# A triangle has two corners with angles of # ( pi ) / 2 # and # ( 5 pi )/ 12 #. If one side of the triangle has a length of #19 #, what is the largest possible area of the triangle?

Largest possible area of triangle is

are the lengths of the sides and opposite angles are

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To find the largest possible area of the triangle, use the formula for the area of a triangle, which is 1/2 times the product of two sides times the sine of the included angle.

Given that two angles are π/2 and 5π/12, the third angle can be found using the fact that the sum of angles in a triangle is π radians (180 degrees).

Let x be the third angle. So, π/2 + 5π/12 + x = π.

Solving for x, we get x = π - π/2 - 5π/12 = π/12.

Now, using the Law of Sines, we can find the length of the other two sides.

Let the side opposite the angle π/2 be a, and the side opposite the angle 5π/12 be b.

So, a/sin(π/2) = b/sin(5π/12).

This simplifies to a = b * (sin(π/2) / sin(5π/12)).

Given that one side (b) has a length of 19, we can find the length of side a.

a = 19 * (sin(π/2) / sin(5π/12)).

Using trigonometric values, sin(π/2) = 1 and sin(5π/12) ≈ 0.9659.

Thus, a ≈ 19 * (1 / 0.9659) ≈ 19.6644.

Now, to find the area of the triangle, use the formula: Area = 1/2 * base * height.

The base of the triangle is 19, and the height can be found using the side length a and the sine of the angle π/12.

Height = a * sin(π/12) ≈ 19.6644 * sin(π/12) ≈ 19.6644 * 0.2588 ≈ 5.086.

Now, calculate the area:

Area = 1/2 * 19 * 5.086 ≈ 48.55 square units.

So, the largest possible area of the triangle is approximately 48.55 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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