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A note on trigonometric identities
Abstract
In this note, we study trigonometric identities involving the angles of an arbitrary triangle and we give algorithms for verifying such identities.
... see [5]. On the other hand, the relation among the cotangents is given by the simple and elegant formula 1 cot A cot B + cot B cot C + cot C cot A = 1. ...
In this paper, we find relations that define the cotangents of the angles of a general triangle, and we use these relations to describe a method for optimizing symmetric functions in the cotangents of such angles and for establishing trigonometric inequalities that involve such functions.
... where r and R are the inradius and circumradius of ABC, and (2.4) follows by squaring sin A sin B = cos C + cos A cos B and using sin 2 θ = 1 − cos 2 θ. For (2.5), see [4, 678, page 166], for (2.6), see [10], and for more on (2.4), see [6]. Conversely, let u, v, and w satisfy (2.2), (2.3), and (2.4), and let s, p and q be as in (2.1). ...
In this note, we describe a method for establishing trigonometric inequalities that involve symmetric functions in the cosines of the angles of a triangle. The method is based on finding a complete set of relations that define the cosines of such angles.
Figure 1 shows a triangle ABC with the midpoints A ′, B ′ and C ′ of its sides. The line segments AA ′, BB ′ and CC ′ are called the medians, and the point G of their intersection the centroid. The line segments AG , BG and CG will be called, for lack of a better name, the semi-medians . It is interesting that the medians of any triangle can serve as the side lengths of some triangle. This property of the medians is referred to as the median triangle theorem in [1, §473, page 282], and is discussed, together with generalisations to tetrahedra and higher dimensional simplices, in [2].
In this paper we attempt to generalize the notion of "unique fac-torization domain" in the spirit of "half-factorial domain". It is shown that this new generalization of UFD implies the now well known notion of half-factorial domain. As a consequence, we discover that the one of the standard axioms for unique factorization domains is slightly redundant.
Trigonometric Identities
- Magid
Andy R. Magid, Trigonometric Identities, Math. Ma9. 47 (1974), 226-227.