SinA + sinB + sinC <= (3 x 3^0.5 )/2

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In summary, the conversation discusses different approaches to proving the inequality sinA + sinB + sinC <= (3 x 3^0.5 )/2, where A, B, and C are angles of a triangle. One approach involves using Jensen's inequality, which states that the arithmetic mean of a convex function is greater or equal to the function of the arithmetic mean. Another approach involves rewriting one of the angles in terms of the other two and showing that sinx is concave. Both approaches ultimately use the fact that the sum of the angles in a triangle is pi.
  • #1
lizzie
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sinA + sinB + sinC <= (3 x 3^0.5 )/2
how do we prove it?
A,B,C are angles of a triangle.
thanks for any help.
 
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  • #3
sorry kurret i am still unable to prove it
 
  • #4
A function is convex (or concace up) if its second derivative is greater than zero. For the full definition of convexity see http://mathworld.wolfram.com/ConvexFunction.html.
Jensens inequality states that the arithmetic mean of a convex function is greater or equal than the function of the arithmetic mean, ie:
[tex]\frac{f(x_1)+f(x_2)+...+f(x_n)}{n} \geq f(\frac{x_1+x_2+...+x_n}{n})[/tex]
For concave functions the inequality is reversed. So consider the function sinx, and show that sinx is concave. After that, set up jensens inequality and use that the sum of the angles in a triangle is pi.
 
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  • #5
Alternative solution:
Rewrite C as pi-A-B, then Sin(C)=Sin(A+B). Now assume that B is any value between 0 and pi, and you can make the LHS a function of A, and find its minimum value. Maybe easier, but I think jensens inequality is really powerful and its really a good idea to learn to master it :)
 
  • #6
Kurret said:
A function is convex (or concace up) if its second derivative is greater than zero. For the full definition of convexity see http://mathworld.wolfram.com/ConvexFunction.html.
Jensens inequality states that the arithmetic mean of a convex function is greater or equal than the function of the arithmetic mean, ie:
[tex]\frac{f(x_1)+f(x_2)+...+f(x_n)}{n} \geq f(\frac{x_1+x_2+...+x_n}{n})[/tex]
So consider the function sinx, and show that sinx is convex. After that, set up jensens inequality and use that the sum of the angles in a triangle is pi.

sin x is concave, and for concave functions Jensen's inequality is reversed giving:
[tex]\frac{f(x_1) + f(x_2) + \ldots + f(x_n)}{n} \leq f\left(\frac{x_1+x_2+...+x_n}{n}\right)[/tex]
 
  • #7
gunch said:
sin x is concave, and for concave functions Jensen's inequality is reversed giving:
[tex]\frac{f(x_1) + f(x_2) + \ldots + f(x_n)}{n} \leq f\left(\frac{x_1+x_2+...+x_n}{n}\right)[/tex]
Asch, sorry, forgot how the original inequality looked like :frown:. Thanks!
 

What is the equation "SinA + sinB + sinC <= (3 x 3^0.5 )/2" used for?

The equation "SinA + sinB + sinC <= (3 x 3^0.5 )/2" is used to determine whether a triangle is acute, right, or obtuse based on the values of its angles.

What do SinA, sinB, and sinC represent in the equation?

SinA, sinB, and sinC represent the sine values of the angles A, B, and C in a triangle, respectively.

What is the significance of the value (3 x 3^0.5 )/2 in the equation?

The value (3 x 3^0.5 )/2 is the maximum possible value for the sum of three sine values in a triangle. If the sum is less than or equal to this value, then the triangle is acute. If the sum is greater than this value, then the triangle is obtuse.

How is the equation derived?

The equation is derived from the Law of Sines, which states that in a triangle, the ratio of each side to the sine of its opposite angle is equal. By substituting the values for the sides and angles in the Law of Sines, the equation "SinA + sinB + sinC <= (3 x 3^0.5 )/2" can be derived.

Can the equation be used for any type of triangle?

Yes, the equation can be used for any type of triangle - acute, right, or obtuse. However, it is most commonly used for acute triangles since the sum of sine values for right and obtuse triangles will always be more than (3 x 3^0.5 )/2.

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