Господин Экзамен
10*sqrt(2)*sin(x)+10*sqrt(2)*cos(x) если x=2
Решение
Вы ввели
[src]
___ ___ 10*\/ 2 *sin(x) + 10*\/ 2 *cos(x)
$$10 \sqrt{2} \sin{\left(x \right)} + 10 \sqrt{2} \cos{\left(x \right)}$$
10*sqrt(2)*sin(x) + 10*sqrt(2)*cos(x)
Общее упрощение
[src]
/ pi\
20*sin|x + --|
\ 4 /
$$20 \sin{\left(x + \frac{\pi}{4} \right)}$$
20*sin(x + pi/4)
Подстановка условия
[src]
10*sqrt(2)*sin(x) + 10*sqrt(2)*cos(x) при x = 2
подставляем
___ ___ 10*\/ 2 *sin(x) + 10*\/ 2 *cos(x)
$$10 \sqrt{2} \sin{\left(x \right)} + 10 \sqrt{2} \cos{\left(x \right)}$$
/ pi\
20*sin|x + --|
\ 4 /
$$20 \sin{\left(x + \frac{\pi}{4} \right)}$$
переменные
x = 2
$$x = 2$$
/ pi\
20*sin|(2) + --|
\ 4 /
$$20 \sin{\left((2) + \frac{\pi}{4} \right)}$$
/ pi\
20*sin|2 + --|
\ 4 /
$$20 \sin{\left(\frac{\pi}{4} + 2 \right)}$$
20*sin(2 + pi/4)
Комбинаторика
[src]
___ 10*\/ 2 *(cos(x) + sin(x))
$$10 \sqrt{2} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)$$
10*sqrt(2)*(cos(x) + sin(x))
Степени
[src]
/ I*x -I*x\
___ |e e | ___ / -I*x I*x\
10*\/ 2 *|---- + -----| - 5*I*\/ 2 *\- e + e /
\ 2 2 /
$$10 \sqrt{2} \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right) - 5 \sqrt{2} i \left(e^{i x} - e^{- i x}\right)$$
10*sqrt(2)*(exp(i*x)/2 + exp(-i*x)/2) - 5*i*sqrt(2)*(-exp(-i*x) + exp(i*x))
Численный ответ
[src]
14.142135623731*cos(x) + 14.142135623731*sin(x)
14.142135623731*cos(x) + 14.142135623731*sin(x)
Тригонометрическая часть
[src]
/ pi\
20*cos|x - --|
\ 4 /
$$20 \cos{\left(x - \frac{\pi}{4} \right)}$$
/ pi\
20*sin|x + --|
\ 4 /
$$20 \sin{\left(x + \frac{\pi}{4} \right)}$$
20 ----------- / pi\ sec|x - --| \ 4 /
$$\frac{20}{\sec{\left(x - \frac{\pi}{4} \right)}}$$
20 ----------- / pi\ csc|x + --| \ 4 /
$$\frac{20}{\csc{\left(x + \frac{\pi}{4} \right)}}$$
___ ___ / pi\
10*\/ 2 *cos(x) + 10*\/ 2 *cos|x - --|
\ 2 /
$$10 \sqrt{2} \cos{\left(x \right)} + 10 \sqrt{2} \cos{\left(x - \frac{\pi}{2} \right)}$$
___ ___ / pi\
10*\/ 2 *sin(x) + 10*\/ 2 *sin|x + --|
\ 2 /
$$10 \sqrt{2} \sin{\left(x \right)} + 10 \sqrt{2} \sin{\left(x + \frac{\pi}{2} \right)}$$
___ ___ 10*\/ 2 10*\/ 2 -------- + -------- csc(x) sec(x)
$$\frac{10 \sqrt{2}}{\sec{\left(x \right)}} + \frac{10 \sqrt{2}}{\csc{\left(x \right)}}$$
___ / 2/x\ \
10*\/ 2 *|-1 + 2*cos |-| + sin(x)|
\ \2/ /
$$10 \sqrt{2} \left(2 \cos^{2}{\left(\frac{x}{2} \right)} + \sin{\left(x \right)} - 1\right)$$
___ ___
10*\/ 2 10*\/ 2
-------- + -----------
sec(x) / pi\
sec|x - --|
\ 2 /
$$\frac{10 \sqrt{2}}{\sec{\left(x - \frac{\pi}{2} \right)}} + \frac{10 \sqrt{2}}{\sec{\left(x \right)}}$$
___ ___
10*\/ 2 10*\/ 2
-------- + -----------
csc(x) /pi \
csc|-- - x|
\2 /
$$\frac{10 \sqrt{2}}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{10 \sqrt{2}}{\csc{\left(x \right)}}$$
___ ___
10*\/ 2 10*\/ 2
-------- + -----------
sec(x) /pi \
sec|-- - x|
\2 /
$$\frac{10 \sqrt{2}}{\sec{\left(- x + \frac{\pi}{2} \right)}} + \frac{10 \sqrt{2}}{\sec{\left(x \right)}}$$
___ ___
10*\/ 2 10*\/ 2
----------- + -----------
csc(pi - x) /pi \
csc|-- - x|
\2 /
$$\frac{10 \sqrt{2}}{\csc{\left(- x + \frac{\pi}{2} \right)}} + \frac{10 \sqrt{2}}{\csc{\left(- x + \pi \right)}}$$
/x pi\
