# A triangle has vertices A, B, and C. Vertex A has an angle of #pi/12 #, vertex B has an angle of #(5pi)/12 #, and the triangle's area is #21 #. What is the area of the triangle's incircle?

Area of triangle's incircle is

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To find the area of the triangle's incircle, we need to first find the lengths of the triangle's sides using the given angles and the area of the triangle.

Let's denote the sides of the triangle as a, b, and c respectively, with a being opposite angle A, b opposite angle B, and c opposite angle C.

Using the formula for the area of a triangle:

Area = (1/2) * base * height

We have:

21 = (1/2) * b * a * sin(C)

We know that sin(C) = sin(π/2) = 1. So, this simplifies to:

21 = (1/2) * b * a

Now, we need to find the lengths of sides a and b. We can use the Law of Sines, which states:

a / sin(A) = b / sin(B) = c / sin(C)

Given angles A and B, we can find the ratio a/sin(A) and b/sin(B).

For angle A:

sin(π/12) = a / b

For angle B:

sin(5π/12) = b / a

We can solve these two equations to find the values of a and b.

Once we have the lengths of the sides, we can use Heron's formula to find the area of the triangle, which is:

Area = sqrt[s(s - a)(s - b)(s - c)]

where s is the semi-perimeter of the triangle given by:

s = (a + b + c) / 2

Once we have the sides of the triangle, we can use the formula for the radius of the incircle of a triangle, which is:

radius = Area / s

This will give us the radius of the incircle, and the area of the incircle can then be calculated using the formula:

Area_incircle = π * radius^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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