Question

Use inverse functions where needed to find all solutions of the equation in the interval [0, 2π). (Enter your answers as a comma-separated list.) tan2 x + tan x − 30 = 0

Answer 1

I'm assuming you meant:


`tan^(2)x + tanx - 30 = 0`
Let
`u = tan(x)`

Thus, we can rewrite it into a quadratic.


`u^(2) + u - 30 = 0`

`(u + 6)(u - 5) = 0`

`u = -6 or u = 5`

Hence,
`tan(x) = -6` or
`tan(x) = 5`

`x = tan^(-1)(-6)` or
`x = tan^(-1)(5)`

Rewriting it in general form, we get:

`x = \pi \cdot n + tan^(-1)(5), n \in Z`

`x = \pi \cdot n - tan^(-1)(6), n \in Z`

From there, you can find your solutions from
`[0, 2\pi)`