40*tan|- + --|
\2 8 /
----------------
2/x pi\
1 + tan |- + --|
\2 8 /
$$\frac{40 \tan{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1}$$
___ ___ /x\
10*\/ 2 *cos(x) + 10*\/ 2 *(1 + cos(x))*tan|-|
\2/
$$10 \sqrt{2} \left(\cos{\left(x \right)} + 1\right) \tan{\left(\frac{x}{2} \right)} + 10 \sqrt{2} \cos{\left(x \right)}$$
___ ___ / 2/x pi\\
10*\/ 2 *cos(x) + 5*\/ 2 *|1 - cot |- + --||*(1 + sin(x))
\ \2 4 //
$$5 \sqrt{2} \cdot \left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(x \right)} + 1\right) + 10 \sqrt{2} \cos{\left(x \right)}$$
___ / 2/x\\ ___ /x\
10*\/ 2 *|1 - tan |-|| 20*\/ 2 *tan|-|
\ \2// \2/
---------------------- + ---------------
2/x\ 2/x\
1 + tan |-| 1 + tan |-|
\2/ \2/
$$\frac{10 \sqrt{2} \cdot \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{20 \sqrt{2} \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}$$
___ /x\ ___ /x pi\
20*\/ 2 *cot|-| 20*\/ 2 *tan|- + --|
\2/ \2 4 /
--------------- + --------------------
2/x\ 2/x pi\
1 + cot |-| 1 + tan |- + --|
\2/ \2 4 /
$$\frac{20 \sqrt{2} \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{20 \sqrt{2} \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1}$$
___ /x\ ___ /x pi\
20*\/ 2 *tan|-| 20*\/ 2 *tan|- + --|
\2/ \2 4 /
--------------- + --------------------
2/x\ 2/x pi\
1 + tan |-| 1 + tan |- + --|
\2/ \2 4 /
$$\frac{20 \sqrt{2} \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} + \frac{20 \sqrt{2} \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1}$$
// / pi\ \ || 0 for |x + --| mod pi = 0| || \ 4 / | 20*|< | || 2/x pi\ /x pi\ | ||2*sin |- + --|*cot|- + --| otherwise | \\ \2 8 / \2 8 / /
$$20 \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{4}\right) \bmod \pi = 0 \\2 \sin^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} \cot{\left(\frac{x}{2} + \frac{\pi}{8} \right)} & \text{otherwise} \end{cases}\right)$$
// / pi\ \ || 0 for |x + --| mod pi = 0| || \ 4 / | || | || /x pi\ | 20*|< 2*cot|- + --| | || \2 8 / | ||---------------- otherwise | || 2/x pi\ | ||1 + cot |- + --| | \\ \2 8 / /
$$20 \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{4}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} + \frac{\pi}{8} \right)}}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{8} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
___ / 1 \
10*\/ 2 *|1 - -------|
| 2/x\|
| cot |-|| ___
\ \2// 20*\/ 2
---------------------- + --------------------
1 / 1 \ /x\
1 + ------- |1 + -------|*cot|-|
2/x\ | 2/x\| \2/
cot |-| | cot |-||
\2/ \ \2//
$$\frac{10 \sqrt{2} \cdot \left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)}{1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}} + \frac{20 \sqrt{2}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right) \cot{\left(\frac{x}{2} \right)}}$$
___ / 2/x\\ ___ / 2/x pi\\
10*\/ 2 *|-1 + cot |-|| 10*\/ 2 *|-1 + tan |- + --||
\ \2// \ \2 4 //
----------------------- + ----------------------------
2/x\ 2/x pi\
1 + cot |-| 1 + tan |- + --|
\2/ \2 4 /
$$\frac{10 \sqrt{2} \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1\right)}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} + \frac{10 \sqrt{2} \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{\cot^{2}{\left(\frac{x}{2} \right)} + 1}$$
