SECTION 6.1 Angles, Arc Length, and Circular Motion 383 OBJECTIVES 1 Angles and Degree Measure (p. 383) 2 Convert between Decimal and Degree, Minute, Second Measures for Angles (p. 385) 3 Find the Length of an Arc of a Circle (p. 386) 4 Convert from Degrees to Radians and from Radians to Degrees (p. 387) 5 Find the Area of a Sector of a Circle (p. 390) 6 Find the Linear Speed of an Object Traveling in Circular Motion (p. 390) 6.1 Angles, Arc Length, and Circular Motion Now Work the ‘Are You Prepared?’ problems on page 392. • Circumference and Area of a Circle (Section A.2, p.A16) • Uniform Motion (Section A.8, pp. A70–A71) PREPARING FOR THIS SECTION Before getting started, review the following: 1 Angles and Degree Measure A ray , or half-line , is a portion of a line that starts at a point V on the line and extends indefinitely in one direction.The starting point V of a ray is called its vertex . See Figure 1. When two rays are drawn with a common vertex, they form an angle .We call one ray of an angle the initial side and the other the terminal side . The angle formed is identified by showing the direction and amount of rotation from the initial side to the terminal side. If the rotation is in the counterclockwise direction, the angle is positive ; if the rotation is clockwise, the angle is negative . See Figure 2. Figure 2 (a) Counterclockwise rotation; Positive angle Terminal side Initial side Vertex a (b) Clockwise rotation; Negative angle Terminal side Initial side Vertex b (c) Counterclockwise rotation; Positive angle Terminal side Initial side Vertex g Lowercase Greek letters, such as α (alpha), β (beta), γ (gamma), and θ (theta), are often used to denote angles. Notice in Figure 2(a) that the angle α is positive because the direction of the rotation from the initial side to the terminal side is counterclockwise. The angle β in Figure 2(b) is negative because the rotation is clockwise.The angle γ in Figure 2(c) is positive. Notice that the angle α in Figure 2(a) and the angle γ in Figure 2(c) have the same initial side and the same terminal side. However, α and γ are unequal, because the amount of rotation required to go from the initial side to the terminal side is greater for angle γ than for angle .α An angle θ is said to be in standard position if its vertex is at the origin of a rectangular coordinate system and its initial side coincides with the positive x -axis. See Figure 3. Figure 3 Standard position of an angle y x Initial side Terminal side Vertex u (a) u is in standard position; u is positive y x Initial side Terminal side Vertex u (b) u is in standard position; u is negative Need to Review? Rectangular Coordinates are discussed in Section 1.1, p. 2. Valerie Thomas (1943–present) Credit: Interim Archives/Contributor/ Archive Photos/Getty Images Valerie Thomas is best known for inventing the Illusion Transmitter in 1980. This technology was the first of its kind in the field of 3D-imagery and is the basis for modern 3D-televisions, video games, and movies. Additionally, Thomas worked for NASA creating real-time computer data systems, and directing the team that developed the Landsat technology that helped the first satellite to transmit images from space. Figure 1 A ray or half-line V Ray Line
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