Question

How to solve `tg^2x-(1+sqrt3)tgx+sqrt3<0`?

Answer 1

`x = pi/3 + kpi`
`x = pi/4 + kpi`

Explanation:

`tan^2 x - (1 + sqrt3)tan x + sqrt3 = 0`
Solve this quadratic equation for tan x.
Use the improved quadratic formula (Socratic, Google Search)
`D = d^2 = b^2 - 4ac = (1 + sqrt3)^2 - 4sqrt3 = 4 - 2sqrt3 = 0.536` --> `d = +- 0.73`
There are 2 real roots:
`tan x = -b/(2a) +- d/(2a) = (1 + sqrt3)/2 +- 0.73/2 = 1.37 +- 0.37`
`tan x = 1.37 + 0.37 = 1.73 = sqrt3`, and
`tan x = 1.37 - 0.37 = 1`
a. `tan x = sqrt3`
Trig table and unit circle give -->
`x = pi/3 + kpi`
b. `tan x = 1`
`x = pi/4 + kpi`