# Two corners of a triangle have angles of # (5 pi )/ 8 # and # ( pi ) / 3 #. If one side of the triangle has a length of # 4 #, what is the longest possible perimeter of the triangle?

The longest possible perimeter is,

Associate the given side with the smallest angle, because that will lead to the longest perimeter:

Let side a = 4

Use the law of sines to compute the other two sides:

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To find the longest possible perimeter of the triangle, use the Law of Sines to determine the lengths of the other sides. Then, calculate the perimeter using the formula: perimeter = side1 + side2 + side3.

Let's denote the given angles as A = (5π)/8 and B = π/3. The third angle, C, can be found using the fact that the sum of angles in a triangle is π radians.

C = π - A - B

Once you have the three angles, you can use the Law of Sines:

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

where a, b, and c are the lengths of the sides opposite to angles A, B, and C, respectively.

Given that one side, say a, has a length of 4, you can use the Law of Sines to find the other sides, b and c. Then, calculate the perimeter using the formula mentioned earlier.

Calculate the values of sides b and c using the Law of Sines:

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

Substitute the values of angles A, B, and C along with the given side length to find the lengths of sides b and c. Once you have all three side lengths, calculate the perimeter.

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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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