If cos ⁡x=−1/2, and 90°<x<180°, find tan⁡(x+30°).

Question

If cos ⁡x=−1/2, and 90°<x<180°, what is tan⁡(x+30°) in radical form?

Answer 1

x = 120 so x+30 = 150 tan(150) = sin(150)/cos(150) = 1/2/(-sqrt(3)/2) = -sqrt(3)/3

Answer 2

Tan (x + 30) = (Tan x + Tan 30) / (1- Tan xTan 30)

Step 1. Tan 30 = -sqr (3)

Step 2:

Cos^2 x + Sin^2 x = 1

(-1/2)^2 + Sin^2 x = 1

Sin^2 x = 1 - 1/4

Sin^2 x = 3/4

Sin x = Sqr (3)/2 and is positive because x is in the second quadrant.

Step 3:

Tan x = Sin x / Cos x = (Sqr(3)/2)/(-1/2) = -sqr (3)

Step 4:

Tan (x + 30) = (Tan x + Tan 30) / (1 - Tan xTan 30) = (-sqr(3) + sqr(3)/3) / ( 1- (-sqr(3)(sqr(3)/3))

= (-sqr(3) + sqr(3)/3) / (1 + 3/3) = ((-3sqr(3) + sqr(3))/3 / 2) =(-2sqr(3) / 3 ) * (1/2) = -sqr(3) / 3

Final Answer:

Tan (x + 30) = -sqr(3) / 3