Question

The equation to cos squared theta equals cos data has Solutions of

Answer 1

Given the domain

`0\le\theta<2\pi`

The equation

`2\cos ^2\theta=\cos \theta`

To simplify,

`\begin{gathered} \text{let} \\ x=\cos \theta \end{gathered}`

so that

`2x^2=x`

Then we will have

`2x^2-x=0`

We will have

`\begin{gathered} x(2x-1)=0 \\ \text{Thus} \\ x=0 \\ \text{and} \\ x=(1)/(2) \end{gathered}`

Hence

`\begin{gathered} \cos \theta=0 \\ \text{and} \\ \cos \theta=(1)/(2) \end{gathered}`

We can find the values of θ as follow

when cosθ = 0

`\theta=(\pi)/(2),(3\pi)/(2)`

When

`\begin{gathered} \cos \theta=(1)/(2) \\ \theta=(\pi)/(3),(5\pi)/(3) \end{gathered}`

Thus, the answer is:

`\begin{gathered} \text{Option A} \\ (\pi)/(3),(\pi)/(2),(3\pi)/(2),(5\pi)/(3) \end{gathered}`