log^2 (2) (cos^2x)-22log2(cosx)-12=0 решить. Created by kristinka2010g. algebra-ru

ОДЗ : cos x > 0Замена переменной

log^2 (2) (cos^2x)-22log2(cosx)-12=0 решить...

kristinka2010g @kristinka2010g

July 2022 1 25 Report

log^2 (2) (cos^2x)-22log2(cosx)-12=0 решить

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$log^2_2 (cos^2x)-22\cdot log_2(cosx)-12=0$

ОДЗ : cos x > 0

$log^2_2 (cos^2x)-11 \cdot log_2(cos^2x)-12=0$

Замена переменной $y = log_2(cos^2x)$

$y^2 - 11y -12 = 0~~~\Leftrightarrow~~~ (y-12)(y+1)=0\\ \\ 1)~~y-12 = 0;~y=12\\ ~~~log_2(cos^2x)=12; ~~cos^2x=2^{12}=4096\\ ~~~cos^2x\leq 1;~~~\Rightarrow~~~x \in \varnothing\\ \\ 2) y+1 =0;~~y=-1\\ ~~~log_2(cos^2x)=-1; ~~cos^2x=2^{-1}=\dfrac{1}{2}\\ \\ ~~~cos~x=\pm \dfrac {1}{\sqrt{2} };~~cos~x>0~(ODZ) ~~\Rightarrow\\ \\ ~~~cos~x= \dfrac {1}{\sqrt{2} };~~\boxed{\boldsymbol {x=\pm \dfrac{\pi}{4}+2\pi n; ~n\in Z}}$

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