# A triangle has two corners with angles of # pi / 4 # and # (5 pi )/ 8 #. If one side of the triangle has a length of #9 #, what is the largest possible area of the triangle?

Largest possible area of the triangle is 69.1378

The remaining angle:

I am assuming that length AB (9) 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 given two sides and the angle between them:

Area = 1/2 * a * b * sin(C)

where a and b are the lengths of the two sides, and C is the angle between them.

Let's denote the angle with π/4 radians as A and the angle with (5π)/8 radians as B. We know that the sum of the angles in a triangle is π radians, so the third angle C can be calculated as:

C = π - A - B

Now, we can calculate the area of the triangle for different configurations where one side is 9 units long and the other two sides are determined by the angles A and B. We can then find the maximum area among these configurations.

Let's start by calculating the area for the given angles A = π/4 and B = (5π)/8:

C = π - π/4 - (5π)/8 = π/8

Now, using the formula for the area of a triangle:

Area = 1/2 * 9 * 9 * sin(π/8)

Calculate the sin(π/8) and then multiply it by 1/2 * 9 * 9 to find the maximum area of the triangle.

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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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