Question

How do you find the exact value of `tanx-3cotx=0` in the interval `0<=x<360^@`?

Answer 1

`x\in {60^@, 120^@, 240^@, 300^@}`

Explanation:

It goes as follows:
`tan x-3cot x=0`
`tan x=3cot x`
`tan x=3\cdot 1/tan x`
`tan^2 x=3`
`tan x=sqrt 3` or `tan x=-sqrt 3`
`x=60^@+k cdot 180^@` or `x=120^@+k cdot 180^@` where `k` is an integer

Now we can simply list all solutions that fall between `0^@` and `360^@`:
`x\in {60^@, 120^@, 240^@, 300^@}`

Note: at the very begining we should also include the domain of this equation; we want the `tan x` and `cot x` to exist so:
`x\in\mathbb{R}\setminus {k cdot 90^@ | k " is integer"}`
Fortunately enough, the solutions that we find fall into the domain.