Дано
$$frac{v}{g} sqrt{h g 2 cos^{2}{left (a right )} + frac{1}{4} sin^{2}{left (2 a right )}} + frac{v^{2}}{g} cos{left (2 a right )}$$
Подстановка условия
$$frac{v}{g} sqrt{h g 2 cos^{2}{left (a right )} + frac{1}{4} sin^{2}{left (2 a right )}} + frac{v^{2}}{g} cos{left (2 a right )}$$
(v/g)*sqrt(sin(2*(-2))^2/4 + ((2*cos((-2))^2)*g)*h) + (v^2*cos(2*(-2)))/g
$$frac{v}{g} sqrt{h g 2 cos^{2}{left ((-2) right )} + frac{1}{4} sin^{2}{left (2 (-2) right )}} + frac{v^{2}}{g} cos{left (2 (-2) right )}$$
(v/g)*sqrt(sin(2*(-2))^2/4 + ((2*cos(-2)^2)*g)*h) + (v^2*cos(2*(-2)))/g
$$frac{v}{g} sqrt{h g 2 cos^{2}{left (-2 right )} + frac{1}{4} sin^{2}{left (-2 cdot 2 right )}} + frac{v^{2}}{g} cos{left (-2 cdot 2 right )}$$
v*sqrt(sin(4)^2/4 + 2*g*h*cos(2)^2)/g + v^2*cos(4)/g
$$frac{v^{2}}{g} cos{left (4 right )} + frac{v}{g} sqrt{2 g h cos^{2}{left (2 right )} + frac{1}{4} sin^{2}{left (4 right )}}$$
Степени
$$frac{v^{2}}{g} cos{left (2 a right )} + frac{v}{g} sqrt{2 g h cos^{2}{left (a right )} + frac{1}{4} sin^{2}{left (2 a right )}}$$
Численный ответ
v*(0.25*sin(2*a)^2 + 2.0*g*h*cos(a)^2)^0.5/g + v^2*cos(2*a)/g
Рациональный знаменатель
$$frac{1}{2 g^{2}} left(2 g v^{2} cos{left (2 a right )} + g v sqrt{8 g h cos^{2}{left (a right )} + sin^{2}{left (2 a right )}}right)$$
Объединение рациональных выражений
$$frac{v}{2 g} left(2 v cos{left (2 a right )} + sqrt{8 g h cos^{2}{left (a right )} + sin^{2}{left (2 a right )}}right)$$
Общее упрощение
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| / 2 2 |
v*/ sin (2*a) + 8*g*h*cos (a) + 2*v*cos(2*a)/
————————————————-
2*g
$$frac{v}{2 g} left(2 v cos{left (2 a right )} + sqrt{8 g h cos^{2}{left (a right )} + sin^{2}{left (2 a right )}}right)$$
Соберем выражение
$$frac{v^{2}}{g} cos{left (2 a right )} + frac{sqrt{2} v}{4 g} sqrt{8 g h cos{left (2 a right )} + 8 g h – cos{left (4 a right )} + 1}$$
Общий знаменатель
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/ 2 2 2
v*/ sin (2*a) + 8*g*h*cos (a) + 2*v *cos(2*a)
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2*g
$$frac{1}{2 g} left(2 v^{2} cos{left (2 a right )} + v sqrt{8 g h cos^{2}{left (a right )} + sin^{2}{left (2 a right )}}right)$$
Тригонометрическая часть
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/ 2 / 2 2
v*/ cos (a)*sin (a) + 2*g*h/ v *cos(2*a)
——————————– + ———–
g g
$$frac{v^{2}}{g} cos{left (2 a right )} + frac{v}{g} sqrt{left(2 g h + sin^{2}{left (a right )}right) cos^{2}{left (a right )}}$$
Комбинаторика
/ ___________________________
| / 2 2 |
v*/ sin (2*a) + 8*g*h*cos (a) + 2*v*cos(2*a)/
————————————————-
2*g
$$frac{v}{2 g} left(2 v cos{left (2 a right )} + sqrt{8 g h cos^{2}{left (a right )} + sin^{2}{left (2 a right )}}right)$$
Раскрыть выражение
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/ 2 2 2 2 / 2 2
v*/ cos (a)*sin (a) + 2*g*h*cos (a) v *cos (a) – sin (a)/
————————————– + ———————-
g g
$$frac{v^{2}}{g} left(- sin^{2}{left (a right )} + cos^{2}{left (a right )}right) + frac{v}{g} sqrt{2 g h cos^{2}{left (a right )} + sin^{2}{left (a right )} cos^{2}{left (a right )}}$$
