A triangle has two corners with angles of # pi / 4 # and # pi / 6 #. If one side of the triangle has a length of #1 #, what is the largest possible area of the triangle?

Answer 1

The largest area is #A ~~ 0.68#

Let #/_ A = pi/6# and #/_ B = pi/4#, then #/_C = pi - pi/4 - pi/6 = (7pi)/12#

Note: We assign the given side to be the side opposite the smallest angle, because that give the largest area.

Let a = the side opposite #/_A# = 1 Let b = the side opposite #/_B# Let c = the side opposite #/_C#

We can use The Law of Sines to write the following equation:

#b/sin(/_B) = (a)/sin(/_A)#

Solve for b:

#b = (a)sin(/_B)/sin(/_A)#
Choose side #a# to be the base, then the height, h, of the triangle is:
#h = bsin(/_C)#

Substituting for b:

#h = ((a)sin(/_B)/sin(/_A))sin(/_C)#

The area, A, with side a as the base is:

#A = 1/2ah#

Substituting for h:

#A = 1/2a((a)sin(/_B)/sin(/_A))sin(/_C)#
#A = (a^2)(sin(/_B)sin(/_C))/(2sin(/_A))#
#A = (1^2)(sin(pi/4)sin((7pi)/12))/(2sin(pi/6))#
#A ~~ 0.68#
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Answer 2

To find the largest possible area of the triangle given the two angles (\frac{\pi}{4}) and (\frac{\pi}{6}), and one side length of 1, we use the formula for the area of a triangle:

[ A = \frac{1}{2} \times \text{base} \times \text{height} ]

Since one side of the triangle is given as 1, and we are trying to maximize the area, we need to maximize the height of the triangle.

Using trigonometric functions, we can find the height of the triangle as follows:

[ \tan(\frac{\pi}{4}) = \frac{\text{height}}{1} ]

[ \text{height} = \tan(\frac{\pi}{4}) = 1 ]

Similarly, for the other angle:

[ \tan(\frac{\pi}{6}) = \frac{\text{height}}{1} ]

[ \text{height} = \tan(\frac{\pi}{6}) = \sqrt{3} ]

Now, we have the height of the triangle as (\sqrt{3}).

Using the formula for the area of a triangle:

[ A = \frac{1}{2} \times 1 \times \sqrt{3} ]

[ A = \frac{\sqrt{3}}{2} ]

Therefore, the largest possible area of the triangle is ( \frac{\sqrt{3}}{2} ) square units.

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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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