# Two corners of a triangle have angles of # ( pi )/ 3 # and # ( pi ) / 4 #. If one side of the triangle has a length of # 1 #, what is the longest possible perimeter of the triangle?

Largest possible area of the triangle is 0.7888

The remaining angle:

I am assuming that length AB (1) is opposite the smallest angle.

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To find the longest possible perimeter of the triangle, you need to maximize the length of the third side.

Given that two angles of the triangle are π/3 and π/4, you can use the fact that the sum of the angles in a triangle is π radians (180 degrees) to find the measure of the third angle:

Third angle = π - (π/3) - (π/4)

With this third angle, you can now determine the longest possible perimeter by maximizing the length of the third side. Since one side of the triangle has a length of 1, the longest possible perimeter occurs when the third side is as long as possible.

By using trigonometric relationships (such as the Law of Sines), you can determine the length of the third side. However, for simplicity, the triangle with angles of π/3, π/4, and the third angle will be an isosceles triangle with equal sides opposite the equal angles.

Now, you can find the length of the third side using trigonometry:

cos(π/3) = Adjacent side / Hypotenuse cos(π/3) = 1 / (side opposite to π/3)

From this, you can find the length of the side opposite to π/3.

cos(π/3) = 1 / (side opposite to π/3) side opposite to π/3 = 1 / cos(π/3)

Similarly, you can find the length of the side opposite to π/4.

cos(π/4) = 1 / (side opposite to π/4) side opposite to π/4 = 1 / cos(π/4)

Now, you have the lengths of two sides of the triangle. Since the triangle is isosceles, the third side will also have the same length.

Calculate the length of the third side using the above calculations.

Once you have the lengths of all three sides, you can find the perimeter of the triangle by adding them together.

This perimeter will be the longest possible perimeter 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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