Господин Экзамен
Другие калькуляторы
-
Как пользоваться?
- y*(2*a+3*b)-y*(2*a-3*b) если y=-2
- 8*x^3+88*x^2+128*x-672 если x=-3
- sin(3*pi+a) если a=-4
- sin(a)*cos(a) если a=1/3
- Разложить многочлен на множители:
- 1-s^2+2*s*c-c^2
- x^2*x/12+24*x+144/x+12
- 49^n-2*21^n+9^n-1
- x^4-9
- Общий знаменатель:
- 17-12*x/x-10/x
- cos(a+pi/2)+sin(a+5*pi/2)
- sin(pi)/16*cos(pi)^(3)/16-sin(pi)^(3)/16*cos(pi)/16
- n^2/n-4-8*n-16/n-4
- Умножение столбиком:
- 23*78
- 42*14
- 44*15
- 66*66
- Деление столбиком:
- 1148/4
- 708/9
- 363/6
- 646/6
- Раскрыть скобки в:
- x*(x^2-9)
- (x-4)*(x+4)-(x+8)*(x-8)
- (8*a+1)*(-6)+(3*a+7)*(-2)
- 2*a*(3*a-b)-(a-2*b)^2
- Разложение числа на простые множители:
- 1050
- 92916
- 2139
- 90
- Производная:
- 1/3
- График функции y =:
- 1/3
- Интеграл d{x}:
-
1/3
-
Идентичные выражения
- sin(a)*cos(a) если a= один / три
- синус от (a) умножить на косинус от (a) если a равно 1 делить на 3
- синус от (a) умножить на косинус от (a) если a равно один делить на три
- sin(a)cos(a) если a=1/3
- sinacosa если a=1/3
- sin(a)*cos(a) при a=1/3
- sin(a)*cos(a) если a=1 разделить на 3
-
Похожие выражения
- -cos(a)^2+sin(a)^2-2*sin(a)*cos(a) если a=-2
sin(a)*cos(a) если a=1/3
Решение
Подстановка условия
[src]
sin(a)*cos(a) при a = 1/3
подставляем
sin(a)*cos(a)
$$\sin{\left(a \right)} \cos{\left(a \right)}$$
sin(2*a) -------- 2
$$\frac{\sin{\left(2 a \right)}}{2}$$
переменные
a = 1/3
$$a = \frac{1}{3}$$
sin(2*(1/3))
------------
2
$$\frac{\sin{\left(2 (1/3) \right)}}{2}$$
sin(2/3) -------- 2
$$\frac{\sin{\left(\frac{2}{3} \right)}}{2}$$
sin(2/3)/2
Степени
[src]
/ I*a -I*a\
|e e | / -I*a I*a\
-I*|---- + -----|*\- e + e /
\ 2 2 /
-----------------------------------
2
$$- \frac{i \left(\frac{e^{i a}}{2} + \frac{e^{- i a}}{2}\right) \left(e^{i a} - e^{- i a}\right)}{2}$$
-i*(exp(i*a)/2 + exp(-i*a)/2)*(-exp(-i*a) + exp(i*a))/2
Тригонометрическая часть
[src]
sin(2*a) -------- 2
$$\frac{\sin{\left(2 a \right)}}{2}$$
1 ---------- 2*csc(2*a)
$$\frac{1}{2 \csc{\left(2 a \right)}}$$
/ pi\
cos|2*a - --|
\ 2 /
-------------
2
$$\frac{\cos{\left(2 a - \frac{\pi}{2} \right)}}{2}$$
1
---------------
/ pi\
2*sec|2*a - --|
\ 2 /
$$\frac{1}{2 \sec{\left(2 a - \frac{\pi}{2} \right)}}$$
1 ------------- csc(a)*sec(a)
$$\frac{1}{\csc{\left(a \right)} \sec{\left(a \right)}}$$
tan(a)
-----------
2
1 + tan (a)
$$\frac{\tan{\left(a \right)}}{\tan^{2}{\left(a \right)} + 1}$$
/ pi\
sin(a)*sin|a + --|
\ 2 /
$$\sin{\left(a \right)} \sin{\left(a + \frac{\pi}{2} \right)}$$
/ pi\
cos(a)*cos|a - --|
\ 2 /
$$\cos{\left(a \right)} \cos{\left(a - \frac{\pi}{2} \right)}$$
1
------------------
/ pi\
sec(a)*sec|a - --|
\ 2 /
$$\frac{1}{\sec{\left(a \right)} \sec{\left(a - \frac{\pi}{2} \right)}}$$
1
------------------
/pi \
sec(a)*sec|-- - a|
\2 /
$$\frac{1}{\sec{\left(a \right)} \sec{\left(- a + \frac{\pi}{2} \right)}}$$
1
------------------
/pi \
csc(a)*csc|-- - a|
\2 /
$$\frac{1}{\csc{\left(a \right)} \csc{\left(- a + \frac{\pi}{2} \right)}}$$
/ 2/a\\ |-1 + 2*cos |-||*sin(a) \ \2//
$$\left(2 \cos^{2}{\left(\frac{a}{2} \right)} - 1\right) \sin{\left(a \right)}$$
