# Two corners of a triangle have angles of # (5 pi )/ 8 # and # ( pi ) / 4 #. If one side of the triangle has a length of # 7 #, what is the longest possible perimeter of the triangle?

Perimeter

Applying Law of Sines,

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To find the longest possible perimeter of the triangle, we need to maximize the length of the third side, given that one side has a length of 7 and two angles are given.

- Use the Triangle Sum Theorem to find the third angle.
- Apply the Law of Sines to find the possible lengths of the third side.
- Choose the maximum value for the perimeter.

The third angle can be found by subtracting the sum of the given angles from 180 degrees (or ( \pi ) radians):

[ \text{Third angle} = \pi - \left(\frac{5\pi}{8} + \frac{\pi}{4}\right) = \pi - \frac{7\pi}{8} = \frac{\pi}{8} ]

Now, using the Law of Sines:

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

Substituting the known values:

[ \frac{\sin\left(\frac{5\pi}{8}\right)}{7} = \frac{\sin\left(\frac{\pi}{4}\right)}{b} = \frac{\sin\left(\frac{\pi}{8}\right)}{c} ]

Solving for ( b ) and ( c ):

[ b = \frac{7\sin\left(\frac{\pi}{4}\right)}{\sin\left(\frac{5\pi}{8}\right)} ] [ c = \frac{7\sin\left(\frac{\pi}{8}\right)}{\sin\left(\frac{\pi}{4}\right)} ]

Using the trigonometric identities ( \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} ) and ( \sin\left(\frac{\pi}{8}\right) = \frac{\sqrt{2-\sqrt{2}}}{2} ), we can calculate ( b ) and ( c ).

[ b \approx 5.33 ] [ c \approx 9.9 ]

The longest possible perimeter occurs when ( b ) and ( c ) are maximized:

[ \text{Perimeter} = 7 + b + c ] [ \text{Perimeter} \approx 7 + 5.33 + 9.9 ] [ \text{Perimeter} \approx 22.23 ]

So, the longest possible perimeter of the triangle is approximately 22.23 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.

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