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Why the laws of Cosines on the sphere and in the plane are equivalent

Mathematics and Computer Education, Spring 2003 by Ayoub, Ayoub B

INTRODUCTION

The Law of Cosines is a fundamental theorem in trigonometry and relates the cosine of any angle in a triangle to its sides. In [Delta]ABC, if we denote the lengths of its sides opposite to [angle]A, [angle]B, [angle]C by a, b, c respectively, then the cosine law, applied to angle C, states that

c^sup 2^ = a^sup 2^ + b^sup 2^ - 2ab cosC

The history of this law could be traced back to Euclid's Elements, about 300 B.C.E. There one finds two geometric theorems, namely propositions 12 and 13 of Book II [3], which are equivalent to the Law of Cosines. We combine them here as

c^sup 2^ = a^sup 2^ + b^sup 2^ ... 2a(CD)

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according to whether angle C is acute or obtuse, and where CD is the orthogonal projection of the side BC on the side AC (Figure 1a and Figure 1b).

It was not until 1620, when Edmund Gunter [14] suggested the use of "cosine of an angle" for the sine of the complementary angle. This meant that cos C = sin(90[degrees] - C). Then the Law of Cosines took the familiar form.

On the other hand, the Law of Cosines for a spherical triangle was first given by the Arab astronomer Al-Battani around 920 A.C.E. [2]. In our current notations the law is

cos c = cos a cos b + sin a sin b cos C

where a , b , and c are the lengths of the sides of a triangle on the surface of a sphere of unit radius and C is an angle of the triangle on which we will elaborate in the next section.

In this article, we will introduce a proof based on vector algebra of the law of cosines for spherical triangles. This proof will show that the law of cosines for plane triangles implies the law of cosines for spherical triangles. Then we will use Maclaurin series and the limit process to deduce the law of cosines for plane triangles from the law of cosines for spherical triangles.

SPHERICAL COSINE LAW VIA VECTOR ALGEBRA

On the unit sphere whose center is O, we consider the triangle ABC whose sides BC , CA , AB are arcs of great circles, i.e. their centers coincide with the center O of the sphere. Since the sphere is of unit radius, the arc lengths a , b , c of the sides BC , CA , AB of the spherical triangle are equal to the central angles BOC, COA, AOB respectively. The angle ACB between the arcs CA and CB is equal to the angle between the planes OCA and OCB and will be referred to as C (Figure 2).

REFERENCES

1. Howard Anton, Calculus, A New Horizon, 6th Edition, John Wiley & Son, New York, NY, pp. 776-778 (1999).

2. Howard Eves, An Introduction to the History of Mathematics, 6th Edition, Saunders College Publishing, Philadelphia, PA, p. 235 (1990).

3. Eleanor Hay es, "Trigonometric Identities," Historical Topics for the Mathematics Classroom, National Council of Teachers of Mathematics, Reston, VA, pp. 374-375 (1993).

4. Roger Lowe and Cynthia Schench, "Sine and Cosine," Historical Topics for the Mathematics Classroom, National Council of Teachers of Mathematics, Reston, VA, pp. 368-371 (1993).

Ayoub B. Ayoub

Mathematics Department

The Pennsylvania State University

Abington College

Abington, Pennsylvania 19001

Why the laws of Cosines on the sphere and in the plane are equivalent

Mathematics and Computer Education, Spring 2003 by Ayoub, Ayoub B

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