If 0 < x < π, and cos x + sin x = 1/2, then tan x is?

A: (4 - root7) / 3
B: -(4 + root7) / 3
C: (1 + root7) / 4
D: (1 - root7) / 4

1 Answer
Aug 2, 2017

tan x = -(4 +- sqrt7)/3

Explanation:

cos x + sin x = 1/2
Divide both sides by cos x
1 + tan x = 1/(2cos x) = (sec x)/2
Square both sides
(1 + tan x) ^2 = (sec^2)/4
4(1 + tan^2 x + 2tan x) = sec^2 x = (1 + tan^2 x)
After simplification:
3tan^2 x + 8tan x + 3 = 0
Solve this quadratic equation for tan x
D = d^2 = b^2 - 4ac = 64 - 36 = 28 --> d = +- 2sqrt7
There are 2 real roots:
tan x = - b/(2a) +- d/(2a) = - 8/6 +- 2sqrt7/6 = - (4 +- sqrt7)/3