526 CHAPTER 5 Trigonometric Functions Basic Terminology Two distinct points A and B determine a line called line AB. The portion of the line between A and B, including points A and B themselves, is line segment AB, or simply segment AB. The portion of line AB that starts at A and continues through B, and on past B, is the ray AB. Point A is the endpoint of the ray. See Figure 1. In trigonometry, an angle consists of two rays in a plane with a common endpoint, or two line segments with a common endpoint. Each of these two rays (or segments) is a side of the angle, and the common endpoint is the vertex of the angle. Associated with an angle is its measure, generated by a rotation about the vertex. See Figure 2. This measure is determined by rotating a ray starting at one side of the angle, the initial side, to the position of the other side, the terminal side. A counterclockwise rotation generates a positive measure, and a clockwise rotation generates a negative measure. The rotation can consist of more than one complete revolution. Figure 3 shows two angles, one positive angle and one negative angle. 5.1 Angles ■ Basic Terminology ■ Degree Measure ■ Standard Position ■ Coterminal Angles Terminal side Vertex A Angle A Initial side Figure 2 Positive angle Figure 3 Negative angle B C A An angle can be named by using the name of its vertex. For example, the angle on the right in Figure 3 can be named angle C. Alternatively, an angle can be named using three letters, with the vertex letter in the middle. Thus, the angle on the right also could be named angle ACB or angle BCA. A complete rotation of a ray gives an angle whose measure is 360°. of a complete rotation gives an angle whose measure is 1°. 360 1 Figure 4 Degree Measure The most common unit for measuring angles is the degree. Degree measure was developed by the Babylonians 4000 yr ago. To use degree measure, we assign 360 degrees to a complete rotation of a ray.* In Figure 4, notice that the terminal side of the angle corresponds to its initial side when it makes a complete rotation. One degree, written 1°, represents 1 360 of a complete rotation. Therefore, 90° represents 90 360 = 1 4 of a complete rotation, and 180° represents 180 360 = 1 2 of a complete rotation. An angle measuring between 0° and 90° is an acute angle. An angle measuring exactly 90° is a right angle. The symbol m is often used at the vertex of a right angle to denote the 90° measure. An angle measuring more than 90° but less than 180° is an obtuse angle, and an angle of exactly 180° is a straight angle. *The Babylonians were the first to subdivide the circumference of a circle into 360 parts. There are various theories about why the number 360 was chosen. One is that it is approximately the number of days in a year, and it has many divisors, which makes it convenient to work with in computations. A B Line AB Segment AB A B Ray AB A B Figure 1
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