___ / 2/x pi\\ ___ / 2/x\\
10*\/ 2 *|1 - cot |- + --|| 10*\/ 2 *|1 - tan |-||
\ \2 4 // \ \2//
--------------------------- + ----------------------
2/x pi\ 2/x\
1 + cot |- + --| 1 + tan |-|
\2 4 / \2/
$$\frac{10 \sqrt{2} \cdot \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} + \frac{10 \sqrt{2} \cdot \left(- \cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1}$$
___ // 0 for x mod pi = 0\ ___ // 1 for x mod 2*pi = 0\
10*\/ 2 *|< | + 10*\/ 2 *|< |
\\sin(x) otherwise / \\cos(x) otherwise /
$$\left(10 \sqrt{2} \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(10 \sqrt{2} \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for x mod pi = 0\
___ || | ___ // 1 for x mod 2*pi = 0\
10*\/ 2 *|< / pi\ | + 10*\/ 2 *|< |
||cos|x - --| otherwise | \\cos(x) otherwise /
\\ \ 2 / /
$$\left(10 \sqrt{2} \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + \left(10 \sqrt{2} \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for x mod 2*pi = 0\
___ // 0 for x mod pi = 0\ ___ || |
10*\/ 2 *|< | + 10*\/ 2 *|< / pi\ |
\\sin(x) otherwise / ||sin|x + --| otherwise |
\\ \ 2 / /
$$\left(10 \sqrt{2} \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(10 \sqrt{2} \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\sin{\left(x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right)$$
___ 2/x\
___ 40*\/ 2 *sin |-|*sin(x)
20*\/ 2 *(-1 - cos(2*x) + 2*cos(x)) \2/
----------------------------------- + -----------------------
2 2 4/x\
1 - cos(2*x) + 2*(1 - cos(x)) sin (x) + 4*sin |-|
\2/
$$\frac{40 \sqrt{2} \sin^{2}{\left(\frac{x}{2} \right)} \sin{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)} + \sin^{2}{\left(x \right)}} + \frac{20 \sqrt{2} \cdot \left(2 \cos{\left(x \right)} - \cos{\left(2 x \right)} - 1\right)}{2 \left(- \cos{\left(x \right)} + 1\right)^{2} - \cos{\left(2 x \right)} + 1}$$
// / 3*pi\ \
___ // 1 for x mod 2*pi = 0\ ___ || 1 for |x + ----| mod 2*pi = 0|
10*\/ 2 *|< | + 10*\/ 2 *|< \ 2 / |
\\cos(x) otherwise / || |
\\sin(x) otherwise /
$$\left(10 \sqrt{2} \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(10 \sqrt{2} \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for x mod pi = 0\
|| | // 1 for x mod 2*pi = 0\
___ || 1 | ___ || |
10*\/ 2 *|<----------- otherwise | + 10*\/ 2 *|< 1 |
|| / pi\ | ||------ otherwise |
||sec|x - --| | \\sec(x) /
\\ \ 2 / /
$$\left(10 \sqrt{2} \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\sec{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(10 \sqrt{2} \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for x mod 2*pi = 0\
// 0 for x mod pi = 0\ || |
___ || | ___ || 1 |
10*\/ 2 *|< 1 | + 10*\/ 2 *|<----------- otherwise |
||------ otherwise | || /pi \ |
\\csc(x) / ||csc|-- - x| |
\\ \2 / /
$$\left(10 \sqrt{2} \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\csc{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(10 \sqrt{2} \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for x mod pi = 0\
|| |
___ ||1 - cos(x) | ___ // 1 for x mod 2*pi = 0\
10*\/ 2 *|<---------- otherwise | + 10*\/ 2 *|< |