1
-----------------------
/pi \
csc(pi - a)*csc|-- - a|
\2 /
$$\frac{1}{\csc{\left(- a + \pi \right)} \csc{\left(- a + \frac{\pi}{2} \right)}}$$
/a\
(1 + cos(a))*cos(a)*tan|-|
\2/
$$\left(\cos{\left(a \right)} + 1\right) \cos{\left(a \right)} \tan{\left(\frac{a}{2} \right)}$$
/ 0 for 2*a mod pi = 0
<
\sin(2*a) otherwise
-----------------------------
2
$$\frac{\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\sin{\left(2 a \right)} & \text{otherwise} \end{cases}}{2}$$
/ 2/a\\ /a\ /a\ 2*|-1 + 2*cos |-||*cos|-|*sin|-| \ \2// \2/ \2/
$$2 \cdot \left(2 \cos^{2}{\left(\frac{a}{2} \right)} - 1\right) \sin{\left(\frac{a}{2} \right)} \cos{\left(\frac{a}{2} \right)}$$
/ 0 for 2*a mod pi = 0
|
| 2*cot(a)
<----------- otherwise
| 2
|1 + cot (a)
\
--------------------------------
2
$$\frac{\begin{cases} 0 & \text{for}\: 2 a \bmod \pi = 0 \\\frac{2 \cot{\left(a \right)}}{\cot^{2}{\left(a \right)} + 1} & \text{otherwise} \end{cases}}{2}$$
/ 2/a\\ /a\
2*|1 - tan |-||*tan|-|
\ \2// \2/
----------------------
2
/ 2/a\\
|1 + tan |-||
\ \2//
$$\frac{2 \cdot \left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \tan{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right)^{2}}$$
/ 1 \
2*|1 - -------|
| 2/a\|
| cot |-||
\ \2//
---------------------
2
/ 1 \ /a\
|1 + -------| *cot|-|
| 2/a\| \2/
| cot |-||
\ \2//
$$\frac{2 \cdot \left(1 - \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \cot{\left(\frac{a}{2} \right)}}$$
2/a\ / 2/a pi\\ / 2/a\\
cos |-|*|1 - cot |- + --||*|1 - tan |-||*(1 + sin(a))
\2/ \ \2 4 // \ \2//
-----------------------------------------------------
2
$$\frac{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(a \right)} + 1\right) \cos^{2}{\left(\frac{a}{2} \right)}}{2}$$
/a\ /a pi\
4*tan|-|*tan|- + --|
\2/ \2 4 /
--------------------------------
/ 2/a\\ / 2/a pi\\
|1 + tan |-||*|1 + tan |- + --||
\ \2// \ \2 4 //
$$\frac{4 \tan{\left(\frac{a}{2} \right)} \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)}$$
/a\ /a pi\
4*cot|-|*tan|- + --|
\2/ \2 4 /
--------------------------------
/ 2/a\\ / 2/a pi\\
|1 + cot |-||*|1 + tan |- + --||
\ \2// \ \2 4 //
$$\frac{4 \tan{\left(\frac{a}{2} + \frac{\pi}{4} \right)} \cot{\left(\frac{a}{2} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
2/a\ / 2 4/a\\
4*sin |-|*|sin (a) - 4*sin |-||*sin(a)
\2/ \ \2//
--------------------------------------
2
/ 2 4/a\\
|sin (a) + 4*sin |-||
\ \2//
$$\frac{4 \left(- 4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}\right) \sin^{2}{\left(\frac{a}{2} \right)} \sin{\left(a \right)}}{\left(4 \sin^{4}{\left(\frac{a}{2} \right)} + \sin^{2}{\left(a \right)}\right)^{2}}$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\ |< |*|< | \\sin(a) otherwise / \\cos(a) otherwise /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\
// 0 for a mod pi = 0\ || |
|< |*|< / pi\ |
\\sin(a) otherwise / ||sin|a + --| otherwise |
\\ \ 2 / /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\sin{\left(a + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ || | // 1 for a mod 2*pi = 0\ |< / pi\ |*|< | ||cos|a - --| otherwise | \\cos(a) otherwise / \\ \ 2 / /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\cos{\left(a - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)$$
