Господин Экзамен
Другие калькуляторы
-
Как пользоваться?
- c^2/(c-9)-81/(c-9) если c=-1
- factorial(x-4)/factorial(x-2) если x=1
- cos(30)*cos(x)-sin(30)*sin(y) если y=-1/2
- 15*x^4-44*x^3+39*x^2+8*x-12 если x=-3
- Разложить многочлен на множители:
- х^n-2*x^3-n*x
- X^3-1
- 3*a^3*b-2*a^2*b^2+7*a^2*b-8*b-11
- 64*x^6-128*x^4+4*x^3+64*x^2-3*x-5
- Общий знаменатель:
- ((sqrt(x)+1)/(1+sqrt(x)+x))/(1/(x^2-sqrt(x)))
- (3*sin(7*x))/(1-6*x)^(3/2)+(7*cos(7*x))/sqrt(1-6*x)
- (cos(pi/2+t))/(sin(pi-t)*tan(-t))
- x*y/x-y*(1/x+1/y)-x+y/x-y
- Умножение столбиком:
- 38*1496
- 6759*61
- 495*574
- 2560*43
- Деление столбиком:
- 39448/8
- 39958/8
- 39960/8
- 41574/8
- Раскрыть скобки в:
- (x^18-10*x^17+43*x^16-102*x^15+141*x^14-108*x^13+36*x^12+2*x^9-10*x^8+18*x^7-12*x^6+1)*(1+x^19-16*x^18+118*x^17-528*x^16+1587*x^15-3348*x^14+5022*x^13-5292*x^12+3753*x^11-1619*x^10+317*x^9+19*x^8-21*x^7-3*x^6+27*x^5-18*x^4)
- 3*b*(2*b-1)-(1-2*b)^2
- y*(-3)*(-5)*x
- x*(x+4)*(x+8)+(x+1)*(x+5)*(x+6)+(x+2)*(x+3)*(x+7)
- Разложение числа на простые множители:
- 58725
- 68818
- 20114
- 56219
- График функции y =:
- -1/2
- Интеграл d{x}:
-
-1/2
- Производная:
- -1/2
-
Идентичные выражения
- cos(тридцать)*cos(x)-sin(тридцать)*sin(y) если y=- один / два
- косинус от (30) умножить на косинус от (x) минус синус от (30) умножить на синус от (y) если y равно минус 1 делить на 2
- косинус от (тридцать) умножить на косинус от (x) минус синус от (тридцать) умножить на синус от (y) если y равно минус один делить на два
- cos(30)cos(x)-sin(30)sin(y) если y=-1/2
- cos30cosx-sin30siny если y=-1/2
- cos(30)*cos(x)-sin(30)*sin(y) при y=-1/2
- cos(30)*cos(x)-sin(30)*sin(y) если y=-1 разделить на 2
-
Похожие выражения
- cos(30)*cos(x)+sin(30)*sin(y) если y=-1/2
- cos(30)*cos(x)-sin(30)*sin(y) если y=+1/2
- cos(30)*cosx-sin(30)*sin(y) если y=-1/2
cos(30)*cos(x)-sin(30)*sin(y) если y=-1/2
Решение
Вы ввели
[src]
cos(30)*cos(x) - sin(30)*sin(y)
$$- \sin{\left(30 \right)} \sin{\left(y \right)} + \cos{\left(30 \right)} \cos{\left(x \right)}$$
cos(30)*cos(x) - sin(30)*sin(y)
Подстановка условия
[src]
cos(30)*cos(x) - sin(30)*sin(y) при y = -1/2
подставляем
cos(30)*cos(x) - sin(30)*sin(y)
$$- \sin{\left(30 \right)} \sin{\left(y \right)} + \cos{\left(30 \right)} \cos{\left(x \right)}$$
cos(30)*cos(x) - sin(30)*sin(y)
$$- \sin{\left(30 \right)} \sin{\left(y \right)} + \cos{\left(30 \right)} \cos{\left(x \right)}$$
переменные
y = -1/2
$$y = - \frac{1}{2}$$
cos(30)*cos(x) - sin(30)*sin((-1/2))
$$- \sin{\left(30 \right)} \sin{\left((-1/2) \right)} + \cos{\left(30 \right)} \cos{\left(x \right)}$$
cos(30)*cos(x) - sin(30)*sin(-1/2)
$$\cos{\left(30 \right)} \cos{\left(x \right)} - \sin{\left(- \frac{1}{2} \right)} \sin{\left(30 \right)}$$
cos(30)*cos(x) + sin(1/2)*sin(30)
$$\cos{\left(30 \right)} \cos{\left(x \right)} + \sin{\left(\frac{1}{2} \right)} \sin{\left(30 \right)}$$
cos(30)*cos(x) + sin(1/2)*sin(30)
Собрать выражение
[src]
cos(-30 + x) cos(30 + x) cos(30 + y) cos(-30 + y)
------------ + ----------- + ----------- - ------------
2 2 2 2
$$\frac{\cos{\left(x - 30 \right)}}{2} + \frac{\cos{\left(x + 30 \right)}}{2} - \frac{\cos{\left(y - 30 \right)}}{2} + \frac{\cos{\left(y + 30 \right)}}{2}$$
cos(-30 + x)/2 + cos(30 + x)/2 + cos(30 + y)/2 - cos(-30 + y)/2
Тригонометрическая часть
[src]
1 1 -------------- - -------------- sec(30)*sec(x) csc(30)*csc(y)
$$\frac{1}{\sec{\left(30 \right)} \sec{\left(x \right)}} - \frac{1}{\csc{\left(30 \right)} \csc{\left(y \right)}}$$
/ pi\ / pi\
cos(30)*cos(x) - cos|30 - --|*cos|y - --|
\ 2 / \ 2 /
$$\cos{\left(30 \right)} \cos{\left(x \right)} - \cos{\left(- \frac{\pi}{2} + 30 \right)} \cos{\left(y - \frac{\pi}{2} \right)}$$
/ pi\ / pi\ sin|30 + --|*sin|x + --| - sin(30)*sin(y) \ 2 / \ 2 /
$$- \sin{\left(30 \right)} \sin{\left(y \right)} + \sin{\left(\frac{\pi}{2} + 30 \right)} \sin{\left(x + \frac{\pi}{2} \right)}$$
1 1 ------------------------- - -------------- / pi\ /pi \ csc(30)*csc(y) csc|-30 + --|*csc|-- - x| \ 2 / \2 /
$$\frac{1}{\csc{\left(-30 + \frac{\pi}{2} \right)} \csc{\left(- x + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(30 \right)} \csc{\left(y \right)}}$$
1 1
-------------- - ------------------------
sec(30)*sec(x) / pi\ / pi\
sec|30 - --|*sec|y - --|
\ 2 / \ 2 /
$$- \frac{1}{\sec{\left(- \frac{\pi}{2} + 30 \right)} \sec{\left(y - \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(30 \right)} \sec{\left(x \right)}}$$
1 1
-------------- - -------------------------
sec(30)*sec(x) / pi\ /pi \
sec|-30 + --|*sec|-- - y|
\ 2 / \2 /
