Solve this quadratic equation for tan x, by using the improved quadratic formula (Socratic Search):
tan^2 x + 3tan x + 1 = 0tan2x+3tanx+1=0
D = d^2 = b^2 - 4ac = 9 - 4 = 5D=d2=b2−4ac=9−4=5 --> d = +- sqrt5d=±√5
There are 2 real roots:
tan x = -b/(2a) +- d/(2a) = -3/2 +- sqrt5/2tanx=−b2a±d2a=−32±√52
tan x1 = (-3 + sqrt5)/2 = - 0.76/2 = - 0.38tanx1=−3+√52=−0.762=−0.38
tan x2 = (-3 - sqrt5)/2 = 22.26/2 = 11.12tanx2=−3−√52=22.262=11.12
Use calculator and unit circle -->
tan x1 = - 0.38 --> x1 = - 20^@90x1=−20∘90 and
x1 = - 20.90 + 180 = 159^@90x1=−20.90+180=159∘90
The arc (-20^@90)(−20∘90) is co-terminal to arc: (360 - 20.90 = 339^@0)(360−20.90=339∘0)
tan x2 = 11.12 --> x2 = 84^@86x2=84∘86 and
x2 = 84.86 + 180 = 264^@86x2=84.86+180=264∘86
Answers for (0, 360)(0,360):
20^@90, 84^@86, 264^@86, 339^@1020∘90,84∘86,264∘86,339∘10
