A triangle has two corners with angles of # pi / 12 # and # pi / 2 #. If one side of the triangle has a length of #6 #, what is the largest possible area of the triangle?

Answer 1

Largest possible area of the triangle is #color(green)(67.1769)#

Three angles are #beta = pi/12, gamma = /_C = pi/2, alpha = (5pi)/12#

It’s a right triangle.and length 6 should be opposite to the vertex having smallest angle #pi/12#

Area of the largest possible triangle

#A_t =(1/2) * b * a = (1/2) * b *( b / tan beta)#

#A_t = (1/2) * 6 * (6/tan (pi/12)) ~~ color (green)(67.1769)#

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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 given two sides and the angle between them:

[ \text{Area} = \frac{1}{2} \times \text{side} \times \text{side} \times \sin(\text{angle}) ]

Given that one side of the triangle has a length of 6 and two angles are ( \frac{\pi}{12} ) and ( \frac{\pi}{2} ), we can use the Law of Sines to find the length of the other side.

Using the Law of Sines:

[ \frac{\text{side}}{\sin(\text{angle})} = \frac{\text{side}}{\sin(\text{angle})} ]

[ \frac{6}{\sin(\frac{\pi}{12})} = \frac{x}{\sin(\frac{\pi}{2})} ]

[ x = 6 \times \frac{\sin(\frac{\pi}{2})}{\sin(\frac{\pi}{12})} ]

[ x = 6 \times \frac{1}{\sin(\frac{\pi}{12})} ]

[ x = 6 \times \frac{1}{\frac{\sqrt{6}+\sqrt{2}}{4}} ]

[ x = \frac{24}{\sqrt{6}+\sqrt{2}} ]

Now, we have the length of the other side. We can use this along with the given side length of 6 to calculate the area of the triangle using the formula mentioned above:

[ \text{Area} = \frac{1}{2} \times 6 \times \frac{24}{\sqrt{6}+\sqrt{2}} \times 6 \times \sin(\frac{\pi}{12}) ]

[ \text{Area} = 18 \times \frac{24}{\sqrt{6}+\sqrt{2}} \times \frac{1}{2} ]

[ \text{Area} = 9 \times \frac{24}{\sqrt{6}+\sqrt{2}} ]

[ \text{Area} = \frac{216}{\sqrt{6}+\sqrt{2}} ]

Therefore, the largest possible area of the triangle is ( \frac{216}{\sqrt{6}+\sqrt{2}} ).

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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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