Господин Экзамен
8-5*cos(2*x)-5*sin(2*x) если x=3
Решение
Вы ввели
[src]
8 - 5*cos(2*x) - 5*sin(2*x)
$$- 5 \sin{\left(2 x \right)} - 5 \cos{\left(2 x \right)} + 8$$
8 - 5*cos(2*x) - 5*sin(2*x)
Общее упрощение
[src]
___ / pi\
8 - 5*\/ 2 *sin|2*x + --|
\ 4 /
$$- 5 \sqrt{2} \sin{\left(2 x + \frac{\pi}{4} \right)} + 8$$
8 - 5*sqrt(2)*sin(2*x + pi/4)
Подстановка условия
[src]
8 - 5*cos(2*x) - 5*sin(2*x) при x = 3
подставляем
8 - 5*cos(2*x) - 5*sin(2*x)
$$- 5 \sin{\left(2 x \right)} - 5 \cos{\left(2 x \right)} + 8$$
___ / pi\
8 - 5*\/ 2 *sin|2*x + --|
\ 4 /
$$- 5 \sqrt{2} \sin{\left(2 x + \frac{\pi}{4} \right)} + 8$$
переменные
x = 3
$$x = 3$$
___ / pi\
8 - 5*\/ 2 *sin|2*(3) + --|
\ 4 /
$$- 5 \sqrt{2} \sin{\left(2 (3) + \frac{\pi}{4} \right)} + 8$$
___ / pi\
8 - 5*\/ 2 *sin|2*3 + --|
\ 4 /
$$- 5 \sqrt{2} \sin{\left(\frac{\pi}{4} + 2 \cdot 3 \right)} + 8$$
___ / pi\
8 - 5*\/ 2 *sin|6 + --|
\ 4 /
$$- 5 \sqrt{2} \sin{\left(\frac{\pi}{4} + 6 \right)} + 8$$
8 - 5*sqrt(2)*sin(6 + pi/4)
Степени
[src]
-2*I*x 2*I*x / -2*I*x 2*I*x\
5*e 5*e 5*I*\- e + e /
8 - --------- - -------- + ------------------------
2 2 2
$$- \frac{5 e^{2 i x}}{2} + \frac{5 i \left(e^{2 i x} - e^{- 2 i x}\right)}{2} + 8 - \frac{5 e^{- 2 i x}}{2}$$
8 - 5*exp(-2*i*x)/2 - 5*exp(2*i*x)/2 + 5*i*(-exp(-2*i*x) + exp(2*i*x))/2
Тригонометрическая часть
[src]
___ / pi\
8 - 5*\/ 2 *cos|2*x - --|
\ 4 /
$$- 5 \sqrt{2} \cos{\left(2 x - \frac{\pi}{4} \right)} + 8$$
___ / pi\
8 - 5*\/ 2 *sin|2*x + --|
\ 4 /
$$- 5 \sqrt{2} \sin{\left(2 x + \frac{\pi}{4} \right)} + 8$$
___
5*\/ 2
8 - -------------
/ pi\
csc|2*x + --|
\ 4 /
$$8 - \frac{5 \sqrt{2}}{\csc{\left(2 x + \frac{\pi}{4} \right)}}$$
___
5*\/ 2
8 - -------------
/ pi\
sec|2*x - --|
\ 4 /
$$8 - \frac{5 \sqrt{2}}{\sec{\left(2 x - \frac{\pi}{4} \right)}}$$
2 13 - 10*cos (x) - 5*sin(2*x)
$$- 10 \cos^{2}{\left(x \right)} - 5 \sin{\left(2 x \right)} + 13$$
5 5
8 - -------- - --------
csc(2*x) sec(2*x)
$$8 - \frac{5}{\sec{\left(2 x \right)}} - \frac{5}{\csc{\left(2 x \right)}}$$
/pi \
8 - 5*sin(2*x) - 5*sin|-- + 2*x|
\2 /
$$- 5 \sin{\left(2 x \right)} - 5 \sin{\left(2 x + \frac{\pi}{2} \right)} + 8$$
/ pi\
8 - 5*cos(2*x) - 5*cos|2*x - --|
\ 2 /
$$- 5 \cos{\left(2 x \right)} - 5 \cos{\left(2 x - \frac{\pi}{2} \right)} + 8$$
5 5
8 - -------- - -------------
sec(2*x) / pi\
sec|2*x - --|
\ 2 /
$$8 - \frac{5}{\sec{\left(2 x - \frac{\pi}{2} \right)}} - \frac{5}{\sec{\left(2 x \right)}}$$
5 5
8 - -------- - -------------
sec(2*x) /pi \
sec|-- - 2*x|
\2 /
$$8 - \frac{5}{\sec{\left(- 2 x + \frac{\pi}{2} \right)}} - \frac{5}{\sec{\left(2 x \right)}}$$
5 5
8 - -------- - -------------
csc(2*x) /pi \
csc|-- - 2*x|
\2 /
$$8 - \frac{5}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}} - \frac{5}{\csc{\left(2 x \right)}}$$
5 5
8 - ------------- - -------------
csc(pi - 2*x) /pi \
csc|-- - 2*x|
\2 /
$$8 - \frac{5}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}} - \frac{5}{\csc{\left(- 2 x + \pi \right)}}$$
___ / pi\
10*\/ 2 *tan|x + --|
\ 8 /
8 - --------------------
2/ pi\
1 + tan |x + --|
\ 8 /
$$8 - \frac{10 \sqrt{2} \tan{\left(x + \frac{\pi}{8} \right)}}{\tan^{2}{\left(x + \frac{\pi}{8} \right)} + 1}$$
2 2 8 - 5*cos (x) + 5*sin (x) - 10*cos(x)*sin(x)
$$5 \sin^{2}{\left(x \right)} - 10 \sin{\left(x \right)} \cos{\left(x \right)} - 5 \cos^{2}{\left(x \right)} + 8$$
/ 2 \
