Question

корень 3cos^2x-0,5sin2x=0

Answer 1

Два разных варианта прочтения условия задачи.

1.

$sqrt{3}cos^2x-0,5sin(2x)=0\sqrt{3}cos^2x-0,5cdot 2 sin x cos x=0\sqrt{3}cos^2x-sin x cos x=0\cos x(sqrt{3}cos x-sin x)=0\\1)~cos x=0;~~boxed{x_1=dfrac{pi}2+pi n,~n in Z}$

$2)~sqrt{3}cos x-sin x=0~~~Leftrightarrow~~~sin x-sqrt{3}cos x=0~~Big|cdot dfrac 12\\~~~~dfrac12 sin x-dfrac{sqrt{3}}2 cos x=0\\~~~~cos Big(dfrac{pi}3Big) sin x-sin Big(dfrac{pi}3Big) cos x=0\\~~~~sin Big(x-dfrac{pi}3 Big)=0;~~~~boxed{x_2=dfrac {pi}3+pi k,~k in Z}$

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2.

$sqrt{3cos^2x-0,5sin2x}=0\3cos^2x-0,5sin2x=0\3cos^2x-0,5cdot 2 sin x cos x=0\3cos^2x-sin x cos x=0\cos x(3cos x-sin x)=0\\1)~cos x=0;~~boxed{x_1=dfrac{pi}2+pi n,~n in Z}\\2)~3cos x-sin x=0~~|:sin x neq 0\\~~~~dfrac{3cos x}{sin x}-dfrac{sin x}{sin x}=0~~~Leftrightarrow~~~3ctg~x-1=0\\~~~~ctg~x=dfrac 13;~~~~boxed{x_2=arcctgBig(dfrac 13Big)+pi k,~k in Z}$

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Использованы формулы

sin (2α) = 2 sin α · cos α

sin α · cos β - sin β · cos α = sin (α - β)

Answer 2

Объяснение:

$sqrt{3}cos^2x-0.5sin 2x=0\ \ sqrt{3}cos^2x-0.5cdot 2sin xcos x=0\ \ sqrt{3}cos^2x-sin xcos x=0\ \ cos x(sqrt{3}cos x-sin x)=0\ \ cos x=0~~~~Longleftrightarrow~~~~ x_1=frac{pi}{2}+pi n,n in Z\ \ sqrt{3}cos x-sin x=0~~~Longleftrightarrow~~~ {rm tg}, x=sqrt{3}~~~Longleftrightarrow~~~ x_2=frac{pi}{3}+pi n,n in Z$