Two opposite sides of a parallelogram each have a length of #8 #. If one corner of the parallelogram has an angle of #( pi)/3 # and the parallelogram's area is #96 #, how long are the other two sides?

Given that the area of the parallelogram is 96 and one corner has an angle of ( \frac{\pi}{3} ), and two opposite sides each have a length of 8, we can use the formula for the area of a parallelogram:

[ \text{Area} = \text{base} \times \text{height} ]

In a parallelogram, the height is the perpendicular distance between the two opposite sides. We can find this height by using the formula:

[ \text{Area} = \text{base} \times \text{height} ]

[ 96 = 8 \times \text{height} ]

[ \text{height} = \frac{96}{8} = 12 ]

Now, we know that one of the angles in the parallelogram is ( \frac{\pi}{3} ). The opposite angle is also ( \frac{\pi}{3} ) since opposite angles in a parallelogram are equal.

Therefore, the height (or the distance between the two opposite sides) forms the hypotenuse of a right triangle where one of the angles is ( \frac{\pi}{3} ). Using trigonometric functions, we can find the lengths of the other two sides.

Using the sine function:

[ \sin\left(\frac{\pi}{3}\right) = \frac{\text{opposite}}{\text{hypotenuse}} ]

[ \sin\left(\frac{\pi}{3}\right) = \frac{x}{12} ]

[ \frac{\sqrt{3}}{2} = \frac{x}{12} ]

[ x = 12 \times \frac{\sqrt{3}}{2} = 6\sqrt{3} ]

So, the length of the other two sides of the parallelogram is ( 6\sqrt{3} ).