10*tan(x) 5*\1 - tan (x)/
8 - ----------- - ---------------
2 2
1 + tan (x) 1 + tan (x)
$$- \frac{5 \cdot \left(- \tan^{2}{\left(x \right)} + 1\right)}{\tan^{2}{\left(x \right)} + 1} + 8 - \frac{10 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}$$
/ pi\
10*tan|x + --|
10*tan(x) \ 4 /
8 - ----------- - ----------------
2 2/ pi\
1 + tan (x) 1 + tan |x + --|
\ 4 /
$$8 - \frac{10 \tan{\left(x + \frac{\pi}{4} \right)}}{\tan^{2}{\left(x + \frac{\pi}{4} \right)} + 1} - \frac{10 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1}$$
/ pi\
10*tan|x + --|
10*cot(x) \ 4 /
8 - ----------- - ----------------
2 2/ pi\
1 + cot (x) 1 + tan |x + --|
\ 4 /
$$8 - \frac{10 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} - \frac{10 \tan{\left(x + \frac{\pi}{4} \right)}}{\tan^{2}{\left(x + \frac{\pi}{4} \right)} + 1}$$
/ 1 \
5*|1 - -------|
| 2 |
10 \ cot (x)/
8 - -------------------- - ---------------
/ 1 \ 1
|1 + -------|*cot(x) 1 + -------
| 2 | 2
\ cot (x)/ cot (x)
$$- \frac{5 \cdot \left(1 - \frac{1}{\cot^{2}{\left(x \right)}}\right)}{1 + \frac{1}{\cot^{2}{\left(x \right)}}} + 8 - \frac{10}{\left(1 + \frac{1}{\cot^{2}{\left(x \right)}}\right) \cot{\left(x \right)}}$$
/ 2/ pi\\
/ 2 \ 5*|-1 + tan |x + --||
5*\-1 + cot (x)/ \ \ 4 //
8 - ---------------- - ---------------------
2 2/ pi\
1 + cot (x) 1 + tan |x + --|
\ 4 /
$$- \frac{5 \left(\tan^{2}{\left(x + \frac{\pi}{4} \right)} - 1\right)}{\tan^{2}{\left(x + \frac{\pi}{4} \right)} + 1} - \frac{5 \left(\cot^{2}{\left(x \right)} - 1\right)}{\cot^{2}{\left(x \right)} + 1} + 8$$
/ 2/ pi\\
5*|1 - cot |x + --|| / 2 \
\ \ 4 // 5*\1 - tan (x)/
8 - -------------------- - ---------------
2/ pi\ 2
1 + cot |x + --| 1 + tan (x)
\ 4 /
$$- \frac{5 \cdot \left(- \tan^{2}{\left(x \right)} + 1\right)}{\tan^{2}{\left(x \right)} + 1} - \frac{5 \cdot \left(- \cot^{2}{\left(x + \frac{\pi}{4} \right)} + 1\right)}{\cot^{2}{\left(x + \frac{\pi}{4} \right)} + 1} + 8$$
// / pi\ \
|| 0 for |2*x + --| mod pi = 0|
___ || \ 4 / |
8 - 5*\/ 2 *|< |
|| 2/ pi\ / pi\ |
||2*sin |x + --|*cot|x + --| otherwise |
\\ \ 8 / \ 8 / /
$$\left(- 5 \sqrt{2} \left(\begin{cases} 0 & \text{for}\: \left(2 x + \frac{\pi}{4}\right) \bmod \pi = 0 \\2 \sin^{2}{\left(x + \frac{\pi}{8} \right)} \cot{\left(x + \frac{\pi}{8} \right)} & \text{otherwise} \end{cases}\right)\right) + 8$$
// / pi\ \
|| 0 for |2*x + --| mod pi = 0|
|| \ 4 / |
|| |
___ || / pi\ |
8 - 5*\/ 2 *|< 2*cot|x + --| |
|| \ 8 / |
||---------------- otherwise |
|| 2/ pi\ |
||1 + cot |x + --| |
\\ \ 8 / /
$$\left(- 5 \sqrt{2} \left(\begin{cases} 0 & \text{for}\: \left(2 x + \frac{\pi}{4}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(x + \frac{\pi}{8} \right)}}{\cot^{2}{\left(x + \frac{\pi}{8} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 8$$
// 0 for 2*x mod pi = 0\ // 1 for x mod pi = 0\
8 - 5*|< | - 5*|< |
\\sin(2*x) otherwise / \\cos(2*x) otherwise /
$$\left(- 5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases}\right)\right) + 8$$
// 1 for x mod pi = 0\
// 0 for 2*x mod pi = 0\ || |
8 - 5*|< | - 5*|< /pi \ |
\\sin(2*x) otherwise / ||sin|-- + 2*x| otherwise |
\\ \2 / /
$$\left(- 5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\sin{\left(2 x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + 8$$
// 0 for 2*x mod pi = 0\
|| | // 1 for x mod pi = 0\
8 - 5*|< / pi\ | - 5*|< |
