Section 8: Basic Trigonometric Functions

Our goal in this section is to define and develop the six basic trigonometric functions. We start with two of the more important transcendental functions, namely the sine function and the cosine function. These are universally denoted by y = sinx and y = cos x respectively. In order to define sinx and cosx, we first need to agree on how we are going to measure angles. An angle, θ, consists of two rays emanating from a common point called the vertex. One ray is called the initial side, the other is the terminal side. Because we can choose where to put the origin in the Cartesian plane and we can decide where the positive x-axis goes, we will assume that the vertex of our angle is at the origin and that the initial side is along the positive x-axis. This is called putting the angle in standard position. Finally, since only the direction of the rays matters, we can chop off both rays at length one. Our angle theta is now determined by the two points at the end of these line segments of length 1, both of which lie on the unit circle whose equation is x + y = 1. The point on the initial side is simply (1, 0). Here we will denote the point on the terminal side by P (θ). All we need to determine how to measure θ is to know how we generated the angle. That is we need to know how we go from (1, 0) to P (θ). If we generate θ by moving from (1, 0) counterclockwise (CCW), we say θ has positive measure. If we generate θ by moving from (1, 0) clockwise (CW), we say that θ has negative measure. The radian measure of θ is the length of the arc on the unit circle we travel from (1, 0) to P (θ) when generating θ. We use a minus sign in front when θ is generated CW. Since the unit circle has circumference 2π · 1 = 2π, once around the circle CCW is 2π radians. Halfway around the circle CCW is therefore π radians. One fourth the way around the circle CCW is 2π 4 = π 2 radians. So we see that P (θ) = P (φ) any time that θ and φ have measures which differ by a multiple of 2π radians. So we write P (θ) = P (θ + 2kπ) for all k ∈ ZZ. Two such angles are called coterminal. We call the x-coordinate of P (θ) the cosine of θ and denote it by cos θ. The y-coordinate of P (θ) is the sine of θ and is denoted by sin θ. It is fairly clear that these are functions of θ where θ, as a length, can be any real number. Therefore both the sine function and cosine function have domain IR, unless restricted for some reason. Since all the points on the unit circle have coordinates bounded between 1 and −1, we have that −1 ≤ sin θ ≤ 1 and −1 ≤ cos θ ≤ 1. By the horizontal line test every one of these