# Two corners of a triangle have angles of #pi / 4 # and # pi / 3 #. If one side of the triangle has a length of #6 #, what is the longest possible perimeter of the triangle?

Longest possible perimeter of the triangle is 21.5447

To get the longest perimeter, we should consider the side corresponding to the angle that is the smallest.

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The longest possible perimeter of the triangle can be found by considering the triangle inequality theorem and the fact that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Given that one side of the triangle has a length of 6, we can find the lengths of the other two sides using trigonometric relationships.

For the angle of ( \frac{\pi}{4} ), the opposite side (let's call it ( a )) can be found using the sine function:

[ \sin\left(\frac{\pi}{4}\right) = \frac{a}{6} ] [ a = 6 \sin\left(\frac{\pi}{4}\right) = 6 \times \frac{\sqrt{2}}{2} = 3\sqrt{2} ]

For the angle of ( \frac{\pi}{3} ), the adjacent side (let's call it ( b )) can be found using the cosine function:

[ \cos\left(\frac{\pi}{3}\right) = \frac{b}{6} ] [ b = 6 \cos\left(\frac{\pi}{3}\right) = 6 \times \frac{1}{2} = 3 ]

Now, we can find the perimeter of the triangle:

[ \text{Perimeter} = 6 + a + b = 6 + 3\sqrt{2} + 3 ]

Therefore, the longest possible perimeter of the triangle is ( 9 + 3\sqrt{2} ).

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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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