# A triangle has two corners with angles of # ( pi ) / 2 # and # ( 5 pi )/ 12 #. If one side of the triangle has a length of #9 #, what is the largest possible area of the triangle?

The area is

The third angle of the triangle is

Therefore,

The area of the triangle is

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To find the largest possible area of the triangle, use the formula for the area of a triangle: ( A = \frac{1}{2} \times b \times h ), where ( b ) is the base and ( h ) is the height.

In this case, one side of the triangle has a length of 9, which can be considered as the base. To find the height, you need to determine the length of the altitude from the vertex opposite the base to the base itself.

To find the height, use trigonometry. The height can be calculated using the formula ( h = \text{side} \times \sin(\text{angle}) ), where the angle is the angle opposite to the side whose length is known.

For the given triangle, the angle opposite to the side with length 9 is ( \frac{5\pi}{12} ). Thus, ( h = 9 \times \sin\left(\frac{5\pi}{12}\right) ).

Calculate the value of ( \sin\left(\frac{5\pi}{12}\right) ), then multiply it by 9 to find the height.

Finally, use the formula for the area of the triangle to calculate the largest possible area.

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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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