# Two opposite sides of a parallelogram each have a length of #18 #. If one corner of the parallelogram has an angle of #(5 pi)/12 # and the parallelogram's area is #132 #, how long are the other two sides?

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Given that two opposite sides of the parallelogram have a length of 18 units and one angle measures ( \frac{5\pi}{12} ), and the area of the parallelogram is 132 square units, we can use the formula for the area of a parallelogram: ( \text{Area} = \text{base} \times \text{height} \times \sin(\theta) ), where ( \theta ) is the angle between the two sides.

We are given the area (( \text{Area} = 132 )), one side length (( \text{base} = 18 )), and an angle (( \theta = \frac{5\pi}{12} )).

Using the formula for the area of a parallelogram, we can solve for the height (( \text{height} )):

[ 132 = 18 \times \text{height} \times \sin\left(\frac{5\pi}{12}\right) ]

[ \text{height} = \frac{132}{18 \times \sin\left(\frac{5\pi}{12}\right)} ]

Once we find the height, we can use the definition of sine to find the other two side lengths:

[ \sin(\theta) = \frac{\text{height}}{\text{opposite side length}} ]

[ \text{opposite side length} = \frac{\text{height}}{\sin(\theta)} ]

So, the other two side lengths of the parallelogram are:

[ \text{opposite side length} = \frac{\text{height}}{\sin\left(\frac{5\pi}{12}\right)} ]

[ \text{opposite side length} = \frac{132}{18 \times \sin\left(\frac{5\pi}{12}\right)} ]

[ \text{opposite side length} \approx \frac{132}{18 \times 0.9659} ]

[ \text{opposite side length} \approx \frac{132}{17.386} ]

[ \text{opposite side length} \approx 7.585 ]

Therefore, the other two sides of the parallelogram are approximately 7.585 units long.

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