$$- \frac{1}{\sec{\left(-30 + \frac{\pi}{2} \right)} \sec{\left(- y + \frac{\pi}{2} \right)}} + \frac{1}{\sec{\left(30 \right)} \sec{\left(x \right)}}$$
cos(-30 + x) cos(30 + x) cos(30 + y) cos(-30 + y)
------------ + ----------- + ----------- - ------------
2 2 2 2
$$\frac{\cos{\left(x - 30 \right)}}{2} + \frac{\cos{\left(x + 30 \right)}}{2} - \frac{\cos{\left(y - 30 \right)}}{2} + \frac{\cos{\left(y + 30 \right)}}{2}$$
1 1 ------------------------- - ------------------------- / pi\ /pi \ csc(-30 + pi)*csc(pi - y) csc|-30 + --|*csc|-- - x| \ 2 / \2 /
$$\frac{1}{\csc{\left(-30 + \frac{\pi}{2} \right)} \csc{\left(- x + \frac{\pi}{2} \right)}} - \frac{1}{\csc{\left(-30 + \pi \right)} \csc{\left(- y + \pi \right)}}$$
cos(-30 + x) + cos(30 + x) /y\
-------------------------- - 2*(1 + cos(y))*cos(15)*sin(15)*tan|-|
2 \2/
$$- 2 \left(\cos{\left(y \right)} + 1\right) \sin{\left(15 \right)} \cos{\left(15 \right)} \tan{\left(\frac{y}{2} \right)} + \frac{\cos{\left(x - 30 \right)} + \cos{\left(x + 30 \right)}}{2}$$
/ 2/ pi\\ / 2/y pi\\
|1 - cot |15 + --||*|1 - cot |- + --||*(1 + sin(30))*(1 + sin(y))
cos(-30 + x) + cos(30 + x) \ \ 4 // \ \2 4 //
-------------------------- - -----------------------------------------------------------------
2 4
$$- \frac{\left(- \cot^{2}{\left(\frac{\pi}{4} + 15 \right)} + 1\right) \left(- \cot^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)} + 1\right) \left(\sin{\left(30 \right)} + 1\right) \left(\sin{\left(y \right)} + 1\right)}{4} + \frac{\cos{\left(x - 30 \right)} + \cos{\left(x + 30 \right)}}{2}$$
// 0 for y mod pi = 0\ 2 / 2 \ // 1 for x mod 2*pi = 0\ - |< |*sin(30) + sin (15)*\-1 + cot (15)/*|< | \\sin(y) otherwise / \\cos(x) otherwise /
$$\left(- \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\sin{\left(y \right)} & \text{otherwise} \end{cases}\right) \sin{\left(30 \right)}\right) + \left(\left(-1 + \cot^{2}{\left(15 \right)}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right) \sin^{2}{\left(15 \right)}\right)$$
// 0 for y mod pi = 0\ 2 / 1 \ // 1 for x mod 2*pi = 0\
- |< |*sin(30) + sin (15)*|-1 + --------|*|< |
\\sin(y) otherwise / | 2 | \\cos(x) otherwise /
\ tan (15)/
$$\left(- \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\sin{\left(y \right)} & \text{otherwise} \end{cases}\right) \sin{\left(30 \right)}\right) + \left(\left(-1 + \frac{1}{\tan^{2}{\left(15 \right)}}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right) \sin^{2}{\left(15 \right)}\right)$$
/ 2 \ / 2/x\\ /y\
\1 - tan (15)/*|1 - tan |-|| 4*tan(15)*tan|-|
\ \2// \2/
---------------------------- - ----------------------------
/ 2 \ / 2/x\\ / 2 \ / 2/y\\
\1 + tan (15)/*|1 + tan |-|| \1 + tan (15)/*|1 + tan |-||
\ \2// \ \2//
$$\frac{\left(- \tan^{2}{\left(15 \right)} + 1\right) \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}{\left(\tan^{2}{\left(15 \right)} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} - \frac{4 \tan{\left(15 \right)} \tan{\left(\frac{y}{2} \right)}}{\left(\tan^{2}{\left(15 \right)} + 1\right) \left(\tan^{2}{\left(\frac{y}{2} \right)} + 1\right)}$$
/y\ / pi\ /x pi\
4*tan(15)*tan|-| 4*tan|15 + --|*tan|- + --|
\2/ \ 4 / \2 4 /
- ---------------------------- + --------------------------------------
/ 2 \ / 2/y\\ / 2/ pi\\ / 2/x pi\\
\1 + tan (15)/*|1 + tan |-|| |1 + tan |15 + --||*|1 + tan |- + --||
\ \2// \ \ 4 // \ \2 4 //
$$\frac{4 \tan{\left(\frac{\pi}{4} + 15 \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{\pi}{4} + 15 \right)} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)} - \frac{4 \tan{\left(15 \right)} \tan{\left(\frac{y}{2} \right)}}{\left(\tan^{2}{\left(15 \right)} + 1\right) \left(\tan^{2}{\left(\frac{y}{2} \right)} + 1\right)}$$
/y\ / pi\ /x pi\
4*cot(15)*cot|-| 4*tan|15 + --|*tan|- + --|
\2/ \ 4 / \2 4 /
- ---------------------------- + --------------------------------------
/ 2 \ / 2/y\\ / 2/ pi\\ / 2/x pi\\
\1 + cot (15)/*|1 + cot |-|| |1 + tan |15 + --||*|1 + tan |- + --||
\ \2// \ \ 4 // \ \2 4 //
$$- \frac{4 \cot{\left(15 \right)} \cot{\left(\frac{y}{2} \right)}}{\left(1 + \cot^{2}{\left(15 \right)}\right) \left(\cot^{2}{\left(\frac{y}{2} \right)} + 1\right)} + \frac{4 \tan{\left(\frac{\pi}{4} + 15 \right)} \tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{\pi}{4} + 15 \right)} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1\right)}$$
/ 2 \ // 1 for x mod 2*pi = 0\
// 0 for y mod pi = 0\ 2 | sin (30) | || |
- |< |*sin(30) + sin (15)*|-1 + ----------|*|< / pi\ |
\\sin(y) otherwise / | 4 | ||sin|x + --| otherwise |
