Question

Find all solutions to the following equations that are on the interval [0,2pie] by solving for theta

Answer 1

Given:

`cos(2\theta)sin(\theta)+sin(\theta)=0`

let us take the common term out,

`\begin{gathered} sin(\theta)(cos(2\theta)+1)=0 \\ \end{gathered}`

Now, let us solve seperately,

`\begin{gathered} \sin (\theta)=0 \\ \theta=2\pi n \end{gathered}`

`\begin{gathered} \cos (2\theta)+1=0 \\ \theta=(\pi)/(2)+\pi n \end{gathered}`

The solutions are,

`\theta=2\pi n,\: \theta=\pi+2\pi n,\: \theta=(\pi)/(2)+\pi n`