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

If the hypotenuse is 1 then

If one of the two catheti is 1 then

In any case we are dealing with a rectangle triangle.

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To find the largest possible area of the triangle, we can use the formula for the area of a triangle, which is ( \frac{1}{2} \times \text{base} \times \text{height} ). Given that one side of the triangle has a length of 1 and the angles are ( \frac{\pi}{3} ) and ( \frac{\pi}{6} ), we can determine the height of the triangle using trigonometry.

Since the angles are ( \frac{\pi}{3} ) and ( \frac{\pi}{6} ), the side opposite the ( \frac{\pi}{3} ) angle is ( \frac{\sqrt{3}}{2} ) times the length of the side with length 1. Therefore, the height of the triangle is ( \frac{\sqrt{3}}{2} ).

Using the formula for the area of a triangle:

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

[ \text{Area} = \frac{1}{2} \times 1 \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{4} ]

So, the largest possible area of the triangle is ( \frac{\sqrt{3}}{4} ).

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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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