# A triangle has vertices A, B, and C. Vertex A has an angle of #pi/3 #, vertex B has an angle of #(5 pi)/12 #, and the triangle's area is #4 #. What is the area of the triangle's incircle?

see a solution step below;

firstly, you have to look for the area of triangle..

Let;

Finding their respective angles..

similarly..

similarly..

squareroot both sides..

smilarly..

smilarly..

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To find the area of the triangle's incircle, we first need to find the lengths of the triangle's sides. We can use the law of sines to do this. The law of sines states:

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

where (a), (b), and (c) are the lengths of the sides opposite to angles (A), (B), and (C) respectively.

Given that (A = \frac{\pi}{3}) and (B = \frac{5\pi}{12}), we can find the measure of angle (C) by using the fact that the sum of angles in a triangle is (180^\circ) or (\pi).

[ C = \pi - A - B = \pi - \frac{\pi}{3} - \frac{5\pi}{12} = \frac{\pi}{4} ]

Now, we can use the law of sines to find the lengths of the sides. Let's denote (a), (b), and (c) as the lengths of sides (BC), (AC), and (AB) respectively.

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

[ \frac{a}{\sin(\frac{\pi}{3})} = \frac{b}{\sin(\frac{5\pi}{12})} = \frac{c}{\sin(\frac{\pi}{4})} ]

Using the known values of the angles, we can solve for (a), (b), and (c).

After finding the lengths of the sides, we can use Heron's formula to find the area of the triangle. Once we have the semi-perimeter ((s)) and the lengths of the sides ((a), (b), and (c)), Heron's formula is:

[ A = \sqrt{s(s - a)(s - b)(s - c)} ]

Given that the area of the triangle is (4), we can solve for (s).

Finally, once we have the lengths of the sides, we can use them to find the radius ((r)) of the incircle using the formula:

[ r = \frac{A}{s} ]

where (A) is the area of the triangle and (s) is the semi-perimeter.

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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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