+15127 If cos 3x + cos x = 0, find all possible values of cos x.
Hello Guest!
\({\color{blue}cos( 3x)=4cos^3( x)-3cos( x)}=(4cos^2(x)-3)\cdot cos(x)\) \(Bartsch\ Leipzig\ 1980\)
\(cos 3x + cos x = 0\)
\((4cos^2(x)-3)\cdot cos(x)+cos(x)=0\)
\(\large cos(x)=u\)
\((4u^2-3)\cdot u+u=0\\ 4u^3-3u+u=0\\ u(4u^2-3+1)=0\)
\(u_1=0\)
\(4u^2-2=0\\ u^2=\frac{1}{2}\)
\(u_2=+\sqrt{\frac{1}{2}}=\frac{1}{2}\sqrt{2}\\ u_3=-\sqrt{\frac{1}{2}}=-\frac{1}{2}\sqrt{2}\)
\(cos(x_1)=0\\ cos(x_2)=\frac{1}{2}\sqrt{2}\\ cos(x_3)=-\frac{1}{2}\sqrt{2}\)
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