Дано
$$frac{6 sin^{2}{left (t right )} cos{left (t right )} sin{left (2 t right )}}{left(cos{left (2 t right )} + 1right)^{2}} + frac{1}{cos{left (2 t right )} + 1} left(- 3 sin^{3}{left (t right )} + sin{left (t right )} 6 cos^{2}{left (t right )}right)$$
Степени
$$frac{1}{cos{left (2 t right )} + 1} left(- 3 sin^{3}{left (t right )} + 6 sin{left (t right )} cos^{2}{left (t right )}right) + frac{6 sin^{2}{left (t right )} sin{left (2 t right )} cos{left (t right )}}{left(cos{left (2 t right )} + 1right)^{2}}$$
Численный ответ
(-3.0*sin(t)^3 + 6.0*cos(t)^2*sin(t))/(1.0 + cos(2*t)) + 6.0*sin(t)^2*cos(t)*sin(2*t)/(1.0 + cos(2*t))^2
Рациональный знаменатель
$$frac{1}{left(cos{left (2 t right )} + 1right)^{3}} left(left(- 3 sin^{3}{left (t right )} + 6 sin{left (t right )} cos^{2}{left (t right )}right) left(cos{left (2 t right )} + 1right)^{2} + 6 left(cos{left (2 t right )} + 1right) sin^{2}{left (t right )} sin{left (2 t right )} cos{left (t right )}right)$$
Объединение рациональных выражений
$$frac{3 sin{left (t right )}}{left(cos{left (2 t right )} + 1right)^{2}} left(left(- sin^{2}{left (t right )} + 2 cos^{2}{left (t right )}right) left(cos{left (2 t right )} + 1right) + 2 sin{left (t right )} sin{left (2 t right )} cos{left (t right )}right)$$
Общее упрощение
/ 1
3*|1 + ——-|*sin(t)
| 2 |
cos (t)/
———————-
2
$$frac{3}{2} left(1 + frac{1}{cos^{2}{left (t right )}}right) sin{left (t right )}$$
Соберем выражение
$$frac{3 sin{left (3 t right )}}{8 cos{left (2 t right )} + 2 cos{left (4 t right )} + 6} – frac{3 sin{left (5 t right )}}{8 cos{left (2 t right )} + 2 cos{left (4 t right )} + 6} + frac{3 sin{left (t right )}}{4 cos{left (2 t right )} + cos{left (4 t right )} + 3} – frac{3 sin{left (t right )}}{4 cos{left (2 t right )} + 4} + frac{9 sin{left (3 t right )}}{4 cos{left (2 t right )} + 4}$$
Общий знаменатель
3 3 2 2 2
– 3*sin (t) – 3*sin (t)*cos(2*t) + 6*cos (t)*sin(t) + 6*cos (t)*cos(2*t)*sin(t) + 6*sin (t)*cos(t)*sin(2*t)
———————————————————————————————————–
2
1 + cos (2*t) + 2*cos(2*t)
$$frac{1}{cos^{2}{left (2 t right )} + 2 cos{left (2 t right )} + 1} left(- 3 sin^{3}{left (t right )} cos{left (2 t right )} – 3 sin^{3}{left (t right )} + 6 sin^{2}{left (t right )} sin{left (2 t right )} cos{left (t right )} + 6 sin{left (t right )} cos^{2}{left (t right )} cos{left (2 t right )} + 6 sin{left (t right )} cos^{2}{left (t right )}right)$$
Тригонометрическая часть
3
3*sin (t)
3*sin(t) + ———
2
2*cos (t)
$$frac{3 sin^{3}{left (t right )}}{2 cos^{2}{left (t right )}} + 3 sin{left (t right )}$$
Комбинаторика
/ 2 2 2 2
3* – sin (t) + 2*cos (t) – sin (t)*cos(2*t) + 2*cos (t)*cos(2*t) + 2*cos(t)*sin(t)*sin(2*t)/*sin(t)
—————————————————————————————————
2
(1 + cos(2*t))
$$frac{3 sin{left (t right )}}{left(cos{left (2 t right )} + 1right)^{2}} left(- sin^{2}{left (t right )} cos{left (2 t right )} – sin^{2}{left (t right )} + 2 sin{left (t right )} sin{left (2 t right )} cos{left (t right )} + 2 cos^{2}{left (t right )} cos{left (2 t right )} + 2 cos^{2}{left (t right )}right)$$
Раскрыть выражение
3 2 2 3
– 3*sin (t) + 6*cos (t)*sin(t) 12*cos (t)*sin (t)
—————————— + ————————
2 2 2
1 + cos (t) – sin (t) / 2 2
1 + cos (t) – sin (t)/
$$frac{- 3 sin^{3}{left (t right )} + 6 sin{left (t right )} cos^{2}{left (t right )}}{- sin^{2}{left (t right )} + cos^{2}{left (t right )} + 1} + frac{12 sin^{3}{left (t right )} cos^{2}{left (t right )}}{left(- sin^{2}{left (t right )} + cos^{2}{left (t right )} + 1right)^{2}}$$
