2 4 / / \\ \ 4 / /
$$\left(5 \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(2 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(x + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(x + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for x mod 2*pi = 0\
// 1 for x mod pi = 0\ || |
// 0 for x mod pi = 0\ || | || 2 |
|| | || 2 | // 0 for 2*x mod pi = 0\ || sin (x) |
|| sin(x) | || sin (2*x) | || | ||-1 + --------- |
||----------------------- otherwise | ||-1 + --------- | || sin(2*x) | || 4/x\ |
||/ 2 \ | || 4 | ||----------------------- otherwise | || 4*sin |-| |
5*|<| sin (x) | 2/x\ |*|< 4*sin (x) | + 5*|
$$\left(5 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \sin^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(2 x \right)}}{4 \sin^{4}{\left(x \right)}}}{1 + \frac{\sin^{2}{\left(2 x \right)}}{4 \sin^{4}{\left(x \right)}}} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{\sin{\left(2 x \right)}}{\left(1 + \frac{\sin^{2}{\left(2 x \right)}}{4 \sin^{4}{\left(x \right)}}\right) \sin^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}}{1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\ || | // 1 for x mod pi = 0\ // 0 for 2*x mod pi = 0\ || | ||/ 0 for x mod pi = 0 | || | || | ||/ 1 for x mod 2*pi = 0 | ||| | ||/ 1 for x mod pi = 0 | ||/ 0 for 2*x mod pi = 0 | ||| | ||| /x\ | ||| | ||| | ||| 2/x\ | 5*|<| 2*cot|-| |*|<| 2 | + 5*|<| 2*cot(x) |*|<|-1 + cot |-| | ||< \2/ otherwise | ||<-1 + cot (x) otherwise | ||<----------- otherwise otherwise | ||< \2/ otherwise | |||----------- otherwise | |||------------ otherwise | ||| 2 | |||------------ otherwise | ||| 2/x\ | ||| 2 | |||1 + cot (x) | ||| 2/x\ | |||1 + cot |-| | \\\1 + cot (x) / \\\ / |||1 + cot |-| | \\\ \2/ / \\\ \2/ /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for x mod 2*pi = 0\
// 1 for x mod pi = 0\ || |
// 0 for x mod pi = 0\ || | || 2/x\ |
|| | || 2 | // 0 for 2*x mod pi = 0\ || cos |-| |
|| /x\ | || cos (x) | || | || \2/ |
|| 2*cos|-| | ||-1 + ------------ | || 2*cos(x) | ||-1 + ------------ |
|| \2/ | || 2/ pi\ | ||------------------------------ otherwise | || 2/x pi\ |
||------------------------------ otherwise | || cos |x - --| | ||/ 2 \ | || cos |- - --| |
5*|
$$\left(5 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{\frac{\cos^{2}{\left(x \right)}}{\cos^{2}{\left(x - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(x \right)}}{\cos^{2}{\left(x - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \cos{\left(x \right)}}{\left(\frac{\cos^{2}{\left(x \right)}}{\cos^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for x mod 2*pi = 0\
// 1 for x mod pi = 0\ || |
// 0 for x mod pi = 0\ || | // 0 for 2*x mod pi = 0\ || 2/x pi\ |
|| | || 2/ pi\ | || | || sec |- - --| |
|| /x pi\ | || sec |x - --| | || / pi\ | || \2 2 / |
|| 2*sec|- - --| | || \ 2 / | || 2*sec|x - --| | ||-1 + ------------ |
|| \2 2 / | ||-1 + ------------ | || \ 2 / | || 2/x\ |
||------------------------- otherwise | || 2 | ||------------------------- otherwise | || sec |-| |
5*|
$$\left(5 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right) \sec{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(x \right)}}}{1 + \frac{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(x \right)}}} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \sec{\left(x - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(x \right)}}\right) \sec{\left(x \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for x mod 2*pi = 0\
// 1 for x mod pi = 0\ || |
// 0 for x mod pi = 0\ || | || 2/x\ |
|| | || 2 | // 0 for 2*x mod pi = 0\ || csc |-| |
|| /x\ | || csc (x) | || | || \2/ |
|| 2*csc|-| | ||-1 + ------------ | || 2*csc(x) | ||-1 + ------------ |
|| \2/ | || 2/pi \ | ||------------------------------ otherwise | || 2/pi x\ |
||------------------------------ otherwise | || csc |-- - x| | ||/ 2 \ | || csc |-- - -| |
5*|
$$\left(5 \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{\frac{\csc^{2}{\left(x \right)}}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(x \right)}}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \csc{\left(x \right)}}{\left(\frac{\csc^{2}{\left(x \right)}}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right)$$
5*Piecewise((0, Mod(x = pi, 0)), (2*csc(x/2)/((1 + csc(x/2)^2/csc(pi/2 - x/2)^2)*csc(pi/2 - x/2)), True))*Piecewise((1, Mod(x = pi, 0)), ((-1 + csc(x)^2/csc(pi/2 - x)^2)/(1 + csc(x)^2/csc(pi/2 - x)^2), True)) + 5*Piecewise((0, Mod(2*x = pi, 0)), (2*csc(x)/((1 + csc(x)^2/csc(pi/2 - x)^2)*csc(pi/2 - x)), True))*Piecewise((1, Mod(x = 2*pi, 0)), ((-1 + csc(x/2)^2/csc(pi/2 - x/2)^2)/(1 + csc(x/2)^2/csc(pi/2 - x/2)^2), True))
Численный ответ
[src]
5.0*cos(x)*sin(2*x) + 5.0*cos(2*x)*sin(x)
5.0*cos(x)*sin(2*x) + 5.0*cos(2*x)*sin(x)
Степени
[src]
/ I*x -I*x\ / -2*I*x 2*I*x\
|e e | / -2*I*x 2*I*x\ |e e | / -I*x I*x\
5*I*|---- + -----|*\- e + e / 5*I*|------- + ------|*\- e + e /
\ 2 2 / \ 2 2 /
- --------------------------------------- - ---------------------------------------
2 2
$$- \frac{5 i \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right) \left(e^{2 i x} - e^{- 2 i x}\right)}{2} - \frac{5 i \left(e^{i x} - e^{- i x}\right) \left(\frac{e^{2 i x}}{2} + \frac{e^{- 2 i x}}{2}\right)}{2}$$
-5*i*(exp(i*x)/2 + exp(-i*x)/2)*(-exp(-2*i*x) + exp(2*i*x))/2 - 5*i*(exp(-2*i*x)/2 + exp(2*i*x)/2)*(-exp(-i*x) + exp(i*x))/2