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

1 Answer
Jun 13, 2018

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