In triangle ABC, A = 30°, C = 105°, a = 8. Find side b.

To determine the length of the side b, we'll apply the law
of sines:
a/sin A = b/sin B
We notice that we do not know the
measure of the angle B.
Since we know two of the three angles of the triangle,
we can determine the 3rd angle, knowing that the sum of the angles is of 180
degrees.
A+B+C = 180
30+B+105 = 180 => B = 180 -
135
B = 45
Now, we can apply the law of sines:
8/sin 30 =
b/sin 45
We'll cross multiply:
b*sin 30 = 8*sin 45
b/2 =
8*sqrt2/2
b = 8sqrt2

The
requested length of the side b is: b = 8sqrt2
units.