# Two rhombuses have sides with lengths of #16 #. If one rhombus has a corner with an angle of #pi/12 # and the other has a corner with an angle of #(5pi)/6 #, what is the difference between the areas of the rhombuses?

Difference between the areas of the rhombuses is

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The area of a rhombus can be calculated using the formula (A = \frac{1}{2} d_1 d_2), where (d_1) and (d_2) are the lengths of its diagonals.

Given that the sides of both rhombuses are 16 units long, we can calculate the lengths of their diagonals using trigonometry.

For the rhombus with an angle of (\frac{\pi}{12}), one diagonal can be found using the Law of Cosines as (d_1 = \sqrt{16^2 + 16^2 - 2(16)(16)\cos(\frac{\pi}{12})}), and since it's a rhombus, both diagonals are equal.

For the rhombus with an angle of (\frac{5\pi}{6}), one diagonal can be found using the Law of Cosines as (d_2 = \sqrt{16^2 + 16^2 - 2(16)(16)\cos(\frac{5\pi}{6})}), and again, both diagonals are equal.

After finding the lengths of the diagonals for both rhombuses, calculate their areas using the formula mentioned earlier. Then find the difference between the areas of the two rhombuses.

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