Дано
$$8 left(sin{left (t + 2 i p right )} + cos{left (t + pi right )} + sin{left (- t + pi right )}right) sin{left (t right )}$$
Подстановка условия
$$8 left(sin{left (t + 2 i p right )} + cos{left (t + pi right )} + sin{left (- t + pi right )}right) sin{left (t right )}$$
(8*(sin((p*(-3))*2 + t) + cos(pi + t) + sin(pi – t)))*sin(t)
$$8 left(sin{left (t + 2 (-3) p right )} + cos{left (t + pi right )} + sin{left (- t + pi right )}right) sin{left (t right )}$$
(8*(sin((p*(-3))*2 + t) + cos(pi + t) + sin(pi – t)))*sin(t)
$$8 left(sin{left (2 cdot -3 p + t right )} + cos{left (t + pi right )} + sin{left (- t + pi right )}right) sin{left (t right )}$$
(-8*cos(t) – 8*sin(-t + 6*p) + 8*sin(t))*sin(t)
$$left(8 sin{left (t right )} – 8 sin{left (6 p – t right )} – 8 cos{left (t right )}right) sin{left (t right )}$$
Степени
$$left(8 sin{left (t right )} + 8 sin{left (2 i p + t right )} – 8 cos{left (t right )}right) sin{left (t right )}$$
Численный ответ
8.0*(cos(pi + t) + sin(pi – t) + sin((p*i)*2 + t))*sin(t)
Рациональный знаменатель
$$8 left(sin{left (t right )} + sin{left (2 i p + t right )} – cos{left (t right )}right) sin{left (t right )}$$
Объединение рациональных выражений
$$left(8 sin{left (t right )} + 8 sin{left (2 i p + t right )} – 8 cos{left (t right )}right) sin{left (t right )}$$
Общее упрощение
(8*cos(pi + t) + 8*sin(t) + 8*sin(t + 2*I*p))*sin(t)
$$left(8 sin{left (t right )} + 8 sin{left (2 i p + t right )} + 8 cos{left (t + pi right )}right) sin{left (t right )}$$
Соберем выражение
$$left(8 sin{left (- t + pi right )} + 8 sin{left (t + 2 i p right )} + 8 cos{left (t + pi right )}right) sin{left (t right )}$$
4 – 4*cos(2*t) – 4*cos(2*t + 2*I*p) – 4*sin(2*t) + 4*cosh(2*p)
$$- 4 sin{left (2 t right )} – 4 cos{left (2 t right )} – 4 cos{left (2 i p + 2 t right )} + 4 cosh{left (2 p right )} + 4$$
Общий знаменатель
8*cos(pi + t)*sin(t) + 8*sin(t)*sin(pi – t) + 8*sin(t)*sin(t + 2*I*p)
$$8 sin{left (t right )} sin{left (- t + pi right )} + 8 sin{left (t right )} sin{left (2 i p + t right )} + 8 sin{left (t right )} cos{left (t + pi right )}$$
Тригонометрическая часть
___ / pi
4 + 4*cosh(2*p) – 4*/ 2 *sin|2*t + –| – 4*cos(2*t)*cosh(2*p) + 4*I*sin(2*t)*sinh(2*p)
4 /
$$4 i sin{left (2 t right )} sinh{left (2 p right )} – 4 sqrt{2} sin{left (2 t + frac{pi}{4} right )} – 4 cos{left (2 t right )} cosh{left (2 p right )} + 4 cosh{left (2 p right )} + 4$$
Комбинаторика
-8*(-sin(t) – sin(t + 2*I*p) + cos(t))*sin(t)
$$- 8 left(- sin{left (t right )} – sin{left (2 i p + t right )} + cos{left (t right )}right) sin{left (t right )}$$
Раскрыть выражение
8*(-cos(t) + cosh(2*p)*sin(t) + I*cos(t)*sinh(2*p) + sin(t))*sin(t)
$$8 left(sin{left (t right )} cosh{left (2 p right )} + sin{left (t right )} + i cos{left (t right )} sinh{left (2 p right )} – cos{left (t right )}right) sin{left (t right )}$$
