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A Taste of Trigonometry Help

By — McGraw-Hill Professional
Updated on Oct 24, 2011

Introduction to Graphing Three Dimensional Space - Vectors and Cartesian Three-Space

Cartesian three-space, also called rectangular three-space or xyz-space, is defined by three number lines that intersect at a common origin point. At the origin, each of the three number lines is perpendicular to the other two. This makes it possible to pictorially relate one variable to another. Most three-dimensional (3D) graphs look like lines, curves, or surfaces. Renditions are enhanced by computer graphics programs.

You will need to know middle-school algebra to understand the material in this chapter.

A Taste of Trigonometry

Before we proceed further, let’s get familiar with some basic trigonometry. In particular, let’s look at angle notation and the sine, cosine , and tangent functions.

It’s Greek To Us

Mathematicians and scientists often use Greek letters to represent angles. The most common symbol for an angle is an italicized, lowercase Greek theta (pronounced “THAY-tuh”). It looks like a numeral zero leaning to the right, with a horizontal line through it ( θ ).

When writing about two different angles, a second Greek letter is used along with θ. Most often, it is the italicized, lowercase letter phi (pronounced “fie” or “fee”). It looks like a lowercase English letter o leaning to the right, with a forward slash through it ( φ ). Numeric or variable subscripts are sometimes used along with the Greek symbols, so don’t be surprised if you see angles denoted θ 1 , θ 2 , θ 3 or θ x , θ y , θ z .

The Unit Circle

Consider a circle in the Cartesian xy -plane with the following equation:

x 2 + y 2 = 1

This equation represents a unit circle because it is centered at the origin and has a radius of one unit. Let θ be an angle whose apex is at the origin, and that is measured counterclockwise from the x axis, as shown in Fig. 9-1. Suppose this angle corresponds to a ray that intersects the unit circle at

 

Vectors and Cartesian Three-Space A Taste of Trigonometry The Unit Circle

Fig. 9-1 . The unit circle is the basis for the trigonometric functions. some point P = ( x 0 , y 0 ). Then we can define three mathematical functions, called circular functions , of the angle θ in a simple way.

The Sine Function

In Fig. 9-1, let ray OP be defined as the ray from the origin (point O ) passing outward through point P on the unit circle. Imagine this ray at first pointing right along the x axis, and then turning around and around in a counterclockwise direction. As the ray turns, the point P, represented by coordinates ( x 0 , y 0 ), revolves around the unit circle.

Imagine what happens to the value of y 0 during one complete revolution of the ray: it starts out at y 0 = 0, then increases until it reaches y 0 = 1 after P has gone 90° or π/2 rad around the circle ( θ = 90° = π/2 rad). After that, y 0 begins to decrease, getting back to y 0 = 0 when P has gone 180° or π rad around the circle ( θ = 180° = π rad). As P continues on its counterclockwise trek, y 0 keeps decreasing until, at θ = 270° = 3π/2 rad, the value of y 0 reaches its minimum of –1. After that, the value of y 0 rises again until, when P has gone completely around the circle, it returns to y 0 = 0 for θ = 360° = rad.

The value of y 0 is defined as the sine of the angle θ . The sine function is abbreviated sin, so we can state this simple equation:

sin θ = y 0

Vectors and Cartesian Three-Space A Taste of Trigonometry The Unit Circle

Fig. 9-1 . The unit circle is the basis for the trigonometric functions. some point P = ( x 0 , y 0 ). Then we can define three mathematical functions, called circular functions , of the angle θ in a simple way.

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