Question

Find the measure of the angle formed between the base of the cone and a line segment that represents the slant height.

Answer 1

Remark.
Thank you for the clear diagram. You need to find AB somehow. When you do you can find the desired angle by using the inverse tan.

Step One
Find AB
AB = the radius of the cone
AB = 1/2 the diameter of the cone
AB = 1/2 * 4
AB = 2 inches.

Step two
Determine The Tangent of the desired angle
Tan(A) = opposite of adjacent
opposite side = h = 6 inches
Adjacent = AB = radius = 2 inches.

Tan(A) = 6/2 = 3 

Step 3
Find the inverse Tan
A = inverse Tan(A)
A =  inverse Tan(3)
A = 71.57 degrees <<<<< Answer.

There are other ways of finding the angle you want, but I think this is the quickest.