# A triangle has vertices A, B, and C. Vertex A has an angle of #pi/12 #, vertex B has an angle of #pi/6 #, and the triangle's area is #4 #. What is the area of the triangle's incircle?

Incircle equations source

Given:

We can compute the measure of angle C:

Using a equation from the source

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To find the area of the triangle's incircle, you can use the formula:

[ \text{Area of incircle} = \pi r^2 ]

where ( r ) is the radius of the incircle.

The radius of the incircle can be found using the formula:

[ r = \frac{\text{Area of the triangle}}{\text{Semiperimeter of the triangle}} ]

The semiperimeter of the triangle (( s )) is calculated as:

[ s = \frac{a + b + c}{2} ]

where ( a ), ( b ), and ( c ) are the lengths of the sides of the triangle.

Given that the area of the triangle is 4, we need to find the lengths of the sides of the triangle using the law of sines:

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

Now, ( \sin(\frac{\pi}{12}) = \frac{1}{2} \sin(\frac{\pi}{6}) = \frac{\sqrt{3}}{2} )

From here, you can calculate the lengths of the sides and then find the semiperimeter. Once you have the semiperimeter, you can find the radius of the incircle and then use that to find the area of the incircle.

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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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