Question

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

A: (4 - `root`7) / 3
B: -(4 + `root`7) / 3
C: (1 + `root`7) / 4
D: (1 - `root`7) / 4

Answer 1

`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`