Господин Экзамен
9+5*sin(a)^2-5*cos(a)^2 если a=-1/4
Решение
Вы ввели
[src]
2 2 9 + 5*sin (a) - 5*cos (a)
$$5 \sin^{2}{\left(a \right)} - 5 \cos^{2}{\left(a \right)} + 9$$
9 + 5*sin(a)^2 - 5*cos(a)^2
Подстановка условия
[src]
9 + 5*sin(a)^2 - 5*cos(a)^2 при a = -1/4
подставляем
2 2 9 + 5*sin (a) - 5*cos (a)
$$5 \sin^{2}{\left(a \right)} - 5 \cos^{2}{\left(a \right)} + 9$$
2 4 + 10*sin (a)
$$10 \sin^{2}{\left(a \right)} + 4$$
переменные
a = -1/4
$$a = - \frac{1}{4}$$
2 4 + 10*sin ((-1/4))
$$10 \sin^{2}{\left((-1/4) \right)} + 4$$
2 4 + 10*sin (-1/4)
$$10 \sin^{2}{\left(- \frac{1}{4} \right)} + 4$$
2 4 + 10*sin (1/4)
$$10 \sin^{2}{\left(\frac{1}{4} \right)} + 4$$
4 + 10*sin(1/4)^2
Степени
[src]
2 2
/ I*a -I*a\ / -I*a I*a\
|e e | 5*\- e + e /
9 - 5*|---- + -----| - -------------------
\ 2 2 / 4
$$- 5 \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right)^{2} - \frac{5 \left(e^{i a} - e^{- i a}\right)^{2}}{4} + 9$$
9 - 5*(exp(i*a)/2 + exp(-i*a)/2)^2 - 5*(-exp(-i*a) + exp(i*a))^2/4
Тригонометрическая часть
[src]
9 - 5*cos(2*a)
$$- 5 \cos{\left(2 a \right)} + 9$$
2 4 + 10*sin (a)
$$10 \sin^{2}{\left(a \right)} + 4$$
10
4 + -------
2
csc (a)
$$4 + \frac{10}{\csc^{2}{\left(a \right)}}$$
2/ pi\
4 + 10*cos |a - --|
\ 2 /
$$10 \cos^{2}{\left(a - \frac{\pi}{2} \right)} + 4$$
10
4 + ------------
2/ pi\
sec |a - --|
\ 2 /
$$4 + \frac{10}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}}$$
5 5
9 - ------- + -------
2 2
sec (a) csc (a)
$$9 - \frac{5}{\sec^{2}{\left(a \right)}} + \frac{5}{\csc^{2}{\left(a \right)}}$$
2/ pi\ 2
9 - 5*sin |a + --| + 5*sin (a)
\ 2 /
$$5 \sin^{2}{\left(a \right)} - 5 \sin^{2}{\left(a + \frac{\pi}{2} \right)} + 9$$
2 2/ pi\
9 - 5*cos (a) + 5*cos |a - --|
\ 2 /
$$- 5 \cos^{2}{\left(a \right)} + 5 \cos^{2}{\left(a - \frac{\pi}{2} \right)} + 9$$
5 5
9 - ------- + ------------
2 2/ pi\
sec (a) sec |a - --|
\ 2 /
$$9 + \frac{5}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} - \frac{5}{\sec^{2}{\left(a \right)}}$$
5 5
9 - ------------ + -------
2/pi \ 2
csc |-- - a| csc (a)
\2 /
$$9 - \frac{5}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} + \frac{5}{\csc^{2}{\left(a \right)}}$$
5 5
9 - ------- + ------------
2 2/pi \
sec (a) sec |-- - a|
\2 /
$$9 + \frac{5}{\sec^{2}{\left(- a + \frac{\pi}{2} \right)}} - \frac{5}{\sec^{2}{\left(a \right)}}$$
5 5
9 - ------------ + ------------
2/pi \ 2
csc |-- - a| csc (pi - a)
\2 /
$$9 - \frac{5}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} + \frac{5}{\csc^{2}{\left(- a + \pi \right)}}$$
2/a\
40*tan |-|
\2/
4 + --------------
2
/ 2/a\\
|1 + tan |-||
\ \2//
$$4 + \frac{40 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
33 2 5*cos(2*a) -- - 10*cos(a) - 5*(1 - cos(a)) - ---------- 2 2
$$- 5 \left(- \cos{\left(a \right)} + 1\right)^{2} - 10 \cos{\left(a \right)} - \frac{5 \cos{\left(2 a \right)}}{2} + \frac{33}{2}$$
// 0 for a mod pi = 0\
|| |
4 + 10*|< 2 |
||sin (a) otherwise |
\\ /
$$\left(10 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + 4$$
2
/ 2/a pi\\ 2
5*|1 - cot |- + --|| *(1 + sin(a))
5*(1 + cos(2*a)) \ \2 4 //
9 - ---------------- + -----------------------------------
2 4
$$\frac{5 \left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \left(\sin{\left(a \right)} + 1\right)^{2}}{4} - \frac{5 \left(\cos{\left(2 a \right)} + 1\right)}{2} + 9$$