\ 4*sin (15)/ \\ \ 2 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\sin{\left(y \right)} & \text{otherwise} \end{cases}\right) \sin{\left(30 \right)}\right) + \left(\left(-1 + \frac{\sin^{2}{\left(30 \right)}}{4 \sin^{4}{\left(15 \right)}}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\sin{\left(x + \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \sin^{2}{\left(15 \right)}\right)$$
// 1 for x mod 2*pi = 0\
(-1 - cos(60) + 2*cos(30))*|< |
// 0 for y mod pi = 0\ \\cos(x) otherwise /
- |< |*sin(30) + --------------------------------------------------------
\\sin(y) otherwise / 2*(1 - cos(30))
$$\left(- \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\sin{\left(y \right)} & \text{otherwise} \end{cases}\right) \sin{\left(30 \right)}\right) + \left(\frac{\left(-1 + 2 \cos{\left(30 \right)} - \cos{\left(60 \right)}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right)}{2 \cdot \left(- \cos{\left(30 \right)} + 1\right)}\right)$$
// / 3*pi\ \
2 / 2 \ // 1 for x mod 2*pi = 0\ || 1 for |y + ----| mod 2*pi = 0|
sin (15)*\-1 + cot (15)/*|< | - 2*|< \ 2 / |*cos(15)*sin(15)
\\cos(x) otherwise / || |
\\sin(y) otherwise /
$$\left(\left(-1 + \cot^{2}{\left(15 \right)}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right) \sin^{2}{\left(15 \right)}\right) - \left(2 \left(\begin{cases} 1 & \text{for}\: \left(y + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\sin{\left(y \right)} & \text{otherwise} \end{cases}\right) \sin{\left(15 \right)} \cos{\left(15 \right)}\right)$$
// 0 for y mod pi = 0\ || | ||1 - cos(y) | 2 / 1 \ // 1 for x mod 2*pi = 0\ - |<---------- otherwise |*sin(30) + sin (15)*|-1 + --------|*|< | || /y\ | | 2 | \\cos(x) otherwise / || tan|-| | \ tan (15)/ \\ \2/ /
$$\left(- \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{- \cos{\left(y \right)} + 1}{\tan{\left(\frac{y}{2} \right)}} & \text{otherwise} \end{cases}\right) \sin{\left(30 \right)}\right) + \left(\left(-1 + \frac{1}{\tan^{2}{\left(15 \right)}}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right) \sin^{2}{\left(15 \right)}\right)$$
/ 1 \ / 1 \
|1 - --------|*|1 - -------|
| 2 | | 2/x\|
\ cot (15)/ | cot |-||
\ \2// 4
---------------------------- - -------------------------------------------
/ 1 \ / 1 \ / 1 \ / 1 \ /y\
|1 + --------|*|1 + -------| |1 + --------|*|1 + -------|*cot(15)*cot|-|
| 2 | | 2/x\| | 2 | | 2/y\| \2/
\ cot (15)/ | cot |-|| \ cot (15)/ | cot |-||
\ \2// \ \2//
$$\frac{\left(- \frac{1}{\cot^{2}{\left(15 \right)}} + 1\right) \left(1 - \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right)}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{x}{2} \right)}}\right) \left(\frac{1}{\cot^{2}{\left(15 \right)}} + 1\right)} - \frac{4}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{y}{2} \right)}}\right) \left(\frac{1}{\cot^{2}{\left(15 \right)}} + 1\right) \cot{\left(15 \right)} \cot{\left(\frac{y}{2} \right)}}$$
2 / 1 \ // 1 for x mod 2*pi = 0\
4*tan (15/2)*|-1 + --------|*|< |
| 2 | \\cos(x) otherwise /
// 0 for y mod pi = 0\ \ tan (15)/
- |< |*sin(30) + ----------------------------------------------------------
\\sin(y) otherwise / 2
/ 2 \
\1 + tan (15/2)/
$$\left(- \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\sin{\left(y \right)} & \text{otherwise} \end{cases}\right) \sin{\left(30 \right)}\right) + \left(\frac{4 \left(-1 + \frac{1}{\tan^{2}{\left(15 \right)}}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right) \tan^{2}{\left(\frac{15}{2} \right)}}{\left(1 + \tan^{2}{\left(\frac{15}{2} \right)}\right)^{2}}\right)$$
// 1 for x mod 2*pi = 0\
/ 2 \ || |
/ 0 for y mod pi = 0 | csc (15) | || 1 |
| |-1 + --------------|*|<----------- otherwise |
< 1 | 2/ pi\| || /pi \ |
|------ otherwise | csc |-15 + --|| ||csc|-- - x| |
\csc(y) \ \ 2 // \\ \2 / /
- ------------------------- + --------------------------------------------------------
csc(30) 2
csc (15)
$$\left(- \frac{\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{1}{\csc{\left(y \right)}} & \text{otherwise} \end{cases}}{\csc{\left(30 \right)}}\right) + \left(\frac{\left(-1 + \frac{\csc^{2}{\left(15 \right)}}{\csc^{2}{\left(-15 + \frac{\pi}{2} \right)}}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\csc{\left(- x + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right)}{\csc^{2}{\left(15 \right)}}\right)$$
// / pi\ \
|| 0 for |x + --| mod pi = 0|
// 0 for y mod pi = 0\ || \ 2 / | / pi\
- |< |*sin(30) + (1 + sin(30))*|< |*cot|15 + --|
\\sin(y) otherwise / || /x pi\ | \ 4 /
||(1 + sin(x))*cot|- + --| otherwise |
\\ \2 4 / /
