cos^2 4x+sin^2 2x-1=0. Created by supermee. algebra-ru

cos^2 4x+sin^2 2x-1=0...

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August 2022 1 16 Report

cos^2 4x+sin^2 2x-1=0

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kirichekov

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Cos²4x+sin²2x-1=0
(cos(2*2x))²+sin²2x-1=0
(1-2sin²2x)²+sin²2x-1=0
1-4sin²2x+4sin⁴2x+sin²2x-1=0
4sin⁴2x-3sin²2x=0
sin²2x*(4sin²2x-3)=0

sin²2x=0 или 4sin²2x-3=0
1. sin²2x=0, sin2x=0. 2x=πn, n∈Z. x=πn/2, n∈Z
2. 4sin²2x-3=0, sin²2x=3/4. sin2x=+-√(3/4). sin2x=+-√3/2
a. sin2x=-√3/2
$2x=(-1) ^{n}*arcsin(- \frac{ \sqrt{3} }{2} )+ \pi n,$ n∈Z
$2x= (-1)^{n+1}* \frac{ \pi }{3} + \pi n,$ n∈Z
$x= (-1)^{n+1}* \frac{ \pi }{6} + \frac{ \pi n}{2} ,$ n∈Z
b. sin2x=√3/2
$x= (-1)^{n}* \frac{ \pi }{6} + \frac{ \pi n}{2},$ n∈z

ответ:
$x_{1} = \frac{ \pi n}{2} ,

 x_{2} = (-1)^{n+1}* \frac{ \pi }{6} + \frac{ \pi n}{2} ,

 x_{3} = (-1)^{n}* \frac{ \pi }{6} + \frac{ \pi n}{2}$ n∈Z

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