A parallelogram has sides A, B, C, and D. Sides A and B have a length of #5 # and sides C and D have a length of # 3 #. If the angle between sides A and C is #(5 pi)/8 #, what is the area of the parallelogram?

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

Area = base × height

The base of the parallelogram can be any of its sides. Since the given sides are labeled as A, B, C, and D, let's consider sides A and B as the base. The height of the parallelogram is the perpendicular distance between the base and the opposite side.

First, we need to find the height. We can use trigonometry to find the height. Considering the given angle between sides A and C, we can find the height using the sine function:

sin(θ) = opposite / hypotenuse

Here, θ = (5π)/8 and the opposite side is the height (h), and the hypotenuse is side A (5).

sin((5π)/8) = h / 5

Solving for h, we get:

h = 5 * sin((5π)/8)

Now, we have the height. To find the area, we multiply the base (5) by the height (h):

Area = 5 * h

Substitute the value of h:

Area = 5 * (5 * sin((5π)/8))

Calculate sin((5π)/8):

sin((5π)/8) ≈ 0.9239

Area ≈ 5 * (5 * 0.9239)

Area ≈ 23.0975

So, the area of the parallelogram is approximately 23.0975 square units.