Уравнение, часть С. . Created by SashaKrasic. algebra-ru

Уравнение, часть С. ...

SashaKrasic @SashaKrasic

August 2022 1 19 Report

Уравнение, часть С.

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kalbim

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а) (1/25)^(sin^2(x)) = 5^(-2sin^2(x))
25^((sin2x)/2) = 5^(sin2x) = 5^(2*sinx*cosx)
5^(cos2x) = 5^(1-sin^2(x))
5^(1-2sin^2(x)) + 4*5^(1-2sin^2(x)) = 5^(2*sinx*cosx)
5^(2-2sin^2(x)) = 5^(2*sinx*cosx) -
т.к. равны основания степени, значит показатели степени тоже равны:
2-2sin^2(x) = 2*sinx*cosx
1 - sin^2(x) = sinx*cosx
sin^2(x) + cos^2(x) - sin^2(x) = sinx*cosx
cos^2(x) = sinx*cosx
cosx*(cosx - sinx) = 0
1) cosx = 0, x=pi/2 + pi*k
2) sinx=cosx, tgx=1, x=pi/4 + pi*k
б) x [0.5; 3pi/2]
k=0, x=pi/2, x=pi/4
k=1, x=pi/2+pi = 3pi/2, x=pi/4+pi = 5pi/4
Ответ: pi/2, pi/4, 5pi/4, 3pi/2

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