|| /x\ | \\cos(x) otherwise /
|| tan|-| |
\\ \2/ /
$$\left(10 \sqrt{2} \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{- \cos{\left(x \right)} + 1}{\tan{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(10 \sqrt{2} \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right)\right)$$
/ 4/x\\
| 4*sin |-||
___ | \2/|
10*\/ 2 *|1 - ---------| ___ 2/x\
| 2 | 40*\/ 2 *sin |-|
\ sin (x) / \2/
------------------------ + ----------------------
4/x\ / 4/x\\
4*sin |-| | 4*sin |-||
\2/ | \2/|
1 + --------- |1 + ---------|*sin(x)
2 | 2 |
sin (x) \ sin (x) /
$$\frac{10 \sqrt{2} \left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)}{\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1} + \frac{40 \sqrt{2} \sin^{2}{\left(\frac{x}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right) \sin{\left(x \right)}}$$
// / pi\ \
|| 0 for |x + --| mod pi = 0|
___ // 0 for x mod pi = 0\ ___ || \ 2 / |
10*\/ 2 *|< | + 10*\/ 2 *|< |
\\sin(x) otherwise / || /x pi\ |
||(1 + sin(x))*cot|- + --| otherwise |
\\ \2 4 / /
$$\left(10 \sqrt{2} \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases}\right)\right) + \left(10 \sqrt{2} \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(x \right)} + 1\right) \cot{\left(\frac{x}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
|| /x\ | || 2/x\ |
___ || 2*cot|-| | ___ ||-1 + cot |-| |
10*\/ 2 *|< \2/ | + 10*\/ 2 *|< \2/ |
||----------- otherwise | ||------------ otherwise |
|| 2/x\ | || 2/x\ |
||1 + cot |-| | ||1 + cot |-| |
\\ \2/ / \\ \2/ /
$$\left(10 \sqrt{2} \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(10 \sqrt{2} \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
|| /x\ | || 2/x\ |
___ || 2*tan|-| | ___ ||1 - tan |-| |
10*\/ 2 *|< \2/ | + 10*\/ 2 *|< \2/ |
||----------- otherwise | ||----------- otherwise |
|| 2/x\ | || 2/x\ |
||1 + tan |-| | ||1 + tan |-| |
\\ \2/ / \\ \2/ /
$$\left(10 \sqrt{2} \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{x}{2} \right)}}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(10 \sqrt{2} \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{x}{2} \right)} + 1}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
___ || | ___ || |
10*\/ 2 *|
$$\left(10 \sqrt{2} \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(10 \sqrt{2} \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
/ 2/x\ \
| sec |-| |
___ | \2/ |
10*\/ 2 *|1 - ------------|
| 2/x pi\| ___ /x\
| sec |- - --|| 20*\/ 2 *sec|-|
\ \2 2 // \2/
--------------------------- + ------------------------------
2/x\ / 2/x\ \
sec |-| | sec |-| |
\2/ | \2/ | /x pi\
1 + ------------ |1 + ------------|*sec|- - --|
2/x pi\ | 2/x pi\| \2 2 /
sec |- - --| | sec |- - --||
\2 2 / \ \2 2 //
$$\frac{10 \sqrt{2} \left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)}{\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1} + \frac{20 \sqrt{2} \sec{\left(\frac{x}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}$$
/ 2/x pi\\
| cos |- - --||
___ | \2 2 /|
10*\/ 2 *|1 - ------------|
| 2/x\ | ___ /x pi\
| cos |-| | 20*\/ 2 *cos|- - --|
\ \2/ / \2 2 /
--------------------------- + -------------------------
