# Two corners of a triangle have angles of # (3 pi )/ 8 # and # ( pi ) / 2 #. If one side of the triangle has a length of # 4 #, what is the longest possible perimeter of the triangle?

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To find the longest possible perimeter of the triangle, we need to determine the length of the remaining side.

The sum of angles in a triangle is always ( \pi ) radians or 180 degrees.

Let ( \theta_1 ) and ( \theta_2 ) be the given angles.

The third angle ( \theta_3 ) can be found by subtracting the given angles from ( \pi ):

( \theta_3 = \pi - \theta_1 - \theta_2 )

( \theta_3 = \pi - \frac{3\pi}{8} - \frac{\pi}{2} )

( \theta_3 = \pi - \frac{3\pi}{8} - \frac{4\pi}{8} )

( \theta_3 = \pi - \frac{7\pi}{8} )

Now, we find the length of the remaining side using the Law of Sines:

( \frac{a}{\sin(\theta_1)} = \frac{b}{\sin(\theta_2)} = \frac{c}{\sin(\theta_3)} )

Given that one side of the triangle has a length of 4, we can set ( a = 4 ):

( \frac{4}{\sin\left(\frac{3\pi}{8}\right)} = \frac{b}{\sin\left(\frac{\pi}{2}\right)} )

( b = 4 \times \frac{\sin\left(\frac{\pi}{2}\right)}{\sin\left(\frac{3\pi}{8}\right)} )

Now, we can calculate the perimeter of the triangle:

( \text{Perimeter} = 4 + 4 + b )

( \text{Perimeter} = 8 + b )

Finally, we substitute the value of ( b ) into the equation:

( \text{Perimeter} = 8 + 4 \times \frac{\sin\left(\frac{\pi}{2}\right)}{\sin\left(\frac{3\pi}{8}\right)} )

( \text{Perimeter} = 8 + 4 \times \frac{1}{\sin\left(\frac{3\pi}{8}\right)} )

By substituting the values and calculating, we can find the numerical value of 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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