Дано
$$frac{sin{left (4 x + 80 right )}}{sin{left (- x + 70 right )} 4 sin{left (x + 20 right )}}$$
Подстановка условия
$$frac{sin{left (4 x + 80 right )}}{sin{left (- x + 70 right )} 4 sin{left (x + 20 right )}}$$
sin(80 + 4*(-3))/((4*sin(20 + (-3)))*sin(70 – (-3)))
$$frac{sin{left (4 (-3) + 80 right )}}{sin{left (- (-3) + 70 right )} 4 sin{left ((-3) + 20 right )}}$$
sin(80 + 4*(-3))/((4*sin(20 – 3))*sin(70 – (-3)))
$$frac{sin{left (-3 cdot 4 + 80 right )}}{4 sin{left (-3 + 20 right )} sin{left (- -3 + 70 right )}}$$
sin(68)/(4*sin(17)*sin(73))
$$frac{sin{left (68 right )}}{4 sin{left (17 right )} sin{left (73 right )}}$$
Степени
$$- frac{sin{left (4 x + 80 right )}}{4 sin{left (x – 70 right )} sin{left (x + 20 right )}}$$
Численный ответ
0.25*sin(80 + 4*x)/(sin(20 + x)*sin(70 – x))
Рациональный знаменатель
$$- frac{sin{left (4 x + 80 right )}}{4 sin{left (x – 70 right )} sin{left (x + 20 right )}}$$
Объединение рациональных выражений
$$- frac{sin{left (4 left(x + 20right) right )}}{4 sin{left (x – 70 right )} sin{left (x + 20 right )}}$$
Общее упрощение
-sin(80 + 4*x)
————————–
4*sin(-70 + x)*sin(20 + x)
$$- frac{sin{left (4 x + 80 right )}}{4 sin{left (x – 70 right )} sin{left (x + 20 right )}}$$
Соберем выражение
$$- frac{1}{4} sin{left (4 x + 80 right )} csc{left (x – 70 right )} csc{left (x + 20 right )}$$
sin(80 + 4*x)
————————-
4*sin(20 + x)*sin(70 – x)
$$frac{sin{left (4 x + 80 right )}}{4 sin{left (- x + 70 right )} sin{left (x + 20 right )}}$$
Общий знаменатель
sin(80 + 4*x)
————————-
4*sin(20 + x)*sin(70 – x)
$$frac{sin{left (4 x + 80 right )}}{4 sin{left (- x + 70 right )} sin{left (x + 20 right )}}$$
Тригонометрическая часть
sin(80 + 4*x)
————————-
4*sin(20 + x)*sin(70 – x)
$$frac{sin{left (4 x + 80 right )}}{4 sin{left (- x + 70 right )} sin{left (x + 20 right )}}$$
Комбинаторика
-sin(80 + 4*x)
————————–
4*sin(-70 + x)*sin(20 + x)
$$- frac{sin{left (4 x + 80 right )}}{4 sin{left (x – 70 right )} sin{left (x + 20 right )}}$$
Раскрыть выражение
sin(80 + 4*x)
————————-
4*sin(20 + x)*sin(70 – x)
$$frac{sin{left (4 x + 80 right )}}{4 sin{left (- x + 70 right )} sin{left (x + 20 right )}}$$
-cos(80)*sin(4*x) – cos(4*x)*sin(80)
———————————————————————–
(cos(70)*sin(x) – cos(x)*sin(70))*(4*cos(20)*sin(x) + 4*cos(x)*sin(20))
$$frac{- sin{left (4 x right )} cos{left (80 right )} – sin{left (80 right )} cos{left (4 x right )}}{left(4 sin{left (x right )} cos{left (20 right )} + 4 sin{left (20 right )} cos{left (x right )}right) left(sin{left (x right )} cos{left (70 right )} – sin{left (70 right )} cos{left (x right )}right)}$$