2/a pi\
20*tan |- + --|
5*(1 - cos(2*a)) \2 4 /
9 + ---------------- - -------------------
2 2
/ 2/a pi\\
|1 + tan |- + --||
\ \2 4 //
$$\frac{5 \cdot \left(- \cos{\left(2 a \right)} + 1\right)}{2} + 9 - \frac{20 \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
// 0 for a mod pi = 0\
|| |
|| 2/a\ |
|| 4*cot |-| |
|| \2/ |
4 + 10*|<-------------- otherwise |
|| 2 |
||/ 2/a\\ |
|||1 + cot |-|| |
||\ \2// |
\\ /
$$\left(10 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + 4$$
2
/ 2/a\\ 2/a\
5*|1 - tan |-|| 20*tan |-|
\ \2// \2/
9 - ---------------- + --------------
2 2
/ 2/a\\ / 2/a\\
|1 + tan |-|| |1 + tan |-||
\ \2// \ \2//
$$- \frac{5 \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} + 9 + \frac{20 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
/ 4 8/a\ 2 4/a\\
4*|sin (a) + 16*sin |-| + 48*sin (a)*sin |-||
\ \2/ \2//
---------------------------------------------
2
/ 2 4/a\\
|sin (a) + 4*sin |-||
\ \2//
$$\frac{4 \cdot \left(16 \sin^{8}{\left(\frac{a}{2} \right)} + 48 \sin^{4}{\left(\frac{a}{2} \right)} \sin^{2}{\left(a \right)} + \sin^{4}{\left(a \right)}\right)}{\left(4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}\right)^{2}}$$
2/a pi\ 2/a\
20*tan |- + --| 20*tan |-|
\2 4 / \2/
9 - ------------------- + --------------
2 2
/ 2/a pi\\ / 2/a\\
|1 + tan |- + --|| |1 + tan |-||
\ \2 4 // \ \2//
$$9 - \frac{20 \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{20 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
2/a pi\ 2/a\
20*tan |- + --| 20*cot |-|
\2 4 / \2/
9 - ------------------- + --------------
2 2
/ 2/a pi\\ / 2/a\\
|1 + tan |- + --|| |1 + cot |-||
\ \2 4 // \ \2//
$$9 + \frac{20 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} - \frac{20 \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2
/ 1 \
5*|1 - -------|
| 2/a\|
| cot |-||
\ \2// 20
9 - ---------------- + ----------------------
2 2
/ 1 \ / 1 \ 2/a\
|1 + -------| |1 + -------| *cot |-|
| 2/a\| | 2/a\| \2/
| cot |-|| | cot |-||
\ \2// \ \2//
$$- \frac{5 \left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)^{2}} + 9 + \frac{20}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \cot^{2}{\left(\frac{a}{2} \right)}}$$
// 1 for a mod 2*pi = 0\ // 0 for a mod pi = 0\
|| | || |
9 - 5*|< 2 | + 5*|< 2 |
||cos (a) otherwise | ||sin (a) otherwise |
\\ / \\ /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + 9$$
2 2
/ 2/a\\ / 2/a pi\\
5*|-1 + cot |-|| 5*|-1 + tan |- + --||
\ \2// \ \2 4 //
9 - ----------------- + ----------------------
2 2
/ 2/a\\ / 2/a pi\\
|1 + cot |-|| |1 + tan |- + --||
\ \2// \ \2 4 //
$$\frac{5 \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} - \frac{5 \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} + 9$$
// 1 for a mod 2*pi = 0\ // 0 for a mod pi = 0\
|| | || |
9 - 5*|< 2/ pi\ | + 5*|< 2 |
||sin |a + --| otherwise | ||sin (a) otherwise |
\\ \ 2 / / \\ /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\sin^{2}{\left(a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + 9$$
2 2
/ 2/a\\ / 2/a pi\\
5*|1 - tan |-|| 5*|1 - cot |- + --||
\ \2// \ \2 4 //
9 - ---------------- + ---------------------
2 2
/ 2/a\\ / 2/a pi\\
|1 + tan |-|| |1 + cot |- + --||
\ \2// \ \2 4 //