2/x pi\ / 2/x pi\\
cos |- - --| | cos |- - --||
\2 2 / | \2 2 /| /x\
1 + ------------ |1 + ------------|*cos|-|
2/x\ | 2/x\ | \2/
cos |-| | cos |-| |
\2/ \ \2/ /
$$\frac{10 \sqrt{2} \cdot \left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right)}{1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}} + \frac{20 \sqrt{2} \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right) \cos{\left(\frac{x}{2} \right)}}$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ || 1 |
|| | ||-1 + ------- |
|| 2 | || 2/x\ |
___ ||-------------------- otherwise | ___ || tan |-| |
10*\/ 2 *|
$$\left(10 \sqrt{2} \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}\right) \tan{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(10 \sqrt{2} \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)$$
/ 2/pi x\\
| csc |-- - -||
___ | \2 2/|
10*\/ 2 *|1 - ------------|
| 2/x\ | ___ /pi x\
| csc |-| | 20*\/ 2 *csc|-- - -|
\ \2/ / \2 2/
--------------------------- + -------------------------
2/pi x\ / 2/pi x\\
csc |-- - -| | csc |-- - -||
\2 2/ | \2 2/| /x\
1 + ------------ |1 + ------------|*csc|-|
2/x\ | 2/x\ | \2/
csc |-| | csc |-| |
\2/ \ \2/ /
$$\frac{10 \sqrt{2} \cdot \left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right)}{1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}} + \frac{20 \sqrt{2} \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \csc{\left(\frac{x}{2} \right)}}$$
// / pi\ \
// 0 for x mod pi = 0\ || 0 for |x + --| mod pi = 0|
|| | || \ 2 / |
|| /x\ | || |
___ || 2*cot|-| | ___ || /x pi\ |
10*\/ 2 *|< \2/ | + 10*\/ 2 *|< 2*cot|- + --| |
||----------- otherwise | || \2 4 / |
|| 2/x\ | ||---------------- otherwise |
||1 + cot |-| | || 2/x pi\ |
\\ \2/ / ||1 + cot |- + --| |
\\ \2 4 / /
$$\left(10 \sqrt{2} \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(10 \sqrt{2} \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// / 3*pi\ \
// 1 for x mod 2*pi = 0\ || 1 for |x + ----| mod 2*pi = 0|
|| | || \ 2 / |
|| 2/x\ | || |
___ ||-1 + cot |-| | ___ || 2/x pi\ |
10*\/ 2 *|< \2/ | + 10*\/ 2 *|<-1 + tan |- + --| |
||------------ otherwise | || \2 4 / |
|| 2/x\ | ||----------------- otherwise |
||1 + cot |-| | || 2/x pi\ |
\\ \2/ / || 1 + tan |- + --| |
\\ \2 4 / /
$$\left(10 \sqrt{2} \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + \left(10 \sqrt{2} \left(\begin{cases} 1 & \text{for}\: \left(x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for x mod pi = 0\
|| |
|| 2*sin(x) | // 1 for x mod 2*pi = 0\
||---------------------------- otherwise | || |
___ || / 2 \ | ___ || 2 |
10*\/ 2 *|< | sin (x) | | + 10*\/ 2 *|< -4 + 4*sin (x) + 4*cos(x) |
||(1 - cos(x))*|1 + ---------| | ||--------------------------- otherwise |
|| | 4/x\| | || 2 2 |
|| | 4*sin |-|| | \\2*(1 - cos(x)) + 2*sin (x) /
|| \ \2// |
\\ /
$$\left(10 \sqrt{2} \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \sin{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \left(- \cos{\left(x \right)} + 1\right)} & \text{otherwise} \end{cases}\right)\right) + \left(10 \sqrt{2} \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{4 \sin^{2}{\left(x \right)} + 4 \cos{\left(x \right)} - 4}{2 \left(- \cos{\left(x \right)} + 1\right)^{2} + 2 \sin^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ || 2 |
