Determine the perimeter of triangle ABC if AB=6, B=pi/4, C=pi/6.

To determine the perimeter of the triangle ABC, we'll have
to determine all the lengths of it's sides.

For the
beginning, we'll determine the angle A. Since the sum of the angles of a triangle is 180
degrees = pi radians, we'll get A:

<A = pi - pi/4 -
pi/6

<A = (12pi - 3pi -
2pi)/12

<A =
7pi/12

<A = pi/2 +
pi/12

sin pi/12 = sqrt[(1 - cos
pi/6)/2]

sin pi/12 = sqrt(2 -
sqrt3)/2

We'll apply sine theorem and we'll
get:

AB/sin C = AC/sin B

6/sin
pi/6 = AC/sin pi/4

AC* 1/2 =
6*sqrt2/2

AC =
6sqrt2

AC/sin B = BC/sin
A

6sqrt2/ sin pi/4 = BC/[sqrt(2 -
sqrt3)/2]

BC*sqrt2/2 = {6sqrt2*[sqrt(2 -
sqrt3)]}/2

BC = 6[sqrt(2 -
sqrt3)]

The perimeter P is:

P
= AB + AC + BC

P = 6 + 6sqrt2 + 6[sqrt(2 -
sqrt3)]

We'll factorize by
6:

P = 6[1+sqrt2+sqrt(2 - sqrt3)]
units