/ 4/a\\
| 4*sin |-||
2/a\ | \2/|
4*sin |-|*|1 - ---------|
\2/ | 2 |
\ sin (a) /
-------------------------
2
/ 4/a\\
| 4*sin |-||
| \2/|
|1 + ---------| *sin(a)
| 2 |
\ sin (a) /
$$\frac{4 \left(- \frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right) \sin^{2}{\left(\frac{a}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} \right)}}{\sin^{2}{\left(a \right)}} + 1\right)^{2} \sin{\left(a \right)}}$$
/ 2/a\\ / 2/a pi\\ |-1 + cot |-||*|-1 + tan |- + --|| \ \2// \ \2 4 // ---------------------------------- / 2/a\\ / 2/a pi\\ |1 + cot |-||*|1 + tan |- + --|| \ \2// \ \2 4 //
$$\frac{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} - 1\right)}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} \right)} + 1\right)}$$
/ 2/a pi\\ / 2/a\\ |1 - cot |- + --||*|1 - tan |-|| \ \2 4 // \ \2// -------------------------------- / 2/a pi\\ / 2/a\\ |1 + cot |- + --||*|1 + tan |-|| \ \2 4 // \ \2//
$$\frac{\left(- \tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(- \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)}{\left(\tan^{2}{\left(\frac{a}{2} \right)} + 1\right) \left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)}$$
// / 3*pi\ \
// 1 for a mod 2*pi = 0\ || 1 for |a + ----| mod 2*pi = 0|
|< |*|< \ 2 / |
\\cos(a) otherwise / || |
\\sin(a) otherwise /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ || | // 1 for a mod 2*pi = 0\ || 1 | || | |<----------- otherwise |*|< 1 | || / pi\ | ||------ otherwise | ||sec|a - --| | \\sec(a) / \\ \ 2 / /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(a \right)}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\
// 0 for a mod pi = 0\ || |
|| | || 1 |
|< 1 |*|<----------- otherwise |
||------ otherwise | || /pi \ |
\\csc(a) / ||csc|-- - a| |
\\ \2 / /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{1}{\csc{\left(a \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- a + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ || | ||1 - cos(a) | // 1 for a mod 2*pi = 0\ |<---------- otherwise |*|< | || /a\ | \\cos(a) otherwise / || tan|-| | \\ \2/ /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{- \cos{\left(a \right)} + 1}{\tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases}\right)$$
/ 2/a\ \
| sec |-| |
| \2/ | /a\
2*|1 - ------------|*sec|-|
| 2/a pi\| \2/
| sec |- - --||
\ \2 2 //
-------------------------------
2
/ 2/a\ \
| sec |-| |
| \2/ | /a pi\
|1 + ------------| *sec|- - --|
| 2/a pi\| \2 2 /
| sec |- - --||
\ \2 2 //
$$\frac{2 \left(- \frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(\frac{a}{2} \right)}}{\left(\frac{\sec^{2}{\left(\frac{a}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right)^{2} \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}$$
// / pi\ \
|| 0 for |a + --| mod pi = 0|
// 0 for a mod pi = 0\ || \ 2 / |
|< |*|< |
\\sin(a) otherwise / || /a pi\ |
||(1 + sin(a))*cot|- + --| otherwise |