$$\left(- \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\sin{\left(y \right)} & \text{otherwise} \end{cases}\right) \sin{\left(30 \right)}\right) + \left(\left(\sin{\left(30 \right)} + 1\right) \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\left(\sin{\left(x \right)} + 1\right) \cot{\left(\frac{x}{2} + \frac{\pi}{4} \right)} & \text{otherwise} \end{cases}\right) \cot{\left(\frac{\pi}{4} + 15 \right)}\right)$$
// 0 for y mod pi = 0\ / 2 \
|| | / pi\ 2/ pi\ | cos (15) | // 1 for x mod 2*pi = 0\
- |< / pi\ |*cos|30 - --| + cos |15 - --|*|-1 + -------------|*|< |
||cos|y - --| otherwise | \ 2 / \ 2 / | 2/ pi\| \\cos(x) otherwise /
\\ \ 2 / / | cos |15 - --||
\ \ 2 //
$$\left(- \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\cos{\left(y - \frac{\pi}{2} \right)} & \text{otherwise} \end{cases}\right) \cos{\left(- \frac{\pi}{2} + 30 \right)}\right) + \left(\left(-1 + \frac{\cos^{2}{\left(15 \right)}}{\cos^{2}{\left(- \frac{\pi}{2} + 15 \right)}}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases}\right) \cos^{2}{\left(- \frac{\pi}{2} + 15 \right)}\right)$$
/ 2 \ / 2/x\\ / 2/ pi\\ / 2/y pi\\
\-1 + cot (15)/*|-1 + cot |-|| |-1 + tan |15 + --||*|-1 + tan |- + --||
\ \2// \ \ 4 // \ \2 4 //
------------------------------ - ----------------------------------------
/ 2 \ / 2/x\\ / 2/ pi\\ / 2/y pi\\
\1 + cot (15)/*|1 + cot |-|| |1 + tan |15 + --||*|1 + tan |- + --||
\ \2// \ \ 4 // \ \2 4 //
$$- \frac{\left(-1 + \tan^{2}{\left(\frac{\pi}{4} + 15 \right)}\right) \left(\tan^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)} - 1\right)}{\left(\tan^{2}{\left(\frac{\pi}{4} + 15 \right)} + 1\right) \left(\tan^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)} + 1\right)} + \frac{\left(-1 + \cot^{2}{\left(15 \right)}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} - 1\right)}{\left(1 + \cot^{2}{\left(15 \right)}\right) \left(\cot^{2}{\left(\frac{x}{2} \right)} + 1\right)}$$
/ 2 \ / 2/x\\ / 2/ pi\\ / 2/y pi\\
\1 - tan (15)/*|1 - tan |-|| |1 - cot |15 + --||*|1 - cot |- + --||
\ \2// \ \ 4 // \ \2 4 //
---------------------------- - --------------------------------------
/ 2 \ / 2/x\\ / 2/ pi\\ / 2/y pi\\
\1 + tan (15)/*|1 + tan |-|| |1 + cot |15 + --||*|1 + cot |- + --||
\ \2// \ \ 4 // \ \2 4 //
$$\frac{\left(- \tan^{2}{\left(15 \right)} + 1\right) \left(- \tan^{2}{\left(\frac{x}{2} \right)} + 1\right)}{\left(\tan^{2}{\left(15 \right)} + 1\right) \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)} - \frac{\left(- \cot^{2}{\left(\frac{\pi}{4} + 15 \right)} + 1\right) \left(- \cot^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)} + 1\right)}{\left(1 + \cot^{2}{\left(\frac{\pi}{4} + 15 \right)}\right) \left(\cot^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)} + 1\right)}$$
/ 0 for y mod pi = 0
| / 2/ pi\\
| 1 | sec |15 - --|| // 1 for x mod 2*pi = 0\
<----------- otherwise | \ 2 /| || |
| / pi\ |-1 + -------------|*|< 1 |
|sec|y - --| | 2 | ||------ otherwise |
\ \ 2 / \ sec (15) / \\sec(x) /
- ------------------------------ + --------------------------------------------------
/ pi\ 2/ pi\
sec|30 - --| sec |15 - --|
\ 2 / \ 2 /
$$\left(- \frac{\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{1}{\sec{\left(y - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}}{\sec{\left(- \frac{\pi}{2} + 30 \right)}}\right) + \left(\frac{\left(-1 + \frac{\sec^{2}{\left(- \frac{\pi}{2} + 15 \right)}}{\sec^{2}{\left(15 \right)}}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{1}{\sec{\left(x \right)}} & \text{otherwise} \end{cases}\right)}{\sec^{2}{\left(- \frac{\pi}{2} + 15 \right)}}\right)$$
// 1 for x mod 2*pi = 0\ // 0 for y mod pi = 0\
|| | || |
|| 2/x\ | || /y\ |
/ 2 \ ||-1 + cot |-| | || 2*cot|-| |
\-1 + cot (15)/*|< \2/ | 2*|< \2/ |*cot(15)
||------------ otherwise | ||----------- otherwise |
|| 2/x\ | || 2/y\ |
||1 + cot |-| | ||1 + cot |-| |
\\ \2/ / \\ \2/ /
--------------------------------------------------- - ------------------------------------------
2 2
1 + cot (15) 1 + cot (15)
$$\left(- \frac{2 \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{y}{2} \right)}}{\cot^{2}{\left(\frac{y}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \cot{\left(15 \right)}}{1 + \cot^{2}{\left(15 \right)}}\right) + \left(\frac{\left(-1 + \cot^{2}{\left(15 \right)}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)}{1 + \cot^{2}{\left(15 \right)}}\right)$$
// 0 for y mod pi = 0\ // 1 for x mod 2*pi = 0\
|| | 2 4 / 2 \ || |
- |
$$\left(- \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\sin{\left(y \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \sin{\left(30 \right)}\right) + \left(4 \left(-1 + \cot^{2}{\left(15 \right)}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\cos{\left(x \right)} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \sin^{4}{\left(\frac{15}{2} \right)} \cot^{2}{\left(\frac{15}{2} \right)}\right)$$
