Chapter 5: Trigonometric Functions

The term "trigonometry" is a combination of the two Greek terms, "trigon" and "metria", which translate as "triangle" and "measurement", and was coined by Bartholomeo Pitiscus in 1595. Thus, one might assume that "trigonometric" functions (such as, sin(x), cos(x), tan(x), etc.) would be defined in terms of measurements of triangles. While this is the way they are naively defined in most High School textbooks, the earliest known calculations related to these functions were by the Greek mathematician Hipparchus in about 140 BC as measurements of circles (specifically, the lengths of chords of a circle). Thus, "trigonometric" functions are also commonly referred to as "circular" functions.

While these functions do relate to measurements of both triangles and circles, from a functional point of view, the most modern perspective on these functions is actually in terms of the exponential from Chapter 3. In 1748 (following the work of Johann Bernoulli, Roger Cotes and Abraham de Moivre), Euler proved the formula:

e^ix = cos(x) + i sin(x).

where i is the "complex" number such that i² = -1. This shows that the basic trigonometric functions, cos and sin (from which all other trigonometric function can be defined) come out of the exponential function (with base e; this is another reason why mathematicians consider e as the most "natural" base for exponentials and logarithms). We will follow this approach, since:

it is the most modern and Calculus-based approach,
it directly relates these functions to one we have already studied, and
it provides the easiest way to derive almost all of the important formulas used in Trigonometry.

To use Euler's formula as a definition, we will first need to discuss complex numbers and their geometry. We will then demonstrate the relationship between the complex exponential e^ix and circles. This will lead to a discussion of different units for angle measurements. We will then define the other trigonometric functions, and discuss their various algebraic properties. We will next practice our transformational graphing technique to plot and model trigonometric functions. We will end with a discussion of the geometric properties of trigonometric functions and their applications.

Go to Complex Numbers and Geometry

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