By selecting certain special triangles, students can learn about the laws of sines and cosines without wrestling with long decimal representations or irrational numbers. Since the law of cosines requires only one of the three angles of a triangle, there are many examples of triangles with integral sides and a cosine that can be represented exactly…

This paper takes a known point from Brocard geometry, a known result from the geometry of the equilateral triangle, and bring in Euler's [empty set] function. It then demonstrates how to obtain new Brocard Geometric number theory results from them. Furthermore, this paper aims to determine a [triangle]ABC whose Crelle-Brocard Point [omega]…

The author has always been fascinated by the title identity. It's charming and simple, as well as easy to believe after pressing a few calculator keys. Several fine proofs have appeared in the literature, including several proofs without words. His own earlier proof is trigonometric, and he has often been dissatisfied with not being able to…

The typical precalculus book contains the obscure trigonometric identities known as the product-to-sum formulas. They usually get short treatment (or none) in a precalculus course because they are so rarely used. This is unfortunate since they have an interesting history. Before the invention of logarithms they were used to perform multiplications…

Proposes that a mathematical treatment of Strong Inequality Triangles (sum of any two sides is greater than the third) within the reach of high school and college students is possible. Invites further explorations of other properties of Strong Inequality Triangles. (ASK)

The Law of Sines and The Law of Cosines are of paramount importance in the field of trigonometry because these two theorems establish relationships satisfied by the three sides and the three angles of any triangle. In this article, the authors use these two laws to discover a host of other trigonometric relationships that exist within any…

The authors draw together a variety of facts concerning a nonlinear differential equation and compare the exact solution with approximate solutions. Then they provide an expository introduction to the elliptic sine function suitable for presentation in undergraduate courses on differential equations. (MNS)

An integral, either definite or improper, cannot always be computed by elementary methods, such as reversed usage of differentiation formulae. Graphical properties, in particular symmetries, can be useful to compute the integral, via an auxiliary computation. We present graded examples, then prove a general result. (Contains 4 figures.)

An old way to determine asymptotes for curves described in polar coordinates is presented. Practice in solving trigonometric equations, in differentiation, and in calculating limits is involved. (MNS)

How to determine when there is a unique solution when two sides and an angle of a triangle are known, using simple algebra and the law of cosines, is described. (MNS)

We examine the behavior of the curvature function associated with most common families of functions and curves, with the focus on establishing where maximum curvature occurs. Many examples are included for student illustrations. (Contains 18 figures.)

This paper is written on a level accessible to college/university students of mathematics who are taking second-year, algebra based, mathematics courses beyond calculus I. This article combines material from geometry, trigonometry, and number theory. This integration of various techniques is an excellent experience for the serious student. The…

Details are given of a simple computer program written in BASIC which calculates the sine of an angle through an application of DeMoivre's Theorem. The program is included in the material, and the program's success is discussed in terms of why the approximation works. (MP)

Presents an approach to Vieta's formula involving pi and infinite product expansions of the sine and cosine functions. Indicates how the formula could be used in computing approximations of pi. (MVL)

Describes the derivation of the parameters incorporated into computer programs that are utilized to draw hypocycloids, which are the loci of points traced out by a point on a disk as it rolls against a circle and its interior. Includes information to obtain copies of the programs described. (JJK)