A triangle has sides A, B, and C. The angle between sides A and B is #(pi)/6#. If side C has a length of #5 # and the angle between sides B and C is #(5pi)/12#, what are the lengths of sides A and B?

To find the lengths of sides A and B in the triangle given that side C has a length of 5, and the angles between sides A and B and between sides B and C are given, we can use the Law of Sines.

The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. In this case, we can set up the following proportion:

(\frac{A}{\sin(\frac{\pi}{6})} = \frac{5}{\sin(\frac{5\pi}{12})})

First, we find the sine values of the angles:

(\sin(\frac{\pi}{6}) = \frac{1}{2}) and (\sin(\frac{5\pi}{12}) \approx 0.9659)

Substituting these values into the proportion, we get:

(\frac{A}{\frac{1}{2}} = \frac{5}{0.9659})

Solving for A:

(A = \frac{5}{0.9659} \times \frac{1}{2})

(A \approx 2.581 \times \frac{1}{2})

(A \approx 1.2905)

Now, we have found the length of side A. To find the length of side B, we can use the Law of Sines again:

(\frac{B}{\sin(\frac{5\pi}{12})} = \frac{5}{\sin(\frac{\pi}{6})})

Substituting the sine values:

(\frac{B}{0.9659} = \frac{5}{\frac{1}{2}})

Solving for B:

(B = 0.9659 \times \frac{5}{\frac{1}{2}})

(B \approx 9.659)

Therefore, the lengths of sides A and B are approximately 1.2905 and 9.659 units, respectively.