Trigonometric Equations
Solve:

In the interval [0,2pi)
We have to find all the values of x that make equality stand. First, divide by 3:

Subtract tan x

Factor tan x out:

One solution comes immediately:
tan x = 0
There are two angles whose tangent is 0:

The other solutions come when equating:

Adding 1, and dividing by 3:

Taking the square root:
![\tan x=\sqrt[\square]{(1)/(3)}=\pm\frac{\sqrt[]{3}}{3}](https://img.qammunity.org/2023/formulas/mathematics/college/jg9df19a6e5b057syfgq2oqbfn1nxjsjd1.png)
The positive answer gives us two solutions:
![\tan x=\frac{\sqrt[]{3}}{3}](https://img.qammunity.org/2023/formulas/mathematics/college/odhlhjmsjjcxs8qt0gwpv7sgbx6ywt2mi5.png)
x=pi/6 and x=7pi/6
The negative answer also gives us two solutions:
![\tan x=-\frac{\sqrt[]{3}}{3}](https://img.qammunity.org/2023/formulas/mathematics/college/i3qm7fd4p4i7snp9x3lxt38elqeqbn11ql.png)
x=5pi/6, 11pi/6
Summarizing the solutions are:
{