Question

Find all theta in the interval [0, 2pi) that satisfy the equation.

2costheta*tantheta + tantheta = 0

Answer 1

`2\cos\theta*\tan\theta + \tan\theta = 0 \\ \\2\cos\theta* (\sin\theta)/(\cos\theta) + (\sin\theta)/(\cos\theta) = 0 \\ \\ 2\sin\theta + (\sin\theta)/(\cos\theta) = 0 \\ \\ (2sin(\theta)cos(\theta))/(cos(\theta)) + (\sin\theta)/(\cos\theta) = 0 \\ \\ (2sin(\theta)cos(\theta)+sin(\theta))/(cos(\theta)) =0 \\ \\ 2sin(\theta)cos(\theta)+sin(\theta)=0 \\ \\ \sin\theta(2\cos\theta+1)=0 \\ \\ \sin\theta=0 \Rightarrow \theta=0, \pi,2\pi \\ \\2\cos\theta=-1 \\ \\\cos\theta=- (1)/(2) \Rightarrow \theta= (2\pi)/(3) , (4\pi)/(3)`


Therefore,
`\theta=0, \pi,2\pi ,(2\pi)/(3) , (4\pi)/(3)`