# A triangle has two corners with angles of # pi / 4 # and # (3 pi )/ 8 #. If one side of the triangle has a length of #2 #, what is the largest possible area of the triangle?

Largest possible area of the triangle is 2.4142

The remaining angle:

I am assuming that length AB (2) 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 included angle.

Given that one side of the triangle has a length of 2 and two angles are π/4 and (3π)/8, we can find the third angle using the fact that the sum of angles in a triangle is π radians.

Let's denote the third angle as θ. Then, we have:

π/4 + (3π)/8 + θ = π

Solving for θ:

θ = π - π/4 - (3π)/8 = 8π/8 - 2π/8 - 3π/8 = 3π/8

Now, we have the three angles of the triangle: π/4, (3π)/8, and 3π/8.

Using the formula for the area of a triangle given two sides and the included angle:

Area = (1/2) * a * b * sin(θ)

where a and b are the lengths of the sides, and θ is the included angle.

We have one side with length 2, and the other two sides can be found using the law of sines:

a/sin(A) = b/sin(B) = c/sin(C)

Let's denote the unknown side lengths as a and b:

a/sin(π/4) = 2/sin(3π/8) b/sin(π/4) = 2/sin(3π/8)

Solving for a and b:

a = 2*sin(π/4)/sin(3π/8)
b = 2*sin(π/4)/sin(3π/8)

Now, we can substitute these values into the formula for the area of the triangle:

Area = (1/2) * 2 * 2 * sin(3π/8)

Calculating the sine of (3π)/8 and simplifying:

sin(3π/8) ≈ 0.3827

Area ≈ (1/2) * 2 * 2 * 0.3827 ≈ 1.527

Therefore, the largest possible area of the triangle is approximately 1.527 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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