\\ \2 4 / /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(a \right)} + 1\right) \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right)$$
/ 2/a pi\\
| cos |- - --||
| \2 2 /| /a pi\
2*|1 - ------------|*cos|- - --|
| 2/a\ | \2 2 /
| cos |-| |
\ \2/ /
--------------------------------
2
/ 2/a pi\\
| cos |- - --||
| \2 2 /| /a\
|1 + ------------| *cos|-|
| 2/a\ | \2/
| cos |-| |
\ \2/ /
$$\frac{2 \cdot \left(1 - \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right) \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \cos{\left(\frac{a}{2} \right)}}$$
/ 2/pi a\\
| csc |-- - -||
| \2 2/| /pi a\
2*|1 - ------------|*csc|-- - -|
| 2/a\ | \2 2/
| csc |-| |
\ \2/ /
--------------------------------
2
/ 2/pi a\\
| csc |-- - -||
| \2 2/| /a\
|1 + ------------| *csc|-|
| 2/a\ | \2/
| csc |-| |
\ \2/ /
$$\frac{2 \cdot \left(1 - \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right) \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{a}{2} \right)}}\right)^{2} \csc{\left(\frac{a}{2} \right)}}$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\ || | || | || /a\ | || 2/a\ | || 2*cot|-| | ||-1 + cot |-| | |< \2/ |*|< \2/ | ||----------- otherwise | ||------------ otherwise | || 2/a\ | || 2/a\ | ||1 + cot |-| | ||1 + cot |-| | \\ \2/ / \\ \2/ /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\ || | || | || /a\ | || 2/a\ | || 2*tan|-| | ||1 - tan |-| | |< \2/ |*|< \2/ | ||----------- otherwise | ||----------- otherwise | || 2/a\ | || 2/a\ | ||1 + tan |-| | ||1 + tan |-| | \\ \2/ / \\ \2/ /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{a}{2} \right)} + 1}{\tan^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\
|| | || |
|
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\sin{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\cos{\left(a \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 1 |
|| | ||-1 + ------- |
|| 2 | || 2/a\ |
||-------------------- otherwise | || tan |-| |
|
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}\right) \tan{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right)$$
// / pi\ \
// 0 for a mod pi = 0\ || 0 for |a + --| mod pi = 0|
|| | || \ 2 / |
|| /a\ | || |
|| 2*cot|-| | || /a pi\ |
|< \2/ |*|< 2*cot|- + --| |
||----------- otherwise | || \2 4 / |
|| 2/a\ | ||---------------- otherwise |
||1 + cot |-| | || 2/a pi\ |
\\ \2/ / ||1 + cot |- + --| |
\\ \2 4 / /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 0 & \text{for}\: \left(a + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// / 3*pi\ \
// 1 for a mod 2*pi = 0\ || 1 for |a + ----| mod 2*pi = 0|
|| | || \ 2 / |
|| 2/a\ | || |
||-1 + cot |-| | || 2/a pi\ |
|< \2/ |*|<-1 + tan |- + --| |
||------------ otherwise | || \2 4 / |
|| 2/a\ | ||----------------- otherwise |
||1 + cot |-| | || 2/a pi\ |
\\ \2/ / || 1 + tan |- + --| |
\\ \2 4 / /
