Question

Solve 3tan^2 x-1=0 for all solutions [0, 2pi]?

Answer 1

`x in {pi/6, (5pi)/6, (7pi)/6, (11pi)/6}`

Explanation:

`3tan^2(x) - 1 = 0`

`=> tan^2(x) = 1/3`

`=> tan(x) = +-1/sqrt(3)`

If we check the unit circle, we find that `|tan(x)| = 1/sqrt(3)` when `|sin(x)| = 1/2` and `|cos(x)| = sqrt(3)/2`, that is, at `x = pi/6+npi` or `x=-pi/6+npi`.

Finding what values for `n` put these within the interval `[0, 2pi)`, we get `n in {0, 1}` for `x=pi/6+npi` and `n in {1, 2}` for `x = -pi/6+npi`. Thus, our total solution set is

`x in {pi/6, (5pi)/6, (7pi)/6, (11pi)/6}`