Дано
$$frac{1}{4} left(- frac{1}{frac{2 pi}{8} n + 6 pi} sin{left (x left(frac{pi n}{8} + 3 piright) right )} + frac{sin{left (x left(- frac{pi n}{8} + 3 piright) right )}}{2 left(- frac{pi n}{8} + 3 piright)}right)$$
Подстановка условия
$$frac{1}{4} left(- frac{1}{frac{2 pi}{8} n + 6 pi} sin{left (x left(frac{pi n}{8} + 3 piright) right )} + frac{sin{left (x left(- frac{pi n}{8} + 3 piright) right )}}{2 left(- frac{pi n}{8} + 3 piright)}right)$$
(sin((3*pi – n*pi/8)*(1/2))/(2*(3*pi – n*pi/8)) – sin(((n*pi)/8 + 3*pi)*(1/2))/2*((n*pi)/8 + 3*pi))/4
$$frac{1}{4} left(- frac{1}{frac{2 pi}{8} n + 6 pi} sin{left ((1/2) left(frac{pi n}{8} + 3 piright) right )} + frac{sin{left ((1/2) left(- frac{pi n}{8} + 3 piright) right )}}{2 left(- frac{pi n}{8} + 3 piright)}right)$$
(sin((3*pi – n*pi/8)/2)/(2*(3*pi – n*pi/8)) – sin(((n*pi)/8 + 3*pi)/2)/2*((n*pi)/8 + 3*pi))/4
$$frac{1}{4} left(- frac{1}{frac{2 pi}{8} n + 6 pi} sin{left (frac{1}{2} left(frac{pi n}{8} + 3 piright) right )} + frac{sin{left (frac{1}{2} left(- frac{pi n}{8} + 3 piright) right )}}{2 left(- frac{pi n}{8} + 3 piright)}right)$$
-cos(pi*n/16)/(4*(6*pi – pi*n/4)) + cos(pi*n/16)/(4*(6*pi + pi*n/4))
$$frac{cos{left (frac{pi n}{16} right )}}{pi n + 24 pi} – frac{cos{left (frac{pi n}{16} right )}}{- pi n + 24 pi}$$
Степени
$$- frac{sin{left (x left(frac{pi n}{8} + 3 piright) right )}}{pi n + 24 pi} + frac{sin{left (x left(- frac{pi n}{8} + 3 piright) right )}}{- pi n + 24 pi}$$
Численный ответ
0.25*sin((3*pi – n*pi/8)*x)/(18.8495559215388 – 0.785398163397448*n) – 0.25*sin(((n*pi)/8 + 3*pi)*x)/(18.8495559215388 + 0.785398163397448*n)
Рациональный знаменатель
$$frac{1}{4 left(2 pi n – 48 piright) left(2 pi n + 48 piright)} left(16 pi n sin{left (frac{pi n}{8} x – 3 pi x right )} – 16 pi n sin{left (frac{pi n}{8} x + 3 pi x right )} + 384 pi sin{left (frac{pi n}{8} x – 3 pi x right )} + 384 pi sin{left (frac{pi n}{8} x + 3 pi x right )}right)$$
Объединение рациональных выражений
$$frac{1}{pi left(- n + 24right) left(n + 24right)} left(- left(- n + 24right) sin{left (frac{pi x}{8} left(n + 24right) right )} + left(n + 24right) sin{left (frac{pi x}{8} left(- n + 24right) right )}right)$$
Общее упрощение
/pi*x*(-24 + n) /pi*x*(24 + n)
(24 + n)*sin|————–| + (24 – n)*sin|————-|
8 / 8 /
———————————————————-
pi*(-24 + n)*(24 + n)
$$frac{1}{pi left(n – 24right) left(n + 24right)} left(left(- n + 24right) sin{left (frac{pi x}{8} left(n + 24right) right )} + left(n + 24right) sin{left (frac{pi x}{8} left(n – 24right) right )}right)$$
