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

1 Answer
Nghi N
Aug 2, 2017

tan x = -(4 +- sqrt7)/3tanx=−4±√73

Explanation:

cos x + sin x = 1/2
Divide both sides by cos x
1 + tan x = 1/(2cos x) = (sec x)/21+tanx=12cosx=secx2
Square both sides
(1 + tan x) ^2 = (sec^2)/4(1+tanx)2=sec24
4(1 + tan^2 x + 2tan x) = sec^2 x = (1 + tan^2 x)4(1+tan2x+2tanx)=sec2x=(1+tan2x)
After simplification:
3tan^2 x + 8tan x + 3 = 03tan2x+8tanx+3=0
Solve this quadratic equation for tan x
D = d^2 = b^2 - 4ac = 64 - 36 = 28D=d2=b2−4ac=64−36=28 --> d = +- 2sqrt7d=±2√7
There are 2 real roots:
tan x = - b/(2a) +- d/(2a) = - 8/6 +- 2sqrt7/6 = - (4 +- sqrt7)/3tanx=−b2a±d2a=−86±2√76=−4±√73

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