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Консультация № 152472

01.12.2008, 09:10

0.00 руб.

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Образование

Помогите пожалуйста.УВажаемые эксперты..
8sin^6 x=4-7sin(5П/2+2x)+3cos 4x, (-П,5П/4)

P.S. 8sin^6 x 8 умножить на синус в 6 степени

Обсуждение

Неизвестный

03.12.2008, 09:29

общий

это ответ

Здравствуйте, Iriha!
8sin⁶x=4-7sin(5pi/2+2x)+3cos4x, -pi<x<5pi/4
8*((1-cos2x)/2)³=4-7sin(2pi+pi/2+2x)+3(cos²2x-sin²2x)
(1-cos2x)³=4-7sin(pi/2+2x)+3cos²2x-3sin²2x
1-3cos2x+3cos²2x-cos³2x=4-7cos2x+3cos²2x-3(1-cos²2x)
cos³2x+3cos²2x-4cos2x=0
cos2x(cos²2x+3cos2x-4)=0
cos2x=0
2x=pi/2+pi*n,
x=pi/4+pi*n/2, n - целое число
cos²2x+3cos2x-4=0
cos2x=t
t²+3t-4=0
D=9+16=25
t₁=(-3+5)/2=1
t₂=(-3-5)/2=-4
Поскольку t=cos2x, |t|>=1, то рассматриваем только t₁=1
cos2x=1
2x=2*pi*n
x=pi*n
Значит решения данного уравнения
x=pi/4+pi*n/2, n - целое число
x=pi*n
В рассматриваемом интервале находятся следующие корни уравнения:
x₁=-3pi/4
x₂=-pi/4
x₃=0
x₄=pi/4
x₅=3pi/4
x₆=pi

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