$$\left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: \left(a + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ || | || 2*sin(a) | // 1 for a mod 2*pi = 0\ ||---------------------------- otherwise | || | || / 2 \ | || 2 | |< | sin (a) | |*|< -4 + 4*sin (a) + 4*cos(a) | ||(1 - cos(a))*|1 + ---------| | ||--------------------------- otherwise | || | 4/a\| | || 2 2 | || | 4*sin |-|| | \\2*(1 - cos(a)) + 2*sin (a) / || \ \2// | \\ /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \left(- \cos{\left(a \right)} + 1\right)} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{4 \sin^{2}{\left(a \right)} + 4 \cos{\left(a \right)} - 4}{2 \left(- \cos{\left(a \right)} + 1\right)^{2} + 2 \sin^{2}{\left(a \right)}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 2 |
|| | || sin (a) |
|| sin(a) | ||-1 + --------- |
||----------------------- otherwise | || 4/a\ |
||/ 2 \ | || 4*sin |-| |
|<| sin (a) | 2/a\ |*|< \2/ |
|||1 + ---------|*sin |-| | ||-------------- otherwise |
||| 4/a\| \2/ | || 2 |
||| 4*sin |-|| | || sin (a) |
||\ \2// | ||1 + --------- |
\\ / || 4/a\ |
|| 4*sin |-| |
\\ \2/ /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{\sin{\left(a \right)}}{\left(1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}\right) \sin^{2}{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sin^{2}{\left(a \right)}}{4 \sin^{4}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right)$$
// 0 for a mod pi = 0\ // 1 for a mod 2*pi = 0\ || | || | ||/ 0 for a mod pi = 0 | ||/ 1 for a mod 2*pi = 0 | ||| | ||| | ||| /a\ | ||| 2/a\ | |<| 2*cot|-| |*|<|-1 + cot |-| | ||< \2/ otherwise | ||< \2/ otherwise | |||----------- otherwise | |||------------ otherwise | ||| 2/a\ | ||| 2/a\ | |||1 + cot |-| | |||1 + cot |-| | \\\ \2/ / \\\ \2/ /
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{a}{2} \right)} - 1}{\cot^{2}{\left(\frac{a}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 2/a\ |
|| | || cos |-| |
|| /a\ | || \2/ |
|| 2*cos|-| | ||-1 + ------------ |
|| \2/ | || 2/a pi\ |
||------------------------------ otherwise | || cos |- - --| |
|
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{a}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{a}{2} \right)}}{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 2/a pi\ |
|| | || sec |- - --| |
|| /a pi\ | || \2 2 / |
|| 2*sec|- - --| | ||-1 + ------------ |
|| \2 2 / | || 2/a\ |
||------------------------- otherwise | || sec |-| |
|
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}\right) \sec{\left(\frac{a}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{a}{2} \right)}}} & \text{otherwise} \end{cases}\right)$$
// 1 for a mod 2*pi = 0\
|| |
// 0 for a mod pi = 0\ || 2/a\ |
|| | || csc |-| |
|| /a\ | || \2/ |
|| 2*csc|-| | ||-1 + ------------ |
|| \2/ | || 2/pi a\ |
||------------------------------ otherwise | || csc |-- - -| |
|
$$\left(\begin{cases} 0 & \text{for}\: a \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{a}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \left(\begin{cases} 1 & \text{for}\: a \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{a}{2} \right)}}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)$$
Piecewise((0, Mod(a = pi, 0)), (2*csc(a/2)/((1 + csc(a/2)^2/csc(pi/2 - a/2)^2)*csc(pi/2 - a/2)), True))*Piecewise((1, Mod(a = 2*pi, 0)), ((-1 + csc(a/2)^2/csc(pi/2 - a/2)^2)/(1 + csc(a/2)^2/csc(pi/2 - a/2)^2), True))