# A triangle has two corners with angles of # pi / 2 # and # ( pi )/ 8 #. If one side of the triangle has a length of #13 #, what is the largest possible area of the triangle?

The area is

If one of the angles of the triangle is

From trigonmetry

So

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To find the largest possible area of the triangle given two angles and one side length, we can use trigonometry.

Let's denote the angles of the triangle as ( \frac{\pi}{2} ), ( \frac{\pi}{8} ), and ( \theta ) (where ( \theta ) is the remaining angle).

Using the fact that the sum of angles in a triangle is ( \pi ), we find ( \theta = \pi - \left(\frac{\pi}{2} + \frac{\pi}{8}\right) = \frac{3\pi}{8} ).

Now, we can use the Law of Sines to find the lengths of the other two sides of the triangle.

[ \frac{a}{\sin(\frac{\pi}{2})} = \frac{13}{\sin(\frac{3\pi}{8})} ] [ a = 13 \cdot \sin(\frac{\pi}{2}) \cdot \csc(\frac{3\pi}{8}) ] [ a \approx 18.44 ]

[ \frac{b}{\sin(\frac{\pi}{8})} = \frac{13}{\sin(\frac{3\pi}{8})} ] [ b = 13 \cdot \sin(\frac{\pi}{8}) \cdot \csc(\frac{3\pi}{8}) ] [ b \approx 35.89 ]

Now, we can find the area of the triangle using the formula ( \frac{1}{2} \cdot a \cdot b ).

[ \text{Area} = \frac{1}{2} \cdot 18.44 \cdot 35.89 ] [ \text{Area} \approx 331.42 ]

So, the largest possible area of the triangle is approximately ( 331.42 ) 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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