Question

Find all solutions of each equation on the interval [0,2pie). 4cos^2x-1=0

Answer 1

\(\bf 4cos^2(x)-1=0\implies 4cos^2(x)=1\implies cos^2(x)=\cfrac{1}{4}\\ \quad \\
cos(x)=\pm\sqrt{\cfrac{1}{4}}\implies cos(x)=\cfrac{\pm\sqrt{1}}{\pm\sqrt{4}}\implies cos(x)=\pm\cfrac{1}{2}\quad {\color{blue}{ take\quad cos^{-1}}}\\ \quad \\
cos^{-1}[cos(x)]=cos^{-1}\left(\pm\cfrac{1}{2}\right)\implies x=cos^{-1}\left(\pm\cfrac{1}{2}\right)\)