What Is the Minimum Value of (a²+b²)/c² in Triangle ABC?

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In summary, the expression (a^2+b^2)/c^2 represents the ratio of the sum of the squares of two sides to the square of the third side in Triangle ABC. Finding the minimum value of this expression is important as it can provide insights into the shape and properties of the triangle. The minimum value can be calculated using the Pythagorean theorem, and it can never be greater than 1 in Triangle ABC. This minimum value can be applied in real-life situations involving right triangles, such as in engineering, construction, navigation, and surveying.
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Here is this week's POTW:

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In a triangle $ABC$, it is given that $\dfrac{\cos A}{1+\sin A}=\dfrac{\sin 2B}{1+\cos 2B}$.

Find the minimum value of $\dfrac{a^2+b^2}{c^2}$.

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I hope I made no errors in calculation.
From laws of sines
[tex]f:=\frac{a^2+b^2}{c^2}=\frac{\sin^2 A+\sin^2 B}{\sin^2 C}=\frac{\sin^2 A+\sin^2 B}{\sin^2 (A+B)}[/tex]
The given condition reads
[tex]\frac{\cos A}{1+\sin A}=\tan B[/tex]
By this we can delete B in the above formula to get
[tex]f(x)=2x-3+\frac{4}{1+x}[/tex]
where
[tex]x=\sin A[/tex]

[tex]f'(x)=2-\frac{4}{(1+x)^2}[/tex]
[tex]f'(\sqrt{2}-1)=0[/tex]
[tex]f(\sqrt{2}-1)=4\sqrt{2}-5 \approx 0.66 [/tex] as minimum.
 

FAQ: What Is the Minimum Value of (a²+b²)/c² in Triangle ABC?

What is the significance of Triangle ABC in this problem?

Triangle ABC is a specific triangle used in this problem to represent a general case. It is not necessary to know the specific values of the sides or angles of Triangle ABC, as the problem is asking for the minimum value of a mathematical expression that can be applied to any triangle.

What does (a^2+b^2)/c^2 represent in this problem?

This expression represents the ratio of the sum of the squares of two sides of a triangle to the square of the remaining side. It is a common mathematical expression used in geometry and can also be interpreted as the tangent of the angle opposite to the side c.

Why is finding the minimum value of (a^2+b^2)/c^2 important?

Finding the minimum value of this expression can help us determine the minimum possible value of the angle opposite to side c in any triangle. It can also be used to find the minimum possible value of the perimeter or area of a triangle, as these values are directly related to the lengths of its sides.

How can I solve this problem?

This problem can be solved using various approaches, such as using trigonometric identities, calculus, or geometric reasoning. It is important to carefully consider the given information and use appropriate mathematical concepts to arrive at the correct solution.

What are some real-life applications of this problem?

The concept of finding the minimum value of a mathematical expression is commonly used in optimization problems, such as minimizing costs or maximizing efficiency. In the case of this problem, it can also be applied in engineering and architecture to determine the minimum angle or length of a side in a triangular structure to ensure stability and strength.

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