$$- \frac{5 \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} + \frac{5 \left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + 9$$
// 1 for a mod 2*pi = 0\ // 0 for a mod pi = 0\
|| | || |
9 - 5*|< 2 | + 5*|< 2/ pi\ |
||cos (a) otherwise | ||cos |a - --| otherwise |
\\ / \\ \ 2 / /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\cos^{2}{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + 9$$
// 1 for a mod 2*pi = 0\ // 0 for a mod pi = 0\
|| | || |
|| 1 | || 1 |
9 - 5*|<------- otherwise | + 5*|<------------ otherwise |
|| 2 | || 2/ pi\ |
||sec (a) | ||sec |a - --| |
\\ / \\ \ 2 / /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec^{2}{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\sec^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) + 9$$
// 1 for a mod 2*pi = 0\ // 0 for a mod pi = 0\
|| | || |
|| 1 | || 1 |
9 - 5*|<------------ otherwise | + 5*|<------- otherwise |
|| 2/pi \ | || 2 |
||csc |-- - a| | ||csc (a) |
\\ \2 / / \\ /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) + 9$$
// / 3*pi\ \
// 1 for a mod 2*pi = 0\ || 1 for |a + ----| mod 2*pi = 0|
|| | || \ 2 / |
9 - 5*|< 2 | + 5*|< |
||cos (a) otherwise | || 4/a\ 2/a\ |
\\ / ||- 4*cos |-| + 4*cos |-| otherwise |
\\ \2/ \2/ /
$$\left(- 5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\- 4 \cos^{4}{\left(\frac{a}{2} \right)} + 4 \cos^{2}{\left(\frac{a}{2} \right)} & \text{otherwise} \end{cases}\right)\right) + 9$$
2
/ 4/a\\
| 4*sin |-||
| \2/|
5*|1 - ---------| 4/a\
| 2 | 80*sin |-|
\ sin (a) / \2/
9 - ------------------ + ------------------------
2 2
/ 4/a\\ / 4/a\\
| 4*sin |-|| | 4*sin |-||
| \2/| | \2/| 2
|1 + ---------| |1 + ---------| *sin (a)
| 2 | | 2 |
\ sin (a) / \ sin (a) /
$$- \frac{5 \left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{2}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{2}} + 9 + \frac{80 \sin^{4}{\left(\frac{a}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{2} \sin^{2}{\left(a \right)}}$$
// / pi\ \
|| 0 for |a + --| mod pi = 0| // 0 for a mod pi = 0\
|| \ 2 / | || |
9 - 5*|< | + 5*|< 2 |
|| 2 2/a pi\ | ||sin (a) otherwise |
||(1 + sin(a)) *cot |- + --| otherwise | \\ /
\\ \2 4 / /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(a \right)} + 1\right)^{2} \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)\right) + 9$$
// 1 for a mod 2*pi = 0\ // 0 for a mod pi = 0\
|| | || |
||/ 1 for a mod 2*pi = 0 | ||/ 0 for a mod pi = 0 |
9 - 5*|<| | + 5*|<| |
||< 2 otherwise | ||< 2 otherwise |
|||cos (a) otherwise | |||sin (a) otherwise |
\\\ / \\\ /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin^{2}{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos^{2}{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + 9$$
// 1 for a mod 2*pi = 0\ // 0 for a mod pi = 0\
|| | || |
|| 2 | || 2/a\ |
||/ 2/a\\ | || 4*cot |-| |
|||-1 + cot |-|| | || \2/ |
9 - 5*|<\ \2// | + 5*|<-------------- otherwise |
||--------------- otherwise | || 2 |
|| 2 | ||/ 2/a\\ |
|| / 2/a\\ | |||1 + cot |-|| |
|| |1 + cot |-|| | ||\ \2// |
\\ \ \2// / \\ /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + 9$$
// 1 for a mod 2*pi = 0\ // 0 for a mod pi = 0\
|| | || |
|| 2 | || 2/a\ |
||/ 2/a\\ | || 4*tan |-| |
|||1 - tan |-|| | || \2/ |
9 - 5*|<\ \2// | + 5*|<-------------- otherwise |
||-------------- otherwise | || 2 |