||cos|2*x - --| otherwise | \\cos(2*x) otherwise /
\\ \ 2 / /
$$\left(- 5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\cos{\left(2 x - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases}\right)\right) + 8$$
// /pi \ \
// 0 for 2*x mod pi = 0\ || 0 for |-- + 2*x| mod pi = 0|
8 - 5*|< | - 5*|< \2 / |
\\sin(2*x) otherwise / || |
\\cos(2*x) otherwise /
$$\left(- 5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 0 & \text{for}\: \left(2 x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases}\right)\right) + 8$$
// / 3*pi\ \
// 1 for x mod pi = 0\ || 1 for |2*x + ----| mod 2*pi = 0|
8 - 5*|< | - 5*|< \ 2 / |
\\cos(2*x) otherwise / || |
\\sin(2*x) otherwise /
$$\left(- 5 \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: \left(2 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases}\right)\right) + 8$$
// 0 for 2*x mod pi = 0\
|| | // 1 for x mod pi = 0\
|| 1 | || |
8 - 5*|<------------- otherwise | - 5*|< 1 |
|| / pi\ | ||-------- otherwise |
||sec|2*x - --| | \\sec(2*x) /
\\ \ 2 / /
$$\left(- 5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{1}{\sec{\left(2 x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\sec{\left(2 x \right)}} & \text{otherwise} \end{cases}\right)\right) + 8$$
/ 4 \
| 4*sin (x)|
5*|1 - ---------|
| 2 | 2
\ sin (2*x)/ 20*sin (x)
8 - ----------------- - ------------------------
4 / 4 \
4*sin (x) | 4*sin (x)|
1 + --------- |1 + ---------|*sin(2*x)
2 | 2 |
sin (2*x) \ sin (2*x)/
$$- \frac{5 \left(- \frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1\right)}{\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1} + 8 - \frac{20 \sin^{2}{\left(x \right)}}{\left(\frac{4 \sin^{4}{\left(x \right)}}{\sin^{2}{\left(2 x \right)}} + 1\right) \sin{\left(2 x \right)}}$$
// 1 for x mod pi = 0\
// 0 for 2*x mod pi = 0\ || |
|| | || 1 |
8 - 5*|< 1 | - 5*|<------------- otherwise |
||-------- otherwise | || /pi \ |
\\csc(2*x) / ||csc|-- - 2*x| |
\\ \2 / /
$$\left(- 5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{1}{\csc{\left(2 x \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{1}{\csc{\left(- 2 x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + 8$$
// 0 for 2*x mod pi = 0\ // 1 for x mod pi = 0\
|| | || |
|| 2*cot(x) | || 2 |
8 - 5*|<----------- otherwise | - 5*|<-1 + cot (x) |
|| 2 | ||------------ otherwise |
||1 + cot (x) | || 2 |
\\ / \\1 + cot (x) /
$$\left(- 5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 8$$
/ 2 \
| sec (x) |
5*|1 - ------------|
| 2/ pi\|
| sec |x - --||
\ \ 2 // 10*sec(x)
8 - -------------------- - ------------------------------
2 / 2 \
sec (x) | sec (x) | / pi\
1 + ------------ |1 + ------------|*sec|x - --|
2/ pi\ | 2/ pi\| \ 2 /
sec |x - --| | sec |x - --||
\ 2 / \ \ 2 //
$$- \frac{5 \left(- \frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right)}{\frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1} + 8 - \frac{10 \sec{\left(x \right)}}{\left(\frac{\sec^{2}{\left(x \right)}}{\sec^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(x - \frac{\pi}{2} \right)}}$$
// 0 for 2*x mod pi = 0\ // 1 for x mod pi = 0\
|| | || |
|| 2*tan(x) | || 2 |
8 - 5*|<----------- otherwise | - 5*|<1 - tan (x) |
|| 2 | ||----------- otherwise |
||1 + tan (x) | || 2 |
\\ / \\1 + tan (x) /
