Find exact value of Tan(11pi/12)?

Question

I cannot solve this problem. Please show all work and explain why on each step. I can get up to Tan(11pi/12)= pi/4+2(pi/3). Unsure of 2(pi/3) refrence triangle quadrant. When using 2nd quadrant for 2(pi/3) my equation looks like this.

1+((-square root of three)/1) / 1-(1)((-square root of three)/1)

Cannot have a radical in the denominator

Answer 1

11/12 = 8/12 + 3/12 = 2/3 + 1/4
tan(11π/12) =
tan(2π/3 + π/4) =
(tan(2π/3) + tan(π/4))/(1 - tan(2π/3)tan(π/4)) =
(- √3 + 1)/(1 + √3) =
(1 - √3)(1 - √3)/(1 + √3)(1 - √3) =
(1 - 2√3 + 3)/(1 - 3) =
(4 - 2√3)/-2 =
- 2 + √3

Answer 2

well you could do that, and use the tangent formula for the sum of two angles, but personally I prefer,
tan (11pi/12) = tan (pi - pi/12) = -tan pi/12

1/V3 = (2tan pi/12) /(1 - tan^2 pi/12)
Now solve the quadratic and accept a solution depending on the quadrant and you're done.

Answer 3

Using a few Trig identities simplifies the problem:

tan (π - x) = -tan(x)
11π/12 = π - π/12
tan (11π/12) = -tan (π/12)

tan (x/2) = sin x / (1 + cos x)
-tan (π/12) = -sin (&pi/6) / (1 + cos π/6)
-tan (π/12) = -1/2 / (1 + sqrt(3)/2)
-tan (π/12) = -1 / (1 + sqrt(3) )

So, tan (11π/12) = -1 / (1 + sqrt(3) )

I suspect you can simplify the equation you have to this, but it seems tougher than needed.

Answer 4

pi = 180 degrees

tan (11 pi/12) = tan (11*15)= tan(165 degrees)
tan(165)= tan(90 + 75)= -cot 75
= -cot(45+30)
= - 1-(cot45. cot30)/cot45 + cot30
= -1-(1)(1/root3)/1 + root 3
=simplify

Answer 5

tan(α + β) = (tanα + tanβ)/[1 - (tanα)(tanβ)]

tan(3π/4 + π/6) = [tan(3π/4) + tan(π/6)]/[1 - (tan 3π/4)(tan π/6)]
= (-1 + 1/√3) / [1 - (-1)(1/√3)] = (-1 + 1/√3) / (1 + 1/√3)
= (1/√3 - 1) / (1/√3 + 1)
= (1/√3 - 1)² / {(1/√3 + 1)(1/√3 - 1)}
= (1/3 - 2/√3 + 1) / (1/3 - 1)
= (4/3 - 2√3/3) / (-2/3)
= (-4 + 2√3) / 2
= -2 + √3

Answer 6

tan(11 pi/12)
= tan(pi/4 + 2pi/3)
= [tan(pi/4)+tan(2pi/3)]
/[1-tan(pi/4)tan(2pi/3)]
= [1-√3]/[1+√3]
= (4-2√3)/(-2)
= -2+√3
------------
More explanation:
tan(2pi/3)
= -tan(pi-2pi/3)
= -tan(pi/3)
= -√3

Answer 7

I could not solve it too . I have a maths lecture today . If teacher allows I will ask this in the lecture.