# A triangle has two corners with angles of # (2 pi ) / 3 # and # ( pi )/ 4 #. If one side of the triangle has a length of #1 #, what is the largest possible area of the triangle?

Largest possible area of the triangle is 2.3458

The remaining angle:

I am assuming that length AB (1) is opposite the smallest angle.

Using the ASA

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To find the largest possible area of the triangle, we can use the formula for the area of a triangle: ( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} ). Given that one side of the triangle has a length of 1, we can use this side as the base of the triangle.

Now, to find the height of the triangle, we need to determine the length of the perpendicular from the vertex opposite the base to the base itself. We can do this by using trigonometric functions.

Let's denote the angles of the triangle as ( \frac{2\pi}{3} ), ( \frac{\pi}{4} ), and ( \theta ), where ( \theta ) is the angle opposite the side of length 1. Using trigonometric ratios, we can express the height in terms of ( \theta ).

For the angle ( \frac{2\pi}{3} ), the corresponding side opposite is 1, and for ( \frac{\pi}{4} ), the corresponding side opposite is the height. Using the tangent function, we can set up the following equation:

[ \tan\left(\frac{\pi}{4}\right) = \frac{\text{height}}{1} ]

Solving for the height:

[ \text{height} = \tan\left(\frac{\pi}{4}\right) ]

Now, the area of the triangle can be expressed as:

[ \text{Area} = \frac{1}{2} \times 1 \times \tan\left(\frac{\pi}{4}\right) ]

[ \text{Area} = \frac{1}{2} \times \tan\left(\frac{\pi}{4}\right) ]

[ \text{Area} = \frac{1}{2} \times 1 ]

[ \text{Area} = \frac{1}{2} ]

Therefore, the largest possible area of the triangle is ( \frac{1}{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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