The problem says "one side of the triangular garden is 42.0 ft. the angles on each end of this side measure 66 degrees and 82 degrees. find length of fence needed to enclose the garden." m

Draw out a triangle with one side labeled 42 feet, and the two adjacent angles 66 and 82. 
When using the law of sines, you want to label the sides of your triangle a, b, c, and the angles of your triangle A, B, C. 
First, label the triangle. To make this easy, I'll tell you what I labeled them so you can follow along. 
The side labeled 42 feet is side "a", and the angle opposite this side is angle A
66 degrees is angle B, and the side opposite this angle is side b
82 degrees is angle C, and the side opposite this angle is side c. 

We know two angles of the triangle, 66 and 82. We also know that a triangle is made up of 180 degrees. To find the angle A, we can subtract angle B and angle C from 180. 

A = 180 - 66 - 82 = 32

a = 42                  b = ?               c = ?
A = 32                 B = 66             C = 82

Now we can start using the Law of Sines to solve for sides b and c. 
the law of sines says:

sin(A) = sin(B) = sin(C)
  a         b            c

we know a, A, and B, and we are looking for "b" so we can substitute those values into sin(A)/a = sin(B)/ b

sin(32) = sin(66)
   42        b

cross multiply: 
b sin(32) = 42sin(66)
divide both sides by sin(32) 
b = 42sin(66)  ≈ 72.4
        sin(32)

We can do the same thing for side c. 
sin(A) = sin(C)
  a           c

sin(32) = sin(82)
   42         c

c sin(32) = 42sin(82)

c = 42sin(82) ≈ 78.5
         sin(32)

The 3 sides are 42.0 feet, 72.4 feet, and 78.5 feet. To find the length of the fence needed to enclose the garden, simply add the three sides up.

42.0 + 72.4 + 78.5 = 192.9 feet