// / pi\ \
|| 0 for |x + --| mod pi = 0|
// 0 for y mod pi = 0\ || \ 2 / |
|| | || |
|| /y\ | || /x pi\ | / pi\
|| 2*cot|-| | 2*|< 2*cot|- + --| |*cot|15 + --|
2*|< \2/ |*cot(15) || \2 4 / | \ 4 /
||----------- otherwise | ||---------------- otherwise |
|| 2/y\ | || 2/x pi\ |
||1 + cot |-| | ||1 + cot |- + --| |
\\ \2/ / \\ \2 4 / /
- ------------------------------------------ + -----------------------------------------------------------
2 2/ pi\
1 + cot (15) 1 + cot |15 + --|
\ 4 /
$$\left(- \frac{2 \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{y}{2} \right)}}{\cot^{2}{\left(\frac{y}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \cot{\left(15 \right)}}{1 + \cot^{2}{\left(15 \right)}}\right) + \left(\frac{2 \left(\begin{cases} 0 & \text{for}\: \left(x + \frac{\pi}{2}\right) \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{\cot^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right) \cot{\left(\frac{\pi}{4} + 15 \right)}}{1 + \cot^{2}{\left(\frac{\pi}{4} + 15 \right)}}\right)$$
// 0 for y mod pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
|| /y\ | || 2/x\ |
|| 2*tan|-| | 2 / 1 \ ||1 - tan |-| |
2*|< \2/ |*tan(15) 4*tan (15/2)*|-1 + --------|*|< \2/ |
||----------- otherwise | | 2 | ||----------- otherwise |
|| 2/y\ | \ tan (15)/ || 2/x\ |
||1 + tan |-| | ||1 + tan |-| |
\\ \2/ / \\ \2/ /
- ------------------------------------------ + ---------------------------------------------------------------
2 2
1 + tan (15) / 2 \
\1 + tan (15/2)/
$$\left(- \frac{2 \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{2 \tan{\left(\frac{y}{2} \right)}}{\tan^{2}{\left(\frac{y}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \tan{\left(15 \right)}}{\tan^{2}{\left(15 \right)} + 1}\right) + \left(\frac{4 \left(-1 + \frac{1}{\tan^{2}{\left(15 \right)}}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{- \tan^{2}{\left(\frac{x}{2} \right)} + 1}{\tan^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right) \tan^{2}{\left(\frac{15}{2} \right)}}{\left(1 + \tan^{2}{\left(\frac{15}{2} \right)}\right)^{2}}\right)$$
// 1 for x mod 2*pi = 0\
|| |
|| 1 |
||-1 + ------- |
|| 2/x\ | // 0 for y mod pi = 0\
/ 1 \ || tan |-| | || |
|-1 + --------|*|< \2/ | || 2 |
| 2 | ||------------ otherwise | ||-------------------- otherwise |
\ tan (15)/ || 1 | 2*|
$$\left(- \frac{2 \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{2}{\left(1 + \frac{1}{\tan^{2}{\left(\frac{y}{2} \right)}}\right) \tan{\left(\frac{y}{2} \right)}} & \text{otherwise} \end{cases}\right)}{\left(1 + \frac{1}{\tan^{2}{\left(15 \right)}}\right) \tan{\left(15 \right)}}\right) + \left(\frac{\left(-1 + \frac{1}{\tan^{2}{\left(15 \right)}}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{-1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}}{1 + \frac{1}{\tan^{2}{\left(\frac{x}{2} \right)}}} & \text{otherwise} \end{cases}\right)}{1 + \frac{1}{\tan^{2}{\left(15 \right)}}}\right)$$
/ 4/x\\
/ 4 \ | 4*sin |-||
| 4*sin (15)| | \2/|
|1 - ----------|*|1 - ---------| 2 2/y\
| 2 | | 2 | 16*sin (15)*sin |-|
\ sin (30) / \ sin (x) / \2/
-------------------------------- - -----------------------------------------------
/ 4/x\\ / 4/y\\
/ 4 \ | 4*sin |-|| / 4 \ | 4*sin |-||
| 4*sin (15)| | \2/| | 4*sin (15)| | \2/|
|1 + ----------|*|1 + ---------| |1 + ----------|*|1 + ---------|*sin(30)*sin(y)
| 2 | | 2 | | 2 | | 2 |
\ sin (30) / \ sin (x) / \ sin (30) / \ sin (y) /
$$\frac{\left(- \frac{4 \sin^{4}{\left(15 \right)}}{\sin^{2}{\left(30 \right)}} + 1\right) \left(- \frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)}{\left(\frac{4 \sin^{4}{\left(15 \right)}}{\sin^{2}{\left(30 \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{x}{2} \right)}}{\sin^{2}{\left(x \right)}} + 1\right)} - \frac{16 \sin^{2}{\left(15 \right)} \sin^{2}{\left(\frac{y}{2} \right)}}{\left(\frac{4 \sin^{4}{\left(15 \right)}}{\sin^{2}{\left(30 \right)}} + 1\right) \left(\frac{4 \sin^{4}{\left(\frac{y}{2} \right)}}{\sin^{2}{\left(y \right)}} + 1\right) \sin{\left(30 \right)} \sin{\left(y \right)}}$$
// / 3*pi\ \
|| 1 for |y + ----| mod 2*pi = 0|
// 1 for x mod 2*pi = 0\ || \ 2 / |
|| | || |
|| 2/x\ | / 2/ pi\\ || 2/y pi\ |
/ 2 \ ||-1 + cot |-| | |-1 + tan |15 + --||*|<-1 + tan |- + --| |
\-1 + cot (15)/*|< \2/ | \ \ 4 // || \2 4 / |
||------------ otherwise | ||----------------- otherwise |
|| 2/x\ | || 2/y pi\ |
||1 + cot |-| | || 1 + tan |- + --| |
\\ \2/ / \\ \2 4 / /
