# Two corners of a triangle have angles of #pi / 12 # and # pi / 3 #. If one side of the triangle has a length of #6 #, 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 consider the Law of Sines, which states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.

Given that two angles of the triangle are (\frac{\pi}{12}) and (\frac{\pi}{3}), we can find the third angle by subtracting the sum of the known angles from (\pi):

[\text{Third angle} = \pi - \left(\frac{\pi}{12} + \frac{\pi}{3}\right) = \pi - \left(\frac{5\pi}{12}\right) = \frac{7\pi}{12}]

Now, we use the Law of Sines to find the lengths of the other two sides:

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

Given that (b = 6) (one side has a length of 6), we can find the lengths of the other sides using the known angles:

For angle (\frac{\pi}{12}) opposite side (a): [\frac{a}{\sin \left(\frac{\pi}{12}\right)} = \frac{6}{\sin \left(\frac{7\pi}{12}\right)}] [a = 6 \cdot \frac{\sin \left(\frac{\pi}{12}\right)}{\sin \left(\frac{7\pi}{12}\right)}]

For angle (\frac{\pi}{3}) opposite side (c): [\frac{c}{\sin \left(\frac{\pi}{3}\right)} = \frac{6}{\sin \left(\frac{7\pi}{12}\right)}] [c = 6 \cdot \frac{\sin \left(\frac{\pi}{3}\right)}{\sin \left(\frac{7\pi}{12}\right)}]

Now, we can find the perimeter of the triangle:

[\text{Perimeter} = a + b + c]

Plug in the values of (a), (b), and (c) and solve for the perimeter.

[a = 6 \cdot \frac{\sin \left(\frac{\pi}{12}\right)}{\sin \left(\frac{7\pi}{12}\right)}] [c = 6 \cdot \frac{\sin \left(\frac{\pi}{3}\right)}{\sin \left(\frac{7\pi}{12}\right)}]

[a \approx 6 \cdot \frac{\sin \left(\frac{\pi}{12}\right)}{\sin \left(\frac{7\pi}{12}\right)} \approx 6.52] [c \approx 6 \cdot \frac{\sin \left(\frac{\pi}{3}\right)}{\sin \left(\frac{7\pi}{12}\right)} \approx 9.62]

[ \text{Perimeter} \approx 6 + 6.52 + 9.62 \approx 22.14]

So, the longest possible perimeter of the triangle is approximately (22.14) 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.

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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