Page 18 - MATINF Nr. 8

On the bounds of a product involving cosines

1
Leonard Mihai Giugiuc and Constantin Mateescu 2

The aim of this paper is to determine the best upper and lower bounds of

cos(A + α) cos(B + α) cos(C + α)

over all non-obtuse-angled and arbitrary triangles ABC, where 0 < α ≤ π is given.
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1 Introduction

Main Problem. Let ABC be a triangle and let α ∈ 0, π be a given angle. Find the best
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numerical lower and upper bounds of

cos(A + α) cos(B + α) cos(C + α)
over

1. all non-obtuse-angled triangles ABC;
2. all arbitrary triangles ABC.

2 Equivalent statement

The identity
cos(x + y) = cos x cos y − sin x sin y
holds for all real numbers x and y, so we have

cos(A + α) = cos A cos α − sin A sin α = sin(α) · (cot α cos A − sin A)

as well as the cyclic counterparts.
√ î √ ä
π
As 0 < α ≤ , it means that cot α ≥ 3 , so by setting k = cot α ∈ 3 , ∞ we get the
3 3 3
following:
√
Equivalent Problem. Let ABC be a triangle and let k be a real number so that k ≥ 3 .
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Find the best numerical lower and upper bounds of
F(k) := (k cos A − sin A) (k cos B − sin B) (k cos C − sin C)

over

1. all non-obtuse-angled triangles ABC;
2. all arbitrary triangles ABC.

1
Teacher, National College ”Traian”, Drobeta-Turnu Severin, leonardgiugiuc@yahoo.com
2
Independent Researcher, Pites , ti, costika1234@gmail.com
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