A parallelogram has sides A, B, C, and D. Sides A and B have a length of #2 # and sides C and D have a length of # 3 #. If the angle between sides A and C is #pi/8 #, what is the area of the parallelogram?

To find the area of the parallelogram, we can use the formula:

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

In this case, we can take sides A and B as the base, and side C as the height, or vice versa. Since we know the length of sides A and B is 2, and the length of side C is 3, we can use the length of side C as the base, and the height as the perpendicular distance from side C to sides A and B.

To find the height, we can use trigonometry. Given that the angle between sides A and C is ( \frac{\pi}{8} ), we can use the sine function:

[ \sin\left(\frac{\pi}{8}\right) = \frac{\text{opposite}}{\text{hypotenuse}} ]

The opposite side is the height we are looking for, and the hypotenuse is side A or B (since they are equal). Thus:

[ \text{Height} = \sin\left(\frac{\pi}{8}\right) \times 2 ]

Now, we have the base and the height, so we can calculate the area of the parallelogram:

[ \text{Area} = 3 \times \sin\left(\frac{\pi}{8}\right) \times 2 ]

Compute the value of ( \sin\left(\frac{\pi}{8}\right) ), then multiply it by 3 and 2 to find the area of the parallelogram.