# Two corners of a triangle have angles of # (5 pi )/ 8 # and # ( pi ) / 6 #. If one side of the triangle has a length of # 7 #, what is the longest possible perimeter of the triangle?

Largest possible area of the triangle is 27.5587

The remaining angle:

I am assuming that length AB (1) is opposite the smallest angle.

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To find the longest possible perimeter of the triangle given that two corners have angles of ( \frac{5\pi}{8} ) and ( \frac{\pi}{6} ), and one side has a length of 7, we can use the law of sines.

First, let's denote the angle between the known side (of length 7) and the side we're looking for as ( \theta ). Then, we can use the law of sines to find the relationship between the sides and angles:

[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} ]

Where ( a ), ( b ), and ( c ) are the lengths of the sides, and ( A ), ( B ), and ( C ) are the corresponding angles.

Given that one side has a length of 7, we have:

[ \frac{7}{\sin(\theta)} = \frac{b}{\sin\left(\frac{5\pi}{8}\right)} ]

Solving for ( b ):

[ b = \frac{7 \cdot \sin\left(\frac{5\pi}{8}\right)}{\sin(\theta)} ]

Now, we want to maximize the perimeter, which is ( P = 7 + b + c ). Since ( b ) is already expressed in terms of ( \theta ), we need to find ( c ) in terms of ( \theta ). We know that the sum of angles in a triangle is ( \pi ), so ( C = \pi - \theta - \frac{5\pi}{8} ).

Now, using the law of sines again:

[ \frac{c}{\sin(C)} = \frac{7}{\sin\left(\frac{\pi}{6}\right)} ]

Solving for ( c ):

[ c = \frac{7 \cdot \sin\left(\frac{\pi}{6}\right)}{\sin\left(\pi - \theta - \frac{5\pi}{8}\right)} ]

Now, substitute ( b ) and ( c ) back into the perimeter formula ( P = 7 + b + c ) to get the perimeter in terms of ( \theta ). Then, find the value of ( \theta ) that maximizes this expression to get the longest possible perimeter 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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