Дано
$$sqrt{sin^{2}{left (x right )} 81 cos^{4}{left (x right )} + 81 sin^{4}{left (x right )} cos^{2}{left (x right )} + 9}$$
Подстановка условия
$$sqrt{sin^{2}{left (x right )} 81 cos^{4}{left (x right )} + 81 sin^{4}{left (x right )} cos^{2}{left (x right )} + 9}$$
sqrt((81*cos((-1/4))^4)*sin((-1/4))^2 + (81*sin((-1/4))^4)*cos((-1/4))^2 + 9)
$$sqrt{sin^{2}{left ((-1/4) right )} 81 cos^{4}{left ((-1/4) right )} + 81 sin^{4}{left ((-1/4) right )} cos^{2}{left ((-1/4) right )} + 9}$$
sqrt((81*cos(-1/4)^4)*sin(-1/4)^2 + (81*sin(-1/4)^4)*cos(-1/4)^2 + 9)
$$sqrt{81 sin^{4}{left (- frac{1}{4} right )} cos^{2}{left (- frac{1}{4} right )} + sin^{2}{left (- frac{1}{4} right )} 81 cos^{4}{left (- frac{1}{4} right )} + 9}$$
sqrt(9 + 81*cos(1/4)^2*sin(1/4)^4 + 81*cos(1/4)^4*sin(1/4)^2)
$$sqrt{81 sin^{4}{left (frac{1}{4} right )} cos^{2}{left (frac{1}{4} right )} + 81 sin^{2}{left (frac{1}{4} right )} cos^{4}{left (frac{1}{4} right )} + 9}$$
Степени
$$sqrt{81 sin^{4}{left (x right )} cos^{2}{left (x right )} + 81 sin^{2}{left (x right )} cos^{4}{left (x right )} + 9}$$
Численный ответ
(9.0 + 81.0*cos(x)^2*sin(x)^4 + 81.0*cos(x)^4*sin(x)^2)^0.5
Рациональный знаменатель
$$sqrt{81 sin^{4}{left (x right )} cos^{2}{left (x right )} + 81 sin^{2}{left (x right )} cos^{4}{left (x right )} + 9}$$
Объединение рациональных выражений
$$3 sqrt{9 left(sin^{2}{left (x right )} + cos^{2}{left (x right )}right) sin^{2}{left (x right )} cos^{2}{left (x right )} + 1}$$
Общее упрощение
___________________________
/ 4 2
3*/ 1 – 9*sin (x) + 9*sin (x)
$$3 sqrt{- 9 sin^{4}{left (x right )} + 9 sin^{2}{left (x right )} + 1}$$
Соберем выражение
$$sqrt{- frac{81}{8} cos{left (4 x right )} + frac{153}{8}}$$
Общий знаменатель
___________________________________________
/ 2 4 4 2
3*/ 1 + 9*cos (x)*sin (x) + 9*cos (x)*sin (x)
$$3 sqrt{9 sin^{4}{left (x right )} cos^{2}{left (x right )} + 9 sin^{2}{left (x right )} cos^{4}{left (x right )} + 1}$$
Тригонометрическая часть
_______________________
/ 81*(1 – cos(4*x))
/ 9 + —————–
/ 8
$$sqrt{frac{1}{8} left(- 81 cos{left (4 x right )} + 81right) + 9}$$
Комбинаторика
___________________________________________
/ 2 4 4 2
3*/ 1 + 9*cos (x)*sin (x) + 9*cos (x)*sin (x)
$$3 sqrt{9 sin^{4}{left (x right )} cos^{2}{left (x right )} + 9 sin^{2}{left (x right )} cos^{4}{left (x right )} + 1}$$
Раскрыть выражение
_____________________________________________
/ 2 4 4 2
/ 9 + 81*cos (x)*sin (x) + 81*cos (x)*sin (x)
$$sqrt{81 sin^{4}{left (x right )} cos^{2}{left (x right )} + 81 sin^{2}{left (x right )} cos^{4}{left (x right )} + 9}$$
