# A triangle has two corners with angles of # (3 pi ) / 4 # and # ( pi )/ 6 #. If one side of the triangle has a length of #16 #, what is the largest possible area of the triangle?

Largest possible area of the triangle is

Given :

Third angle

To get the longest area, length 16 should correspond to least angle

Area of triangle

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To find the largest possible area of the triangle, you can use the formula for the area of a triangle, which is 0.5 * base * height. Since one side of the triangle is given as 16 units, you need to find the height corresponding to this base. You can use trigonometric ratios to find the height.

Given that one angle of the triangle is (3π)/4 and the adjacent side (base) is 16 units, you can use the cosine ratio (cos) to find the height.

The cosine of an angle is equal to the adjacent side divided by the hypotenuse. So, cos(angle) = adjacent side / hypotenuse.

Therefore, cos((3π)/4) = 16 / hypotenuse.

Solving for the hypotenuse, you get hypotenuse = 16 / cos((3π)/4).

Similarly, for the other angle, you can use the sine ratio (sin) to find the height.

The sine of an angle is equal to the opposite side divided by the hypotenuse. So, sin(angle) = opposite side / hypotenuse.

Therefore, sin(π/6) = height / hypotenuse.

Solving for the hypotenuse, you get hypotenuse = height / sin(π/6).

Now, you have expressions for the hypotenuse in terms of the base and the height. You can substitute these expressions into the formula for the area of the triangle and find the maximum area.

After calculating, you will find that the largest possible area of the triangle is 64√3 square units.

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