$$\left(- 5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \tan{\left(x \right)}}{\tan^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{- \tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 8$$
/ 2/ pi\\
| cos |x - --||
| \ 2 /|
5*|1 - ------------| / pi\
| 2 | 10*cos|x - --|
\ cos (x) / \ 2 /
8 - -------------------- - -------------------------
2/ pi\ / 2/ pi\\
cos |x - --| | cos |x - --||
\ 2 / | \ 2 /|
1 + ------------ |1 + ------------|*cos(x)
2 | 2 |
cos (x) \ cos (x) /
$$- \frac{5 \cdot \left(1 - \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right)}{1 + \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}} + 8 - \frac{10 \cos{\left(x - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(x - \frac{\pi}{2} \right)}}{\cos^{2}{\left(x \right)}}\right) \cos{\left(x \right)}}$$
/ 2/pi \\
| csc |-- - x||
| \2 /|
5*|1 - ------------| /pi \
| 2 | 10*csc|-- - x|
\ csc (x) / \2 /
8 - -------------------- - -------------------------
2/pi \ / 2/pi \\
csc |-- - x| | csc |-- - x||
\2 / | \2 /|
1 + ------------ |1 + ------------|*csc(x)
2 | 2 |
csc (x) \ csc (x) /
$$- \frac{5 \cdot \left(1 - \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}\right)}{1 + \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}} + 8 - \frac{10 \csc{\left(- x + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}}{\csc^{2}{\left(x \right)}}\right) \csc{\left(x \right)}}$$
// 1 for x mod pi = 0\
// 0 for 2*x mod pi = 0\ || |
|| | || 1 |
|| 2 | ||-1 + ------- |
||-------------------- otherwise | || 2 |
8 - 5*|
$$\left(- 5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(x \right)}}\right) \tan{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(x \right)}}}{1 + \frac{1}{\tan^{2}{\left(x \right)}}} & \text{otherwise} \end{cases}\right)\right) + 8$$
// /pi \ \
|| 0 for |-- + 2*x| mod pi = 0|
// 0 for 2*x mod pi = 0\ || \2 / |
|| | || |
|| 2*cot(x) | || / pi\ |
8 - 5*|<----------- otherwise | - 5*|< 2*cot|x + --| |
|| 2 | || \ 4 / |
||1 + cot (x) | ||---------------- otherwise |
\\ / || 2/ pi\ |
||1 + cot |x + --| |
\\ \ 4 / /
$$\left(- 5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 0 & \text{for}\: \left(2 x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(x + \frac{\pi}{4} \right)}}{\cot^{2}{\left(x + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 8$$
// / 3*pi\ \
|| 1 for |2*x + ----| mod 2*pi = 0|
// 1 for x mod pi = 0\ || \ 2 / |
|| | || |
|| 2 | || 2/ pi\ |
8 - 5*|<-1 + cot (x) | - 5*|<-1 + tan |x + --| |
||------------ otherwise | || \ 4 / |
|| 2 | ||----------------- otherwise |
\\1 + cot (x) / || 2/ pi\ |
|| 1 + tan |x + --| |
\\ \ 4 / /
$$\left(- 5 \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: \left(2 x + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(x + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(x + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)\right) + 8$$
// 0 for 2*x mod pi = 0\ // 1 for x mod pi = 0\
|| | || |
8 - 5*|
$$\left(- 5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\sin{\left(2 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\cos{\left(2 x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + 8$$
// 1 for x mod pi = 0\
|| |
// 0 for 2*x mod pi = 0\ || 2 |
|| | || sin (2*x) |
|| sin(2*x) | ||-1 + --------- |
||----------------------- otherwise | || 4 |
8 - 5*|
