sqrt((8*sin(2*t)*cos(t))^2+(-4*cos(t)^3)^2+(64*cos(t)^2)^2) если t=-1 (упростите выражение)

Вы ввели

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   /                                    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

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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

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  /       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

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(16.0*cos(t)^6 + 4096.0*cos(t)^4 + 64.0*cos(t)^2*sin(2*t)^2)^0.5

Рациональный знаменатель

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  /       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

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    /    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

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     ____________________________
    /    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

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    _____________________________________________________
   /             2                                       
  /  /      3   \                       2           4    
\/   \-4*cos (t)/  + (8*sin(2*t)*cos(t))  + 4096*cos (t) 

$$\sqrt{\left(8 \sin{\left (2 t \right )} \cos{\left (t \right )}\right)^{2} + 4096 \cos^{4}{\left (t \right )} + \left(- 4 \cos^{3}{\left (t \right )}\right)^{2}}$$

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   /                       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

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    /    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

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  /          6              4    
\/  - 240*cos (t) + 4352*cos (t) 

$$\sqrt{- 240 \cos^{6}{\left (t \right )} + 4352 \cos^{4}{\left (t \right )}}$$

Комбинаторика

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    /    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

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  /       6              4         2          2      
\/  16*cos (t) + 4096*cos (t) + cos (t)*64*sin (2*t) 

$$\sqrt{64 \sin^{2}{\left (2 t \right )} \cos^{2}{\left (t \right )} + 16 \cos^{6}{\left (t \right )} + 4096 \cos^{4}{\left (t \right )}}$$

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  /       6              4             4       2    
\/  16*cos (t) + 4096*cos (t) + 256*cos (t)*sin (t) 

$$\sqrt{256 \sin^{2}{\left (t \right )} \cos^{4}{\left (t \right )} + 16 \cos^{6}{\left (t \right )} + 4096 \cos^{4}{\left (t \right )}}$$