- #71
harry654
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soo I see that it apply but I don't know mathematically explain :(
harry654 said:sin²α + sin²β = 1 when α+β=90 so inequality doesn't apply I think
harry654 said:apply sin²α + sin²β = 1 when α+β = 90° and from that α+β > 90 so sin²α + sin²β >1
so when I prove sin²α + sin²β >1, I proved that (a²+b²)cos(α-β) ≤ 2abcos(α-β)?
harry654 said:I have (a²+b²)cos(α-β) ≤ (a²+b²-h²)/cos(α-β)
and from it I get
(a²+b²)cos²(α-β) ≤ a²+b²-h²
but how can I tidy up later?
harry654 said:Assume that apply sin(α-β) is not 0 then question is : Is cos(α-β) positive?
harry654 said:Hi tiny-tim!
If α=135 and β=30 then sin²α + sin²β < 1 and apply α+β>90 so what isn't correct?
tiny-tim said:so (tidying-up time! ) actually we need to prove that, if α + β > 90°, then:
sin²α + sin²β > 1 if cos(α-β) > 0, ie if |α-β| < 90°, and
sin²α + sin²β < 1 if cos(α-β) < 0, ie if |α-β| > 90°;
(alternatively, we could simply point out that we needn't bother with the cos(α-β) < 0 case, since the originally given inequality, (a² + b²)cos(α-β) ≤ 2ab, is obviously true in that case, since the LHS is negative and the RHS is positive! )
The inequality (a^2+b^2)cos(α-β)<=2ab is used to determine when equality occurs in a trigonometric expression involving two variables, a and b, and two angles, α and β.
To solve for equality, you must first simplify the expression by expanding the trigonometric function using the cosine difference formula. Then, you can solve for the values of a and b that make the inequality true.
Yes, equality can occur for any values of a and b as long as they satisfy the trigonometric expression. However, there may be multiple solutions or no solution at all.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. In this inequality, the left side represents the sum of the squares of the two sides, while the right side represents the square of the hypotenuse. Therefore, equality occurs when the triangle is a right triangle.
Yes, this inequality can be used in various fields such as engineering, physics, and navigation to calculate the distance between two points or the angle between two objects. It can also be used in trigonometric identities and proofs.