Question

What is the exact value of Tangent (StartFraction 19 pi Over 12 EndFraction)?

please explain how you got it

Answer 1

9514 1404 393

Answer:

tan(19π/12) = -2-√3

Explanation:

It is convenient to use a half-angle identity for this. One that minimizes the work involved is ...

tan(α/2) = (1 -cos(α))/sin(α)

Here, we can let α = 2(19π/12) = 19π/6 ≅ 7π/6. This is a 3rd-quadrant angle, where sine and cosine are both negative.

cos(7π/6) = -√3/2

sin(7π/6) = -1/2

Then our relation is ...

tan(α/2) = tan(19π/12) = (1 -(-√3/2))/(-1/2)

tan(19π/12) = -2-√3

_____

Your calculator can verify this.

Answer 2

(answer at the bottom)

19π/12 radians = 285°

skipping the degree symbol ...
and 285 = 240+45
we know tan 240 = +tan60 = √3 by the CAST rule
tan 45 = 1

tan(285)
= tan(240 + 45)
= (tan240 + tan45)/(1 - tan240tan45)
= (√3+1)/(1-√3)

ANSWER: tan(19π/12) = (√3+1)/(1-√3)