--------------------------------------------------- - ----------------------------------------------------------------------
2 2/ pi\
1 + cot (15) 1 + tan |15 + --|
\ 4 /
$$\left(\frac{\left(-1 + \cot^{2}{\left(15 \right)}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases}\right)}{1 + \cot^{2}{\left(15 \right)}}\right) - \left(\frac{\left(-1 + \tan^{2}{\left(\frac{\pi}{4} + 15 \right)}\right) \left(\begin{cases} 1 & \text{for}\: \left(y + \frac{3 \pi}{2}\right) \bmod 2 \pi = 0 \\\frac{\tan^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)} - 1}{\tan^{2}{\left(\frac{y}{2} + \frac{\pi}{4} \right)} + 1} & \text{otherwise} \end{cases}\right)}{\tan^{2}{\left(\frac{\pi}{4} + 15 \right)} + 1}\right)$$
// 0 for y mod pi = 0\ // 1 for x mod 2*pi = 0\
|| | || |
||/ 0 for y mod pi = 0 | ||/ 1 for x mod 2*pi = 0 |
||| | ||| |
||| /y\ | 2 / 2 \ ||| 2/x\ |
2*|<| 2*cot|-| |*cot(15) 4*cot (15/2)*\-1 + cot (15)/*|<|-1 + cot |-| |
||< \2/ otherwise | ||< \2/ otherwise |
|||----------- otherwise | |||------------ otherwise |
||| 2/y\ | ||| 2/x\ |
|||1 + cot |-| | |||1 + cot |-| |
\\\ \2/ / \\\ \2/ /
- ------------------------------------------------------------- + -------------------------------------------------------------------------------------
2 2
1 + cot (15) / 2 \
\1 + cot (15/2)/
$$\left(- \frac{2 \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{2 \cot{\left(\frac{y}{2} \right)}}{\cot^{2}{\left(\frac{y}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \cot{\left(15 \right)}}{1 + \cot^{2}{\left(15 \right)}}\right) + \left(\frac{4 \left(-1 + \cot^{2}{\left(15 \right)}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\cot^{2}{\left(\frac{x}{2} \right)} - 1}{\cot^{2}{\left(\frac{x}{2} \right)} + 1} & \text{otherwise} \end{cases} & \text{otherwise} \end{cases}\right) \cot^{2}{\left(\frac{15}{2} \right)}}{\left(\cot^{2}{\left(\frac{15}{2} \right)} + 1\right)^{2}}\right)$$
/ 2/ pi\\ / 2/x pi\\
| cos |15 - --|| | cos |- - --||
| \ 2 /| | \2 2 /|
|1 - -------------|*|1 - ------------|
| 2 | | 2/x\ | / pi\ /y pi\
\ cos (15) / | cos |-| | 4*cos|15 - --|*cos|- - --|
\ \2/ / \ 2 / \2 2 /
-------------------------------------- - -----------------------------------------------------
/ 2/ pi\\ / 2/x pi\\ / 2/ pi\\ / 2/y pi\\
| cos |15 - --|| | cos |- - --|| | cos |15 - --|| | cos |- - --||
| \ 2 /| | \2 2 /| | \ 2 /| | \2 2 /| /y\
|1 + -------------|*|1 + ------------| |1 + -------------|*|1 + ------------|*cos(15)*cos|-|
| 2 | | 2/x\ | | 2 | | 2/y\ | \2/
\ cos (15) / | cos |-| | \ cos (15) / | cos |-| |
\ \2/ / \ \2/ /
$$\frac{\left(1 - \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right) \left(- \frac{\cos^{2}{\left(- \frac{\pi}{2} + 15 \right)}}{\cos^{2}{\left(15 \right)}} + 1\right)}{\left(1 + \frac{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} \right)}}\right) \left(\frac{\cos^{2}{\left(- \frac{\pi}{2} + 15 \right)}}{\cos^{2}{\left(15 \right)}} + 1\right)} - \frac{4 \cos{\left(- \frac{\pi}{2} + 15 \right)} \cos{\left(\frac{y}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(\frac{y}{2} - \frac{\pi}{2} \right)}}{\cos^{2}{\left(\frac{y}{2} \right)}}\right) \left(\frac{\cos^{2}{\left(- \frac{\pi}{2} + 15 \right)}}{\cos^{2}{\left(15 \right)}} + 1\right) \cos{\left(15 \right)} \cos{\left(\frac{y}{2} \right)}}$$
/ 2/x\ \
/ 2 \ | sec |-| |
| sec (15) | | \2/ |
|1 - -------------|*|1 - ------------|
| 2/ pi\| | 2/x pi\| /y\
| sec |15 - --|| | sec |- - --|| 4*sec(15)*sec|-|
\ \ 2 // \ \2 2 // \2/
-------------------------------------- - ---------------------------------------------------------------
/ 2/x\ \ / 2/y\ \
/ 2 \ | sec |-| | / 2 \ | sec |-| |
| sec (15) | | \2/ | | sec (15) | | \2/ | / pi\ /y pi\
|1 + -------------|*|1 + ------------| |1 + -------------|*|1 + ------------|*sec|15 - --|*sec|- - --|
| 2/ pi\| | 2/x pi\| | 2/ pi\| | 2/y pi\| \ 2 / \2 2 /
| sec |15 - --|| | sec |- - --|| | sec |15 - --|| | sec |- - --||
\ \ 2 // \ \2 2 // \ \ 2 // \ \2 2 //
$$\frac{\left(- \frac{\sec^{2}{\left(15 \right)}}{\sec^{2}{\left(- \frac{\pi}{2} + 15 \right)}} + 1\right) \left(- \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)}{\left(\frac{\sec^{2}{\left(15 \right)}}{\sec^{2}{\left(- \frac{\pi}{2} + 15 \right)}} + 1\right) \left(\frac{\sec^{2}{\left(\frac{x}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1\right)} - \frac{4 \sec{\left(15 \right)} \sec{\left(\frac{y}{2} \right)}}{\left(\frac{\sec^{2}{\left(15 \right)}}{\sec^{2}{\left(- \frac{\pi}{2} + 15 \right)}} + 1\right) \left(\frac{\sec^{2}{\left(\frac{y}{2} \right)}}{\sec^{2}{\left(\frac{y}{2} - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(- \frac{\pi}{2} + 15 \right)} \sec{\left(\frac{y}{2} - \frac{\pi}{2} \right)}}$$
