- #1
lizzie
- 25
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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.
how do we prove it?
A,B,C are angles of a triangle.
thanks for any help.
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.
Asch, sorry, forgot how the original inequality looked like . Thanks!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]
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.
SinA, sinB, and sinC represent the sine values of the angles A, B, and C in a triangle, respectively.
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.
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.
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.