Section 9.1 Angles, Arcs, & Their Measures (Part I)

Transcription

1 Week 1 Handout MAC 1114 Professor Niraj Wagh J Section 9.1 Angles, Arcs, & Their Measures (Part I) Basic Terminology Line: Two distinct points A and B determine a line called line AB. Segment: The portion of the line between A and B, including points A and B is called segment AB. Ray: The portion of the line AB that starts at A and continues to B and on past B is called ray AB. Point A is the endpoint of the ray. Angle: An angle consists of two rays in a plane with a common endpoint, or two line segments with a common endpoint. These two rays are called sides of the angle, and the common endpoint is called the vertex of the angle. N. Wagh 1

2 Degree Measure The most common unit used to measure the size of the angles is the degree. To use the degree measure, we assign 360 degrees to a complete rotation of a ray. For example, 1 (degree) measures 1/360 th of the rotation. Similarly, 90 measures 90/360 th of the rotation. Special Types of Angles Complementary: If the sum of the measures of two positive angles is 90 degrees, the angles are called complementary and the angles are called complements. Supplementary: If the sum of the measures of two positive angles is 180 degrees, the angles are called supplementary and the angles are called supplements. Conversions between Time Portions of a degree have been measured with minutes and seconds. (Think of a clock!) N. Wagh 2

3 One minute, written 1, is 1/60 th of a degree. One second, written 1, is 1/60 th of a minute. 9.1 Examples (Part I) 1. Find the measure of each angle. N. Wagh 3

4 2. Perform each calculation. (a) 62 18, (b) 47 29, N. Wagh 4

5 3. Convert each angle measure to decimal degrees. (a) (b) 91 35, Convert each angle measure to degrees, minutes, and seconds. (a) (b) Wonderful! We ve completed the examples for the first part of 9.1! Begin working on HW 9.1 in MyMathLab. If you have any questions, please let me know! I will be more than happy to help you! My student engagement hours are listed in the syllabus. J N. Wagh 5

6 Section 9.1 Angles, Arcs, & Their Measures (Part II) Standard Position An angle is in standard position if its vertex is at the origin and its initial side is along the positive x-axis. An angle in standard position is said to lie in the quadrant in which its terminal side lies. Coterminal Angles A complete rotation of a ray results in an angle measuring 360. By continuing the rotation, angles of measure larger than 360 degrees can be produced. Coterminal angles differ by a multiple of 360. N. Wagh 6

7 Radian Measure Angles are often measured in degrees. In more theoretical work in mathematics, radian measure of angles is preferred. Angles are usually denoted using θ (theta). One radian is equivalent to about = 2π radians Conversion between degrees and radians Arc Length The length s of an arc intercepted on a circle by radius r by a central angle of measure θ radians is given by the product of the radius and the radian measure of the angle. s = rθ, θ in radians N. Wagh 7

8 Area of a Sector A sector of a circle is the portion of the interior of a circle intercepted by a central angle. Linear and Angular Speed A r@ θ, θ in radians In many situations we need to know how fast a point on a circular disk is moving or how fast the central of such a disk is changing. Some examples occur with machinery involving gears or pulleys and with the speed of a car around a curved portion of a highway. Suppose that point P moves at a constant speed along a circle of radius r and center O. The measure of how fast the position of P is changing is the linear speed. v = B C The measure of how fast angle POB is changing is the angular speed (ω, pronounced omega). ω = E, θ in radians C The formula v = rω relates linear and angular speeds. N. Wagh 8

9 9.1 Examples (Part II) 1. Find the angle of least positive measure that is coterminal with the given angle. (a) 40 (b) 450 (c) π 4 2. Give an expression that generates all angles coterminal with each angle. Let n represent any integer. (a) 30 (b) 135 (c) 7π 6 N. Wagh 9

10 3. Convert each degree measure to radians. Leave answers as rational multiples of pi. (a) 39 (b) Convert each radian measure to degrees. (a) π 3 (b) 11π 6 N. Wagh 10

11 5. Find the length of each arc intercepted by central angle θ in a circle of radius r. (a) (b) r = 4.82 meters, θ = Use the formula ω = E C to find the value of the missing variable. θ = 3π 4 radians, t = 8 seconds N. Wagh 11

12 7. Find the area of a sector of a circle having radius r and central angle θ. r = 12.7cm., θ = Radar is used to identify severe weather. If Doppler radar can detect weather within a 240-mile radius and creates a new image every 48 seconds, find the area scanned by the radar in 1 second. Wonderful! We ve completed the examples for this section! Now work on HW 9.1 in MyMathLab. If you have any questions, please let me know! I will be more than happy to help you! My student engagement hours are listed in the syllabus. J N. Wagh 12