# A triangle has two corners with angles of # (2 pi ) / 3 # and # ( pi )/ 6 #. If one side of the triangle has a length of #8 #, what is the largest possible area of the triangle?

Largest possible area of the triangle is 27.7128

The remaining angle:

I am assuming that length AB (8) is opposite the smallest angle.

Using the ASA

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To find the largest possible area of the triangle, we can use the formula for the area of a triangle, which is given by ( \frac{1}{2} \times \text{base} \times \text{height} ).

Since we know one side of the triangle has a length of 8, we can consider this side as the base of the triangle.

To find the height, we need to drop a perpendicular from the vertex opposite to the 8-unit side to the base. This forms a right triangle with angles ( \frac{\pi}{3} ) and ( \frac{\pi}{6} ).

Using trigonometric ratios, we can determine the height of the triangle in terms of the base. Since the side opposite to ( \frac{\pi}{3} ) is the height, we'll use the tangent function:

[ \tan\left(\frac{\pi}{3}\right) = \frac{\text{height}}{8} ]

[ \text{height} = 8 \tan\left(\frac{\pi}{3}\right) ]

Now, we can use the formula for the area of the triangle:

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

[ \text{Area} = \frac{1}{2} \times 8 \times 8 \tan\left(\frac{\pi}{3}\right) ]

[ \text{Area} = 32 \tan\left(\frac{\pi}{3}\right) ]

Since ( \tan\left(\frac{\pi}{3}\right) = \sqrt{3} ), we have:

[ \text{Area} = 32 \sqrt{3} ]

Therefore, the largest possible area of the triangle is ( 32 \sqrt{3} ) 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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