# A triangle has sides A, B, and C. Sides A and B have lengths of 4 and 1, respectively. The angle between A and C is #(3pi)/8# and the angle between B and C is # (5pi)/12#. What is the area of the triangle?

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The clue here is to realize that you can split up this triangle into two right triangles. Then, by using the appropriate trig identities, you can find the lengths needed to compute the area of both triangles. Adding both areas will then give you the total area.

Let's first work with the "A" side of the triangle:

The area of one side of this triangle, then, is:

Now let's compute the area of the other side. Since we know the height, which is universal for both triangles, we just need to compute the base.

The area of the other side of this triangle, then, is:

Thus, the total area is:

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

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

where 'a' and 'b' are the lengths of two sides of the triangle, and 'C' is the angle between them.

Given:

- Side A (a) has a length of 4.
- Side B (b) has a length of 1.
- The angle between side A and side C is (3π)/8.
- The angle between side B and side C is (5π)/12.

Substitute the given values into the formula:

Area = (1/2) * 4 * 1 * sin((3π)/8)

Calculate the sine of (3π)/8 using a calculator.

Once you have the value of sin((3π)/8), multiply it by (1/2) * 4 * 1 to find the area of the triangle.

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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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