Соберем выражение
$$- frac{frac{1}{2}}{4} frac{1}{frac{pi n}{8} + 3 pi} sin{left (x left(frac{pi n}{8} + 3 piright) right )} + frac{sin{left (x left(- frac{pi n}{8} + 3 piright) right )}}{8 left(- frac{pi n}{8} + 3 piright)}$$
/ pi*n*x 2 / / pi*n*x 2 / pi*n*x
64*sin|-3*pi*x + ——| + pi *|576*sin|3*pi*x + ——| – n *sin|3*pi*x + ——||
8 / 8 / 8 //
———————————————————————————–
pi*(-1536 + 64*n)
$$frac{1}{pi left(64 n – 1536right)} left(pi^{2} left(- n^{2} sin{left (frac{pi n}{8} x + 3 pi x right )} + 576 sin{left (frac{pi n}{8} x + 3 pi x right )}right) + 64 sin{left (frac{pi n}{8} x – 3 pi x right )}right)$$
Комбинаторика
/ / pi*n*x / pi*n*x / pi*n*x / pi*n*x
-|- 24*sin|-3*pi*x + ——| – 24*sin|3*pi*x + ——| + n*sin|3*pi*x + ——| – n*sin|-3*pi*x + ——||
8 / 8 / 8 / 8 //
———————————————————————————————————–
pi*(-24 + n)*(24 + n)
$$- frac{1}{pi left(n – 24right) left(n + 24right)} left(- n sin{left (frac{pi n}{8} x – 3 pi x right )} + n sin{left (frac{pi n}{8} x + 3 pi x right )} – 24 sin{left (frac{pi n}{8} x – 3 pi x right )} – 24 sin{left (frac{pi n}{8} x + 3 pi x right )}right)$$
Общий знаменатель
/ / pi*n*x / pi*n*x / pi*n*x / pi*n*x
-|- 24*sin|-3*pi*x + ——| – 24*sin|3*pi*x + ——| + n*sin|3*pi*x + ——| – n*sin|-3*pi*x + ——||
8 / 8 / 8 / 8 //
———————————————————————————————————–
2
-576*pi + pi*n
$$- frac{1}{pi n^{2} – 576 pi} left(- n sin{left (frac{pi n}{8} x – 3 pi x right )} + n sin{left (frac{pi n}{8} x + 3 pi x right )} – 24 sin{left (frac{pi n}{8} x – 3 pi x right )} – 24 sin{left (frac{pi n}{8} x + 3 pi x right )}right)$$
Тригонометрическая часть
/ / n / / n
sin|pi*x*|-3 + -|| sin|pi*x*|3 + -||
8// 8//
– —————— – —————–
/ n*pi /n*pi
8*|3*pi – —-| 8*|—- + 3*pi|
8 / 8 /
$$- frac{sin{left (pi x left(frac{n}{8} + 3right) right )}}{frac{8 pi}{8} n + 24 pi} – frac{sin{left (pi x left(frac{n}{8} – 3right) right )}}{- pi n + 24 pi}$$
Раскрыть выражение
//n*pi // n*pi
sin||—- + 3*pi|*x| sin||3*pi – —-|*x|
8 / / 8 / /
– ——————– + ——————–
/n*pi / n*pi
8*|—- + 3*pi| 8*|3*pi – —-|
8 / 8 /
$$- frac{sin{left (x left(frac{pi n}{8} + 3 piright) right )}}{frac{8 pi}{8} n + 24 pi} + frac{sin{left (x left(- frac{pi n}{8} + 3 piright) right )}}{- pi n + 24 pi}$$
/ pi*n / / pi*n / / pi*n
|3*pi + —-|*sin|x*|3*pi + —-|| sin|x*|3*pi – —-||
8 / 8 // 8 //
– ———————————- + ——————–
8 / pi*n
4*|6*pi – —-|
4 /
$$- frac{1}{8} left(frac{pi n}{8} + 3 piright) sin{left (x left(frac{pi n}{8} + 3 piright) right )} + frac{sin{left (x left(- frac{pi n}{8} + 3 piright) right )}}{- pi n + 24 pi}$$
