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If cos 3x + cos x = 0, find all possible values of cos x.

 Dec 11, 2019
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If cos 3x + cos x = 0, find all possible values of cos x.

 

Hello Guest!

 

\({\color{blue}cos( 3x)=4cos^3( x)-3cos( x)}=(4cos^2(x)-3)\cdot cos(x)\)  \(Bartsch\ Leipzig\ 1980\)

\(cos 3x + cos x = 0\)

\((4cos^2(x)-3)\cdot cos(x)+cos(x)=0\)

 

\(\large cos(x)=u\)

 

\((4u^2-3)\cdot u+u=0\\ 4u^3-3u+u=0\\ u(4u^2-3+1)=0\)

\(u_1=0\)

\(4u^2-2=0\\ u^2=\frac{1}{2}\)

\(u_2=+\sqrt{\frac{1}{2}}=\frac{1}{2}\sqrt{2}\\ u_3=-\sqrt{\frac{1}{2}}=-\frac{1}{2}\sqrt{2}\)

\(cos(x_1)=0\\ cos(x_2)=\frac{1}{2}\sqrt{2}\\ cos(x_3)=-\frac{1}{2}\sqrt{2}\)

 

laugh  !

 Dec 11, 2019
edited by asinus  Dec 11, 2019
edited by asinus  Dec 11, 2019
edited by asinus  Dec 11, 2019

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