A triangle has two corners with angles of # ( pi ) / 4 # and # ( 7 pi )/ 12 #. If one side of the triangle has a length of #1 #, what is the largest possible area of the triangle?

Answer 1

Largest possible area = 0.683

Given are the two angles #(7pi)/12# and #pi/4# and the length 1

The remaining angle:

#= pi - (((7pi)/12) + pi/4) = pi/6#

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

Using the ASA

Area#=(c^2*sin(A)*sin(B))/(2*sin(C)#
Area#=( 1^2*sin(pi/4)*sin((7pi)/12))/(2*sin(pi/6))#

Area#=0.683

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Answer 2

To find the largest possible area of the triangle, we can use the formula for the area of a triangle:

Area = (1/2) * base * height.

First, we need to find the length of the other two sides of the triangle using trigonometric ratios. Let's denote the length of the side opposite the angle (π/4) as a, and the length of the side opposite the angle (7π/12) as b.

Using the given angles and the side length of 1, we can set up the following trigonometric relationships:

  1. For the angle (π/4): tan(π/4) = a / 1 So, a = tan(π/4).

  2. For the angle (7π/12): tan(7π/12) = b / 1 So, b = tan(7π/12).

Now, we have the lengths of all three sides of the triangle.

Next, we can find the height of the triangle. The height is the perpendicular distance from the vertex with the angle (7π/12) to the base (the side with length 1). We can find the height using the formula:

height = side * sin(angle)

Using the side with length 1 and the angle (7π/12), we get:

height = 1 * sin(7π/12).

Now, we have all the necessary values to calculate the area of the triangle using the formula mentioned earlier. After finding the area, we can maximize it by adjusting the values of a and b, while keeping the angle measures constant.

Once the area is calculated, we can compare it with other possible configurations to ensure that it is indeed the largest possible area.

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Answer from HIX Tutor

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.

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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