/ 2/ pi\\ / 2/pi x\\
| csc |-15 + --|| | csc |-- - -||
| \ 2 /| | \2 2/|
|1 - --------------|*|1 - ------------|
| 2 | | 2/x\ | / pi\ /pi y\
\ csc (15) / | csc |-| | 4*csc|-15 + --|*csc|-- - -|
\ \2/ / \ 2 / \2 2/
--------------------------------------- - ------------------------------------------------------
/ 2/ pi\\ / 2/pi x\\ / 2/ pi\\ / 2/pi y\\
| csc |-15 + --|| | csc |-- - -|| | csc |-15 + --|| | csc |-- - -||
| \ 2 /| | \2 2/| | \ 2 /| | \2 2/| /y\
|1 + --------------|*|1 + ------------| |1 + --------------|*|1 + ------------|*csc(15)*csc|-|
| 2 | | 2/x\ | | 2 | | 2/y\ | \2/
\ csc (15) / | csc |-| | \ csc (15) / | csc |-| |
\ \2/ / \ \2/ /
$$\frac{\left(1 - \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \left(- \frac{\csc^{2}{\left(-15 + \frac{\pi}{2} \right)}}{\csc^{2}{\left(15 \right)}} + 1\right)}{\left(1 + \frac{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{x}{2} \right)}}\right) \left(\frac{\csc^{2}{\left(-15 + \frac{\pi}{2} \right)}}{\csc^{2}{\left(15 \right)}} + 1\right)} - \frac{4 \csc{\left(-15 + \frac{\pi}{2} \right)} \csc{\left(- \frac{y}{2} + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(- \frac{y}{2} + \frac{\pi}{2} \right)}}{\csc^{2}{\left(\frac{y}{2} \right)}}\right) \left(\frac{\csc^{2}{\left(-15 + \frac{\pi}{2} \right)}}{\csc^{2}{\left(15 \right)}} + 1\right) \csc{\left(15 \right)} \csc{\left(\frac{y}{2} \right)}}$$
// 0 for y mod pi = 0\
|| |
|| 2*sin(y) |
||---------------------------- otherwise |
// 1 for x mod 2*pi = 0\ || / 2 \ |
/ 2 \ || | 4*|< | sin (y) | |*sin(30)
| sin (30) | || 2 | ||(1 - cos(y))*|1 + ---------| |
4*|-1 + ----------|*|< -4 + 4*sin (x) + 4*cos(x) | || | 4/y\| |
| 4 | ||--------------------------- otherwise | || | 4*sin |-|| |
\ 4*sin (15)/ || 2 2 | || \ \2// |
\\2*(1 - cos(x)) + 2*sin (x) / \\ /
---------------------------------------------------------------------- - -----------------------------------------------------------
2 / 2 \
sin (30) | sin (30)| 2
4 + -------- |4 + --------|*sin (15)
4 | 4 |
sin (15) \ sin (15)/
$$\left(- \frac{4 \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{2 \sin{\left(y \right)}}{\left(1 + \frac{\sin^{2}{\left(y \right)}}{4 \sin^{4}{\left(\frac{y}{2} \right)}}\right) \left(- \cos{\left(y \right)} + 1\right)} & \text{otherwise} \end{cases}\right) \sin{\left(30 \right)}}{\left(4 + \frac{\sin^{2}{\left(30 \right)}}{\sin^{4}{\left(15 \right)}}\right) \sin^{2}{\left(15 \right)}}\right) + \left(\frac{4 \left(-1 + \frac{\sin^{2}{\left(30 \right)}}{4 \sin^{4}{\left(15 \right)}}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{4 \sin^{2}{\left(x \right)} + 4 \cos{\left(x \right)} - 4}{2 \left(- \cos{\left(x \right)} + 1\right)^{2} + 2 \sin^{2}{\left(x \right)}} & \text{otherwise} \end{cases}\right)}{4 + \frac{\sin^{2}{\left(30 \right)}}{\sin^{4}{\left(15 \right)}}}\right)$$
// 1 for x mod 2*pi = 0\
|| |
|| 2 |
|| sin (x) |
||-1 + --------- | // 0 for y mod pi = 0\
/ 2 \ || 4/x\ | || |
| sin (30) | || 4*sin |-| | || sin(y) |
|-1 + ----------|*|< \2/ | ||----------------------- otherwise |
| 4 | ||-------------- otherwise | ||/ 2 \ |
\ 4*sin (15)/ || 2 | |<| sin (y) | 2/y\ |*sin(30)
|| sin (x) | |||1 + ---------|*sin |-| |
||1 + --------- | ||| 4/y\| \2/ |
|| 4/x\ | ||| 4*sin |-|| |
|| 4*sin |-| | ||\ \2// |
\\ \2/ / \\ /
------------------------------------------------------- - ----------------------------------------------------
2 / 2 \
sin (30) | sin (30) | 2
1 + ---------- |1 + ----------|*sin (15)
4 | 4 |
4*sin (15) \ 4*sin (15)/
$$\left(- \frac{\left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{\sin{\left(y \right)}}{\left(1 + \frac{\sin^{2}{\left(y \right)}}{4 \sin^{4}{\left(\frac{y}{2} \right)}}\right) \sin^{2}{\left(\frac{y}{2} \right)}} & \text{otherwise} \end{cases}\right) \sin{\left(30 \right)}}{\left(1 + \frac{\sin^{2}{\left(30 \right)}}{4 \sin^{4}{\left(15 \right)}}\right) \sin^{2}{\left(15 \right)}}\right) + \left(\frac{\left(-1 + \frac{\sin^{2}{\left(30 \right)}}{4 \sin^{4}{\left(15 \right)}}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}}{1 + \frac{\sin^{2}{\left(x \right)}}{4 \sin^{4}{\left(\frac{x}{2} \right)}}} & \text{otherwise} \end{cases}\right)}{1 + \frac{\sin^{2}{\left(30 \right)}}{4 \sin^{4}{\left(15 \right)}}}\right)$$
// 1 for x mod 2*pi = 0\
|| |
|| 2/x\ |
|| cos |-| |
|| \2/ | // 0 for y mod pi = 0\
||-1 + ------------ | || |
/ 2 \ || 2/x pi\ | || /y\ |
| cos (15) | || cos |- - --| | || 2*cos|-| |