|| 2 | ||/ 2/a\\ |
||/ 2/a\\ | |||1 + tan |-|| |
|||1 + tan |-|| | ||\ \2// |
\\\ \2// / \\ /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \tan^{2}{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + 9$$
2
/ 2/a pi\\
| cos |- - --||
| \2 2 /|
5*|1 - ------------|
| 2/a\ | 2/a pi\
| cos |-| | 20*cos |- - --|
\ \2/ / \2 2 /
9 - --------------------- + ---------------------------
2 2
/ 2/a pi\\ / 2/a pi\\
| cos |- - --|| | cos |- - --||
| \2 2 /| | \2 2 /| 2/a\
|1 + ------------| |1 + ------------| *cos |-|
| 2/a\ | | 2/a\ | \2/
| cos |-| | | cos |-| |
\ \2/ / \ \2/ /
$$- \frac{5 \left(1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)^{2}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)^{2}} + 9 + \frac{20 \cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \cos^{2}{\left(\frac{a}{2} \right)}}$$
2
/ 2/a\ \
| sec |-| |
| \2/ |
5*|1 - ------------|
| 2/a pi\| 2/a\
| sec |- - --|| 20*sec |-|
\ \2 2 // \2/
9 - --------------------- + --------------------------------
2 2
/ 2/a\ \ / 2/a\ \
| sec |-| | | sec |-| |
| \2/ | | \2/ | 2/a pi\
|1 + ------------| |1 + ------------| *sec |- - --|
| 2/a pi\| | 2/a pi\| \2 2 /
| sec |- - --|| | sec |- - --||
\ \2 2 // \ \2 2 //
$$- \frac{5 \left(- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} + 9 + \frac{20 \sec^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}$$
2
/ 2/pi a\\
| csc |-- - -||
| \2 2/|
5*|1 - ------------|
| 2/a\ | 2/pi a\
| csc |-| | 20*csc |-- - -|
\ \2/ / \2 2/
9 - --------------------- + ---------------------------
2 2
/ 2/pi a\\ / 2/pi a\\
| csc |-- - -|| | csc |-- - -||
| \2 2/| | \2 2/| 2/a\
|1 + ------------| |1 + ------------| *csc |-|
| 2/a\ | | 2/a\ | \2/
| csc |-| | | csc |-| |
\ \2/ / \ \2/ /
$$- \frac{5 \left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)^{2}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)^{2}} + 9 + \frac{20 \csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \csc^{2}{\left(\frac{a}{2} \right)}}$$
// 1 for a mod 2*pi = 0\
|| |
|| 2 | // 0 for a mod pi = 0\
||/ 1 \ | || |
|||-1 + -------| | || 4 |
||| 2/a\| | ||---------------------- otherwise |
||| tan |-|| | || 2 |
9 - 5*|<\ \2// | + 5*|
$$\left(5 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \tan^{2}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{2}}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + 9$$
// / pi\ \
|| 0 for |a + --| mod pi = 0| // 0 for a mod pi = 0\
|| \ 2 / | || |
|| | || 2/a\ |
|| 2/a pi\ | || 4*cot |-| |
|| 4*cot |- + --| | || \2/ |
9 - 5*|< \2 4 / | + 5*|<-------------- otherwise |
||------------------- otherwise | || 2 |
|| 2 | ||/ 2/a\\ |
||/ 2/a pi\\ | |||1 + cot |-|| |
|||1 + cot |- + --|| | ||\ \2// |
||\ \2 4 // | \\ /
\\ /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + 9$$
// / 3*pi\ \
// 1 for a mod 2*pi = 0\ || 1 for |a + ----| mod 2*pi = 0|
|| | || \ 2 / |
|| 2 | || |
||/ 2/a\\ | || 2 |
|||-1 + cot |-|| | ||/ 2/a pi\\ |
9 - 5*|<\ \2// | + 5*|<|-1 + tan |- + --|| |
||--------------- otherwise | ||\ \2 4 // |
|| 2 | ||-------------------- otherwise |
|| / 2/a\\ | || 2 |
|| |1 + cot |-|| | ||/ 2/a pi\\ |
\\ \ \2// / |||1 + tan |- + --|| |
\\\ \2 4 // /
$$\left(- 5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + \left(5 \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + 9$$
