Two rhombuses have sides with lengths of #1 #. If one rhombus has a corner with an angle of #pi/12 # and the other has a corner with an angle of #(11pi)/12 #, what is the difference between the areas of the rhombuses?
Difference between the areas of the rhombuses is
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The difference between the areas of the two rhombuses can be found by first determining their diagonals.
For a rhombus with side length ( s ) and an angle ( \theta ) between adjacent sides, the length of each diagonal is given by ( \frac{s}{2} \times \sqrt{4 + \sec^2(\theta)} ).
Using this formula:
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For the rhombus with an angle of ( \frac{\pi}{12} ): ( \text{Diagonal}_1 = \frac{1}{2} \times \sqrt{4 + \sec^2\left(\frac{\pi}{12}\right)} )
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For the rhombus with an angle of ( \frac{11\pi}{12} ): ( \text{Diagonal}_2 = \frac{1}{2} \times \sqrt{4 + \sec^2\left(\frac{11\pi}{12}\right)} )
Calculate these diagonal lengths and then use the area formula for a rhombus, which is ( \text{Area} = \frac{1}{2} \times \text{Diagonal}_1 \times \text{Diagonal}_2 ), to find the areas of the two rhombuses. Finally, subtract the smaller area from the larger area to get the difference.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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