|-1 + -------------|*|< \2 2 / | || \2/ |
| 2/ pi\| ||----------------- otherwise | ||------------------------------ otherwise |
| cos |15 - --|| || 2/x\ | 2*|
$$\left(- \frac{2 \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{2 \cos{\left(\frac{y}{2} \right)}}{\left(\frac{\cos^{2}{\left(\frac{y}{2} \right)}}{\cos^{2}{\left(\frac{y}{2} - \frac{\pi}{2} \right)}} + 1\right) \cos{\left(\frac{y}{2} - \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \cos{\left(15 \right)}}{\left(1 + \frac{\cos^{2}{\left(15 \right)}}{\cos^{2}{\left(- \frac{\pi}{2} + 15 \right)}}\right) \cos{\left(- \frac{\pi}{2} + 15 \right)}}\right) + \left(\frac{\left(-1 + \frac{\cos^{2}{\left(15 \right)}}{\cos^{2}{\left(- \frac{\pi}{2} + 15 \right)}}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} - 1}{\frac{\cos^{2}{\left(\frac{x}{2} \right)}}{\cos^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)}{1 + \frac{\cos^{2}{\left(15 \right)}}{\cos^{2}{\left(- \frac{\pi}{2} + 15 \right)}}}\right)$$
// 1 for x mod 2*pi = 0\
|| |
|| 2/x pi\ |
|| sec |- - --| |
|| \2 2 / | // 0 for y mod pi = 0\
/ 2/ pi\\ ||-1 + ------------ | || |
| sec |15 - --|| || 2/x\ | || /y pi\ |
| \ 2 /| || sec |-| | || 2*sec|- - --| |
|-1 + -------------|*|< \2/ | || \2 2 / |
| 2 | ||----------------- otherwise | ||------------------------- otherwise | / pi\
\ sec (15) / || 2/x pi\ | 2*|
$$\left(- \frac{2 \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{2 \sec{\left(\frac{y}{2} - \frac{\pi}{2} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{y}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{y}{2} \right)}}\right) \sec{\left(\frac{y}{2} \right)}} & \text{otherwise} \end{cases}\right) \sec{\left(- \frac{\pi}{2} + 15 \right)}}{\left(1 + \frac{\sec^{2}{\left(- \frac{\pi}{2} + 15 \right)}}{\sec^{2}{\left(15 \right)}}\right) \sec{\left(15 \right)}}\right) + \left(\frac{\left(-1 + \frac{\sec^{2}{\left(- \frac{\pi}{2} + 15 \right)}}{\sec^{2}{\left(15 \right)}}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{-1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}}{1 + \frac{\sec^{2}{\left(\frac{x}{2} - \frac{\pi}{2} \right)}}{\sec^{2}{\left(\frac{x}{2} \right)}}} & \text{otherwise} \end{cases}\right)}{1 + \frac{\sec^{2}{\left(- \frac{\pi}{2} + 15 \right)}}{\sec^{2}{\left(15 \right)}}}\right)$$
// 1 for x mod 2*pi = 0\
|| |
|| 2/x\ |
|| csc |-| |
|| \2/ | // 0 for y mod pi = 0\
||-1 + ------------ | || |
/ 2 \ || 2/pi x\ | || /y\ |
| csc (15) | || csc |-- - -| | || 2*csc|-| |
|-1 + --------------|*|< \2 2/ | || \2/ |
| 2/ pi\| ||----------------- otherwise | ||------------------------------ otherwise |
| csc |-15 + --|| || 2/x\ | 2*|
$$\left(- \frac{2 \left(\begin{cases} 0 & \text{for}\: y \bmod \pi = 0 \\\frac{2 \csc{\left(\frac{y}{2} \right)}}{\left(\frac{\csc^{2}{\left(\frac{y}{2} \right)}}{\csc^{2}{\left(- \frac{y}{2} + \frac{\pi}{2} \right)}} + 1\right) \csc{\left(- \frac{y}{2} + \frac{\pi}{2} \right)}} & \text{otherwise} \end{cases}\right) \csc{\left(15 \right)}}{\left(1 + \frac{\csc^{2}{\left(15 \right)}}{\csc^{2}{\left(-15 + \frac{\pi}{2} \right)}}\right) \csc{\left(-15 + \frac{\pi}{2} \right)}}\right) + \left(\frac{\left(-1 + \frac{\csc^{2}{\left(15 \right)}}{\csc^{2}{\left(-15 + \frac{\pi}{2} \right)}}\right) \left(\begin{cases} 1 & \text{for}\: x \bmod 2 \pi = 0 \\\frac{\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} - 1}{\frac{\csc^{2}{\left(\frac{x}{2} \right)}}{\csc^{2}{\left(- \frac{x}{2} + \frac{\pi}{2} \right)}} + 1} & \text{otherwise} \end{cases}\right)}{1 + \frac{\csc^{2}{\left(15 \right)}}{\csc^{2}{\left(-15 + \frac{\pi}{2} \right)}}}\right)$$
(-1 + csc(15)^2/csc(-15 + pi/2)^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))/(1 + csc(15)^2/csc(-15 + pi/2)^2) - 2*Piecewise((0, Mod(y = pi, 0)), (2*csc(y/2)/((1 + csc(y/2)^2/csc(pi/2 - y/2)^2)*csc(pi/2 - y/2)), True))*csc(15)/((1 + csc(15)^2/csc(-15 + pi/2)^2)*csc(-15 + pi/2))
Численный ответ
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0.154251449887584*cos(x) + 0.988031624092862*sin(y)
0.154251449887584*cos(x) + 0.988031624092862*sin(y)
Степени
[src]
/ -30*I 30*I\ / I*x -I*x\ / -30*I 30*I\ / -I*y I*y\ |e e | |e e | \- e + e /*\- e + e / |------ + -----|*|---- + -----| + ----------------------------------- \ 2 2 / \ 2 2 / 4
$$\left(\frac{e^{30 i}}{2} + \frac{e^{- 30 i}}{2}\right) \left(\frac{e^{i x}}{2} + \frac{e^{- i x}}{2}\right) + \frac{\left(- e^{- 30 i} + e^{30 i}\right) \left(e^{i y} - e^{- i y}\right)}{4}$$
(exp(-30*i)/2 + exp(30*i)/2)*(exp(i*x)/2 + exp(-i*x)/2) + (-exp(-30*i) + exp(30*i))*(-exp(-i*y) + exp(i*y))/4