sqrt((8*sin(2*t)*cos(t))^2+(-4*cos(t)^3)^2+(64*cos(t)^2)^2) если t=-1 (упростите выражение)
Решение
Вы ввели
[LaTeX]
______________________________________________________ / 2 2 / 2 / 3 \ / 2 \ \/ (8*sin(2*t)*cos(t)) + \-4*cos (t)/ + \64*cos (t)/
$$\sqrt{\left(8 \sin{\left (2 t \right )} \cos{\left (t \right )}\right)^{2} + \left(- 4 \cos^{3}{\left (t \right )}\right)^{2} + \left(64 \cos^{2}{\left (t \right )}\right)^{2}}$$
Подстановка условия
[LaTeX]
sqrt(((8*sin(2*t))*cos(t))^2 + (-4*cos(t)^3)^2 + (64*cos(t)^2)^2) при t = -1
sqrt(((8*sin(2*t))*cos(t))^2 + (-4*cos(t)^3)^2 + (64*cos(t)^2)^2)
$$\sqrt{\left(8 \sin{\left (2 t \right )} \cos{\left (t \right )}\right)^{2} + \left(- 4 \cos^{3}{\left (t \right )}\right)^{2} + \left(64 \cos^{2}{\left (t \right )}\right)^{2}}$$
sqrt(((8*sin(2*(-1)))*cos((-1)))^2 + (-4*cos((-1))^3)^2 + (64*cos((-1))^2)^2)
$$\sqrt{\left(8 \sin{\left (2 (-1) \right )} \cos{\left ((-1) \right )}\right)^{2} + \left(- 4 \cos^{3}{\left ((-1) \right )}\right)^{2} + \left(64 \cos^{2}{\left ((-1) \right )}\right)^{2}}$$
sqrt(((8*sin(2*(-1)))*cos(-1))^2 + (-4*cos(-1)^3)^2 + (64*cos(-1)^2)^2)
$$\sqrt{\left(- 4 \cos^{3}{\left (-1 \right )}\right)^{2} + \left(8 \sin{\left (-1 \cdot 2 \right )} \cos{\left (-1 \right )}\right)^{2} + \left(64 \cos^{2}{\left (-1 \right )}\right)^{2}}$$
sqrt(16*cos(1)^6 + 4096*cos(1)^4 + 64*cos(1)^2*sin(2)^2)
$$\sqrt{16 \cos^{6}{\left (1 \right )} + 64 \sin^{2}{\left (2 \right )} \cos^{2}{\left (1 \right )} + 4096 \cos^{4}{\left (1 \right )}}$$
Степени
[LaTeX]
__________________________________________________ / 6 4 2 2 \/ 16*cos (t) + 4096*cos (t) + 64*cos (t)*sin (2*t)
$$\sqrt{64 \sin^{2}{\left (2 t \right )} \cos^{2}{\left (t \right )} + 16 \cos^{6}{\left (t \right )} + 4096 \cos^{4}{\left (t \right )}}$$
Численный ответ
[LaTeX]
(16.0*cos(t)^6 + 4096.0*cos(t)^4 + 64.0*cos(t)^2*sin(2*t)^2)^0.5
Рациональный знаменатель
[LaTeX]
__________________________________________________ / 6 4 2 2 \/ 16*cos (t) + 4096*cos (t) + 64*cos (t)*sin (2*t)
$$\sqrt{64 \sin^{2}{\left (2 t \right )} \cos^{2}{\left (t \right )} + 16 \cos^{6}{\left (t \right )} + 4096 \cos^{4}{\left (t \right )}}$$
Объединение рациональных выражений
[LaTeX]
_______________________________________________
/ 2 / 4 2 2 \
4*\/ cos (t)*\cos (t) + 4*sin (2*t) + 256*cos (t)/
$$4 \sqrt{\left(4 \sin^{2}{\left (2 t \right )} + \cos^{4}{\left (t \right )} + 256 \cos^{2}{\left (t \right )}\right) \cos^{2}{\left (t \right )}}$$
Общее упрощение
[LaTeX]
____________________________
/ 4 / 2 \
4*\/ cos (t)*\272 - 15*cos (t)/
$$4 \sqrt{\left(- 15 \cos^{2}{\left (t \right )} + 272\right) \cos^{4}{\left (t \right )}}$$
Собрать выражение
[LaTeX]
_____________________________________________________ / 2 / / 3 \ 2 4 \/ \-4*cos (t)/ + (8*sin(2*t)*cos(t)) + 4096*cos (t)
$$\sqrt{\left(8 \sin{\left (2 t \right )} \cos{\left (t \right )}\right)^{2} + 4096 \cos^{4}{\left (t \right )} + \left(- 4 \cos^{3}{\left (t \right )}\right)^{2}}$$
___________________________________________________ / 15*cos(6*t) 4127*cos(2*t) / 1557 + 499*cos(4*t) - ----------- + ------------- \/ 2 2
$$\sqrt{\frac{4127}{2} \cos{\left (2 t \right )} + 499 \cos{\left (4 t \right )} - \frac{15}{2} \cos{\left (6 t \right )} + 1557}$$
Общий знаменатель
[LaTeX]
_____________________________________________
/ 6 4 2 2
4*\/ cos (t) + 256*cos (t) + 4*cos (t)*sin (2*t)
$$4 \sqrt{4 \sin^{2}{\left (2 t \right )} \cos^{2}{\left (t \right )} + \cos^{6}{\left (t \right )} + 256 \cos^{4}{\left (t \right )}}$$
Тригонометрическая часть
[LaTeX]
______________________________ / 6 4 \/ - 240*cos (t) + 4352*cos (t)
$$\sqrt{- 240 \cos^{6}{\left (t \right )} + 4352 \cos^{4}{\left (t \right )}}$$
Комбинаторика
[LaTeX]
_______________________________________________
/ 2 / 4 2 2 \
4*\/ cos (t)*\cos (t) + 4*sin (2*t) + 256*cos (t)/
$$4 \sqrt{\left(4 \sin^{2}{\left (2 t \right )} + \cos^{4}{\left (t \right )} + 256 \cos^{2}{\left (t \right )}\right) \cos^{2}{\left (t \right )}}$$
Раскрыть выражение
[LaTeX]
__________________________________________________ / 6 4 2 2 \/ 16*cos (t) + 4096*cos (t) + cos (t)*64*sin (2*t)
$$\sqrt{64 \sin^{2}{\left (2 t \right )} \cos^{2}{\left (t \right )} + 16 \cos^{6}{\left (t \right )} + 4096 \cos^{4}{\left (t \right )}}$$
_________________________________________________ / 6 4 4 2 \/ 16*cos (t) + 4096*cos (t) + 256*cos (t)*sin (t)
$$\sqrt{256 \sin^{2}{\left (t \right )} \cos^{4}{\left (t \right )} + 16 \cos^{6}{\left (t \right )} + 4096 \cos^{4}{\left (t \right )}}$$