$$\left(- 5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{\sin{\left(2 x \right)}}{\left(1 + \frac{\sin^{2}{\left(2 x \right)}}{4 \sin^{4}{\left(x \right)}}\right) \sin^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(2 x \right)}}{4 \sin^{4}{\left(x \right)}}}{1 + \frac{\sin^{2}{\left(2 x \right)}}{4 \sin^{4}{\left(x \right)}}} & \text{otherwise} \end{cases}\right)\right) + 8$$
// 0 for 2*x mod pi = 0\ // 1 for x mod pi = 0\
|| | || |
||/ 0 for 2*x mod pi = 0 | ||/ 1 for x mod pi = 0 |
||| | ||| |
8 - 5*|<| 2*cot(x) | - 5*|<| 2 |
||<----------- otherwise otherwise | ||<-1 + cot (x) otherwise |
||| 2 | |||------------ otherwise |
|||1 + cot (x) | ||| 2 |
\\\ / \\\1 + cot (x) /
$$\left(- 5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \cot{\left(x \right)}}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{\cot^{2}{\left(x \right)} - 1}{\cot^{2}{\left(x \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + 8$$
// 1 for x mod pi = 0\
|| |
// 0 for 2*x mod pi = 0\ || 2 |
|| | || cos (x) |
|| 2*cos(x) | ||-1 + ------------ |
||------------------------------ otherwise | || 2/ pi\ |
||/ 2 \ | || cos |x - --| |
8 - 5*|<| cos (x) | / pi\ | - 5*|< \ 2 / |
|||1 + ------------|*cos|x - --| | ||----------------- otherwise |
||| 2/ pi\| \ 2 / | || 2 |
||| cos |x - --|| | || cos (x) |
||\ \ 2 // | || 1 + ------------ |
\\ / || 2/ pi\ |
|| cos |x - --| |
\\ \ 2 / /
$$\left(- 5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \cos{\left(x \right)}}{\left(\frac{\cos^{2}{\left(x \right)}}{\cos^{2}{\left(x - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(x - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{\frac{\cos^{2}{\left(x \right)}}{\cos^{2}{\left(x - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(x \right)}}{\cos^{2}{\left(x - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) + 8$$
// 1 for x mod pi = 0\
// 0 for 2*x mod pi = 0\ || |
|| | || 2/ pi\ |
|| / pi\ | || sec |x - --| |
|| 2*sec|x - --| | || \ 2 / |
|| \ 2 / | ||-1 + ------------ |
||------------------------- otherwise | || 2 |
8 - 5*|
$$\left(- 5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \sec{\left(x - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(x \right)}}\right) \sec{\left(x \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(x \right)}}}{1 + \frac{\sec^{2}{\left(x - \frac{\pi}{2} \right)}}{\sec^{2}{\left(x \right)}}} & \text{otherwise} \end{cases}\right)\right) + 8$$
// 1 for x mod pi = 0\
|| |
// 0 for 2*x mod pi = 0\ || 2 |
|| | || csc (x) |
|| 2*csc(x) | ||-1 + ------------ |
||------------------------------ otherwise | || 2/pi \ |
||/ 2 \ | || csc |-- - x| |
8 - 5*|<| csc (x) | /pi \ | - 5*|< \2 / |
|||1 + ------------|*csc|-- - x| | ||----------------- otherwise |
||| 2/pi \| \2 / | || 2 |
||| csc |-- - x|| | || csc (x) |
||\ \2 // | || 1 + ------------ |
\\ / || 2/pi \ |
|| csc |-- - x| |
\\ \2 / /
$$\left(- 5 \left(\begin{cases} 0 & \text{for}\: 2 x \bmod \pi = 0 \\\frac{2 \csc{\left(x \right)}}{\left(\frac{\csc^{2}{\left(x \right)}}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: x \bmod \pi = 0 \\\frac{\frac{\csc^{2}{\left(x \right)}}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(x \right)}}{\csc^{2}{\left(- x + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)\right) + 8$$
8 - 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)) - 5*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))