|| | || sin (x) |
|| sin(x) | ||-1 + --------- |
||----------------------- otherwise | || 4/x\ |
___ ||/ 2 \ | ___ || 4*sin |-| |
10*\/ 2 *|<| sin (x) | 2/x\ | + 10*\/ 2 *|< \2/ |
|||1 + ---------|*sin |-| | ||-------------- otherwise |
||| 4/x\| \2/ | || 2 |
||| 4*sin |-|| | || sin (x) |
||\ \2// | ||1 + --------- |
\\ / || 4/x\ |
|| 4*sin |-| |
\\ \2/ /
$$\left(10 \sqrt{2} \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{\sin{\left(x \right)}}{\left(1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}\right) \sin^{2}{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(10 \sqrt{2} \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}}{1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)$$
// 0 for x mod pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
||/ 0 for x mod pi = 0 | ||/ 1 for x mod 2*pi = 0 |
||| | ||| |
___ ||| /x\ | ___ ||| 2/x\ |
10*\/ 2 *|<| 2*cot|-| | + 10*\/ 2 *|<|-1 + cot |-| |
||< \2/ otherwise | ||< \2/ otherwise |
|||----------- otherwise | |||------------ otherwise |
||| 2/x\ | ||| 2/x\ |
|||1 + cot |-| | |||1 + cot |-| |
\\\ \2/ / \\\ \2/ /
$$\left(10 \sqrt{2} \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} \right)}}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + \left(10 \sqrt{2} \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ || 2/x\ |
|| | || cos |-| |
|| /x\ | || \2/ |
|| 2*cos|-| | ||-1 + ------------ |
|| \2/ | || 2/x pi\ |
___ ||------------------------------ otherwise | ___ || cos |- - --| |
10*\/ 2 *|
$$\left(10 \sqrt{2} \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{x}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(10 \sqrt{2} \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ || 2/x pi\ |
|| | || sec |- - --| |
|| /x pi\ | || \2 2 / |
|| 2*sec|- - --| | ||-1 + ------------ |
|| \2 2 / | || 2/x\ |
___ ||------------------------- otherwise | ___ || sec |-| |
10*\/ 2 *|
$$\left(10 \sqrt{2} \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}\right) \sec{\left(\frac{x}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(10 \sqrt{2} \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}} & \text{otherwise} \end{cases}\right)\right)$$
// 1 for x mod 2*pi = 0\
|| |
// 0 for x mod pi = 0\ || 2/x\ |
|| | || csc |-| |
|| /x\ | || \2/ |
|| 2*csc|-| | ||-1 + ------------ |
|| \2/ | || 2/pi x\ |
___ ||------------------------------ otherwise | ___ || csc |-- - -| |
10*\/ 2 *|
$$\left(10 \sqrt{2} \left(\begin{cases} 0 & \text{for}\: x \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{x}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + \left(10 \sqrt{2} \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right)$$
10*sqrt(2)*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)) + 10*sqrt(2)*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))
Объединение рациональных выражений
[src]
___ 10*\/ 2 *(cos(x) + sin(x))
$$10 \sqrt{2} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)$$
10*sqrt(2)*(cos(x) + sin(x))