A triangle has two corners with angles of # (2 pi ) / 3 # and # ( pi )/ 4 #. If one side of the triangle has a length of #2 #, what is the largest possible area of the triangle?
Area of the largest possible triangle is
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To find the largest possible area of the triangle given that it has angles of ( \frac{2\pi}{3} ) and ( \frac{\pi}{4} ) and one side of length 2, you can use the formula for the area of a triangle:
[ A = \frac{1}{2}ab\sin(C) ]
where ( a ) and ( b ) are the lengths of two sides of the triangle and ( C ) is the angle between them.
Given that one side has a length of 2, let's call it ( a ), and let ( b ) be another side. We're looking for the largest possible area, so we want to maximize ( b ).

Find the Third Angle: The sum of angles in a triangle is ( \pi ) radians. So, to find the third angle, subtract the given angles from ( \pi ).
[ \text{Third angle} = \pi  \frac{2\pi}{3}  \frac{\pi}{4} ]

Use Law of Sines: Use the Law of Sines to find the ratio of sides ( a ) and ( b ).

Maximize ( b ): The side ( b ) should be maximized to get the largest area. It happens when ( b ) is opposite to the largest angle.

Calculate Area: Plug the values of ( a ), ( b ), and ( C ) into the area formula.
By following these steps, you can find the largest possible area of the triangle.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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