# 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 # 4 #. If the angle between sides A and C is #pi/12 #, what is the area of the parallelogram?

The area of the parallelogram is

To calculate the height use

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The area of a parallelogram can be calculated using the formula: [ \text{Area} = base \times height ]

Given sides (A) and (C) with lengths 5 and 4 respectively, and the angle between them is (\frac{\pi}{12}), we can consider (A) as the base. To find the height (perpendicular distance from the opposite side to the base), we can use the sine of the given angle, since the height (h) is the opposite side of the angle when considering (C) as the hypotenuse of a right-angled triangle formed by the height, half of (C), and a line segment from (A) to the height's top.

However, there's a more direct approach to find the area of the parallelogram using the formula: [ \text{Area} = ab \sin(\theta) ]

Where:

- (a) and (b) are the lengths of any two adjacent sides,
- (\theta) is the angle between those sides.

In this case, (a = 5), (b = 4), and (\theta = \frac{\pi}{12}).

So, the area is: [ \text{Area} = 5 \times 4 \times \sin\left(\frac{\pi}{12}\right) ]

[ \sin\left(\frac{\pi}{12}\right) ] can be calculated using the half-angle formula, given that (\sin\left(\frac{\pi}{12}\right) = \sin\left(\frac{\pi}{6}/2\right)), but since it's known that (\sin\left(\frac{\pi}{12}\right) = \sin(15^\circ) = \frac{\sqrt{6} - \sqrt{2}}{4}),

[ \text{Area} = 20 \times \frac{\sqrt{6} - \sqrt{2}}{4} ]

[ \text{Area} = 5(\sqrt{6} - \sqrt{2}) ]

So, the area of the parallelogram is (5(\sqrt{6} - \sqrt{2})) square units.

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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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- How do you find the area of a square with sides 6 centimeters long?
- A parallelogram has sides A, B, C, and D. Sides A and B have a length of #8 # and sides C and D have a length of # 9 #. If the angle between sides A and C is #pi/12 #, what is the area of the parallelogram?
- Two opposite sides of a parallelogram each have a length of #6 #. If one corner of the parallelogram has an angle of #( pi)/3 # and the parallelogram's area is #36 #, how long are the other two sides?

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