# A triangle has two corners with angles of # pi / 4 # and # pi / 3 #. If one side of the triangle has a length of #8 #, what is the largest possible area of the triangle?

21.85252

To get the largest possible area of the triangle with any one side with 8, we must have the shortest side as 8.

Then b must be 8

Using the law of Sines,

= 21.85252

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The largest possible area of the triangle can be found when the two given angles form a right triangle. This happens when one angle is ( \frac{\pi}{4} ) and the other is ( \frac{\pi}{3} ). In a right triangle, the side opposite the right angle (hypotenuse) is the longest side.

Given that one side of the triangle has a length of 8, and it's opposite to one of the given angles, which is ( \frac{\pi}{4} ), this side would be the hypotenuse in the right triangle.

Using trigonometric ratios, we can find the lengths of the other two sides of the triangle:

[ \text{Opposite side to } \frac{\pi}{4} = 8 \sin(\frac{\pi}{4}) = 8 \cdot \frac{\sqrt{2}}{2} = 4\sqrt{2} ] [ \text{Adjacent side to } \frac{\pi}{4} = 8 \cos(\frac{\pi}{4}) = 8 \cdot \frac{\sqrt{2}}{2} = 4\sqrt{2} ]

Now, we have a right triangle with one leg of length (4\sqrt{2}) and the other leg of length (4\sqrt{2}). The area of a triangle can be calculated using the formula:

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

Since both legs of the triangle are equal, the base and height are also equal. So, the area of the triangle would be:

[ \text{Area} = \frac{1}{2} \times 4\sqrt{2} \times 4\sqrt{2} ]

[ \text{Area} = 16 ]

So, the largest possible area of the triangle is (16).

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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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