# Two opposite sides of a parallelogram have lengths of #3 #. If one corner of the parallelogram has an angle of #pi/8 # and the parallelogram's area is #36 #, how long are the other two sides?

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We can use the formula for the area of a parallelogram, which is given by the product of the length of one side and the perpendicular distance to that side from the opposite side. Let's denote the lengths of the other two sides as (a) and (b), and the angle between the known side and one of the unknown sides as (\theta).

Given that the area of the parallelogram is 36, we have:

[36 = 3 \times h]

where (h) is the perpendicular distance between the two opposite sides of length 3.

To find (h), we can use the sine of the given angle, (\frac{\pi}{8}):

[h = 3 \sin\left(\frac{\pi}{8}\right)]

Now, we can find the lengths of the other two sides using the area formula again:

[36 = a \times 3 \sin\left(\frac{\pi}{8}\right)]

[36 = b \times 3 \sin\left(\frac{\pi}{8}\right)]

Solving for (a) and (b):

[a = \frac{36}{3 \sin\left(\frac{\pi}{8}\right)}]

[b = \frac{36}{3 \sin\left(\frac{\pi}{8}\right)}]

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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.

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