14 - Trigonometry

Summary

Introduction

This chapter begins the consideration of mathematical topics expected to be known for the AMC 12 exam but not for the AMC 10. There have not been many problems involving trigonometry on the more recent exams because the use of calculators makes many of the traditional problems trivial. However, the topic is important and the subject matter dealing with this subject is quite general. Students taking the AMC 10 examinations will not see problems involving trigonometry.

Definitions and Results

The two very basic definitions in trigonometry are the sine and the cosine of a given number or given angle. There are two standard and equivalent ways to define these concepts; one uses right triangles, and the other uses the unit circle. Defining the sine and cosine for angles using right triangles is generally the first definition that is presented, but the unit circle approach is more appropriate when the sine and cosine are needed for functional representation. We will give the unit circle definition, since it is more general, and may not be as familiar.

Place a unit circle in the xy-plane. For each positive real number t, define P(t) as the point on this unit circle that is a distance t along the circle, measured counterclockwise, from the point (1, 0). For each negative number t, define P(t) as the point on this unit circle that is a distance |t| along the circle, measured clockwise, from the point (1, 0). Finally, define P(0) = (1, 0). In this way we have, for each real number t, a unique pair (x(t), y(t)) of xy-coordinates on the unit circle to describe the point P(t). These coordinates provide the two basic trigonometric functions.

Definition 1 The Sine and Cosine: Suppose that the coordinates of a point P(t) on the unit circle are (x(t), y(t)). Then the sine of t, written sin t, and the cosine of t, written cos t, are defined by

sin t = y(t) and cos t = x(t).

These definitions are also used for the sine and cosine of an angle with radian measure t. So the trigonometric functions serve two purposes, directly as functions with domain the set of real numbers and indirectly as functions with domain the set of all possible angles, where the angles are given in radian measure.