# A triangle has two corners with angles of # (3 pi ) / 4 # and # ( pi )/ 6 #. If one side of the triangle has a length of #1 #, what is the largest possible area of the triangle?

The area is

The angles are

To have the greatest area, the length

So,

We apply the sine rule to the triangle

The area of the triangle is

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To find the largest possible area of the triangle, we can use the formula for the area of a triangle given its side lengths and included angle. The formula is:

[A = \frac{1}{2} \times a \times b \times \sin(C)]

Where (a) and (b) are the lengths of two sides of the triangle, and (C) is the included angle between them.

Given that one side of the triangle has a length of 1, let's denote this side as (a). We also have the angles (\frac{3\pi}{4}) and (\frac{\pi}{6}). To find the area, we need to determine the length of the other side, (b), and the included angle, (C).

The side (b) can be found using the Law of Sines, which states:

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

We know (a = 1) and (A = \frac{3\pi}{4}). We can solve for (b) using the Law of Sines:

[\frac{\sin\left(\frac{\pi}{6}\right)}{1} = \frac{\sin\left(\frac{3\pi}{4}\right)}{b}]

[b = \frac{\sin\left(\frac{3\pi}{4}\right)}{\sin\left(\frac{\pi}{6}\right)}]

Now, we can calculate (b) and then find (C), the included angle:

[b = \frac{\sin\left(\frac{3\pi}{4}\right)}{\sin\left(\frac{\pi}{6}\right)}]

[b = \frac{\sqrt{2}/2}{1/2}]

[b = \sqrt{2}]

Now that we have (a), (b), and their included angle, we can calculate the area using the formula:

[A = \frac{1}{2} \times 1 \times \sqrt{2} \times \sin\left(\frac{\pi}{6}\right)]

[A = \frac{1}{2} \times \sqrt{2} \times \frac{1}{2}]

[A = \frac{\sqrt{2}}{4}]

So, the largest possible area of the triangle is (\frac{\sqrt{2}}{4}) square 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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