# A triangle has two corners with angles of # ( pi ) / 2 # and # ( 5 pi )/ 12 #. If one side of the triangle has a length of #14 #, what is the largest possible area of the triangle?

To get the largest possible areaof the right triangle,

Applying the law of sines,

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To find the largest possible area of the triangle given two angles and one side length, we can use the formula for the area of a triangle:

[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} ]

For a triangle with two given angles and one side length, we'll need to find the length of the other two sides using trigonometric ratios.

Given that one angle is ( \frac{\pi}{2} ) and the other is ( \frac{5\pi}{12} ), we can find the third angle by subtracting the sum of the other two angles from ( \pi ).

[ \text{Third angle} = \pi - \left(\frac{\pi}{2} + \frac{5\pi}{12}\right) = \pi - \frac{6\pi}{12} - \frac{5\pi}{12} = \pi - \frac{11\pi}{12} = \frac{\pi}{12} ]

Now, let's find the length of the other two sides using trigonometric ratios. Since we have one side length of 14, we can use the sine and cosine functions to find the other side lengths.

For the side opposite the ( \frac{5\pi}{12} ) angle:

[ \sin\left(\frac{5\pi}{12}\right) = \frac{\text{opposite side}}{14} ]

[ \text{opposite side} = 14 \times \sin\left(\frac{5\pi}{12}\right) ]

Similarly, for the side opposite the ( \frac{\pi}{12} ) angle:

[ \cos\left(\frac{5\pi}{12}\right) = \frac{\text{adjacent side}}{14} ]

[ \text{adjacent side} = 14 \times \cos\left(\frac{5\pi}{12}\right) ]

Now that we have all three side lengths, we can calculate the area using the formula for the area of a triangle.

[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} ]

[ \text{Area} = \frac{1}{2} \times 14 \times \left(14 \times \sin\left(\frac{5\pi}{12}\right)\right) ]

[ \text{Area} = \frac{1}{2} \times 14 \times \left(14 \times \sin\left(\frac{5\pi}{12}\right)\right) ]

[ \text{Area} \approx 73.77 ]

So, the largest possible area of the triangle is approximately ( 73.77 ) 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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