Question

How do you solve `tan^2(3x)=3` and find all exact general solutions?

Answer 1

`x = pi/9 + (kpi)/3`
`x = (2pi)/9 + (kpi)/3`

Explanation:

`tan^2 (3x) = 3`
`tan (3x) = +- sqrt3`
There are 2 solutions. Use trig table and unit circle

a. `tan (3x) = sqrt3 = tan (pi/3)`
`3x = (pi/3) + kpi`
`x = (pi/9) + (kpi)/3`

b. `tan (3x) = -sqrt3 = tan (2pi)/3`
`3x = (2pi)/3 + kpi`
`x = (2pi)/9 + (kpi)/3`
Check by calculator:
`x = (pi/9) = 20^@` --> `3x = 60^@` --> `tan 3x = sqrt3`-->
`tan^2 3x = 3`. OK
`x = (2pi)/9 = 40^@` --> `3x = 120^@` --> `tan 3x = tan 120 = - sqrt3` --> `tan^2 3x = 3`. OK