Question

HELP ON THIS PROBLEM!!

Solve the following 0deg< x< 360deg: cos x= cos 2x

Answer 1

`x=0^\circ,120^\circ,240^\circ,360^\circ`

Explanation:

Use identities to set the equation up as a quadratic

`\cos x=\cos 2x;\:0^\circ \leq x \leq 360^\circ\\\\\cos x=2\cos^2 x-1\\\\0=2\cos^2 x-\cos x-1`

Make the substitution u=cos(x) and solve the quadratic

`0=2u^2-u-1\\\\0=(2u+1)(u-1)`

`\displaystyle 0=2u+1\\\\0=2\cos x+1\\\\-1=2\cos x\\\\-(1)/(2)=\cos x\\\\120^\circ,240^\circ=x`

`0=u-1\\\\0=\cos x-1\\\\1=\cos x\\\\0^\circ,360^\circ=x`

Hence,
`x=0^\circ,120^\circ,240^\circ,360^\circ`