// 0 for a mod pi = 0\
// 1 for a mod 2*pi = 0\ || |
|| | || 2 |
|| 2 | || sin (a) |
||/ 2 4/a\\ | ||------------------------ otherwise |
|||sin (a) - 4*sin |-|| | || 2 |
9 - 5*|<\ \2// | + 5*|
$$\left(5 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin^{2}{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(- 4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}\right)^{2}}{\left(4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + 9$$
// 1 for a mod 2*pi = 0\
|| |
|| 2 | // 0 for a mod pi = 0\
||/ 2 \ | || |
||| sin (a) | | || 2 |
|||-1 + ---------| | || sin (a) |
||| 4/a\| | ||------------------------ otherwise |
||| 4*sin |-|| | || 2 |
9 - 5*|<\ \2// | + 5*|
$$\left(5 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin^{2}{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{2} \sin^{4}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + 9$$
// 1 for a mod 2*pi = 0\ // 0 for a mod pi = 0\
|| | || |
||/ 1 for a mod 2*pi = 0 | ||/ 0 for a mod pi = 0 |
||| | ||| |
||| 2 | ||| 2/a\ |
|||/ 2/a\\ | ||| 4*cot |-| |
9 - 5*|<||-1 + cot |-|| | + 5*|<| \2/ |
||<\ \2// otherwise | ||<-------------- otherwise otherwise |
|||--------------- otherwise | ||| 2 |
||| 2 | |||/ 2/a\\ |
||| / 2/a\\ | ||||1 + cot |-|| |
||| |1 + cot |-|| | |||\ \2// |
\\\ \ \2// / \\\ /
$$\left(5 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \cot^{2}{\left(\frac{a}{2} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)\right) + 9$$
// 1 for a mod 2*pi = 0\
|| |
|| 2 | // 0 for a mod pi = 0\
||/ 2/a\ \ | || |
||| cos |-| | | || 2/a\ |
||| \2/ | | || 4*cos |-| |
|||-1 + ------------| | || \2/ |
||| 2/a pi\| | ||-------------------------------- otherwise |
||| cos |- - --|| | || 2 |
9 - 5*|<\ \2 2 // | + 5*|
$$\left(5 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \cos^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + 9$$
// 1 for a mod 2*pi = 0\
|| |
|| 2 | // 0 for a mod pi = 0\
||/ 2/a pi\\ | || |
||| sec |- - --|| | || 2/a pi\ |
||| \2 2 /| | || 4*sec |- - --| |
|||-1 + ------------| | || \2 2 / |
||| 2/a\ | | ||--------------------------- otherwise |
||| sec |-| | | || 2 |
9 - 5*|<\ \2/ / | + 5*|
$$\left(5 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \sec^{2}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right)^{2}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + 9$$
// 1 for a mod 2*pi = 0\
|| |
|| 2 | // 0 for a mod pi = 0\
||/ 2/a\ \ | || |
||| csc |-| | | || 2/a\ |
||| \2/ | | || 4*csc |-| |
|||-1 + ------------| | || \2/ |
||| 2/pi a\| | ||-------------------------------- otherwise |
||| csc |-- - -|| | || 2 |
9 - 5*|<\ \2 2// | + 5*|
$$\left(5 \left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{4 \csc^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2} \csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)\right) - \left(5 \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1\right)^{2}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right)^{2}} & \text{otherwise} \end{cases}\right)\right) + 9$$
9 - 5*Piecewise((1, Mod(a = 2*pi, 0)), ((-1 + csc(a/2)^2/csc(pi/2 - a/2)^2)^2/(1 + csc(a/2)^2/csc(pi/2 - a/2)^2)^2, True)) + 5*Piecewise((0, Mod(a = pi, 0)), (4*csc(a/2)^2/((1 + csc(a/2)^2/csc(pi/2 - a/2)^2)^2*csc(pi/2 - a/2)^2), True))