Question

Find all solutions to the equationin the interval [O, 27). Enter thesolutions in increasing order.cos 2x = cos X[?]Tx = 0,2Remember: cos 20 = cos20 – sin20

Answer 1

From

`\begin{gathered} \cos 2x=\cos x \\ \cos ^2x-\sin ^2x=\cos x \\ \cos ^2x-(1-\cos ^2x)=\cos x \\ 2\cos ^2x-1=\cos x \\ 2\cos ^2x-\cos x-1=0 \\ \text{From the quadratic formula} \\ \cos x=\frac{1\pm\sqrt[]{1-(-8)}}{4} \\ \\ \cos x=(1\pm3)/(4) \\ \cos x=\text{ 1 or -}(1)/(2) \\ \text{Taking the cos}^(-1)of\text{ 1 and -}(1)/(2) \\ We\text{ have }\theta\text{ = 0, }(2\pi)/(3),(4\pi)/(3),(8\pi)/(3)\ldots\ldots\ldots2\pi \end{gathered}`

So your answer is

`0,\text{ }(2\pi)/(3),\text{ }(4\pi)/(3)`