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Chapter 1 Trigonometric Functions Copyright © 2005 Pearson Education, Inc. Section1.1- Angles Objective: SWBAT learn the basic terminology of angles and their degree measures. In addition students will be able to determine an angle in standard position with coterminal angles. Copyright © 2005 Pearson Education, Inc. Basic Terms Two distinct points determine a line called line AB. A B Line segment AB—a portion of the line between A and B, including points A and B. A B Ray AB—portion of line AB that starts at A and continues through B, and on past B. A Copyright © 2005 Pearson Education, Inc. B Slide 1-3 Basic Terms continued… Angle-formed by rotating a ray around its endpoint. The ray in its initial position is called the initial side of the angle. The ray in its location after the rotation is the terminal side of the angle. Vertex-the endpoint of the ray Copyright © 2005 Pearson Education, Inc. Slide 1-4 Basic Terms continued… Positive angle: The rotation of the terminal side of an angle counterclockwise. Copyright © 2005 Pearson Education, Inc. Negative angle: The rotation of the terminal side is clockwise. Slide 1-5 Naming an Angle An angle can be named using its vertex. Ex: angle C C . . A B . An angle also can be named using 3 letters with the vertex in the middle. Ex: angle ACB or angle BCA Copyright © 2005 Pearson Education, Inc. Slide 1-6 Types of Angles The most common unit for measuring angles is the degree. 360° for a complete rotation of a ray. 1 ° = 1/360 of a rotation. θ is used to name an angle. Copyright © 2005 Pearson Education, Inc. Slide 1-7 Complementary & Supplementary Angles Two positive angles whose sum is 90° are called complementary angles. Two positive angles whose sum is 180° are called supplementary angles. Copyright © 2005 Pearson Education, Inc. Slide 1-8 Complementary Angles Find the measure of each angle. Since the two angles form a right angle, they are complementary angles. Thus, k 20 k 16 90 2k 4 90 2k 86 k 43 Copyright © 2005 Pearson Education, Inc. k +20 k 16 The two angles have measures of 43 + 20 = 63 and 43 16 = 27 Slide 1-9 Supplementary Angles Find the measure of each angle. Since the two angles form a straight angle, they are supplementary angles. Thus, 6 x 7 3 x 2 180 9 x 9 180 9 x 171 x 19 Copyright © 2005 Pearson Education, Inc. 6x + 7 3x + 2 These angle measures are 6(19) + 7 = 121 and 3(19) + 2 = 59 Slide 1-10 Degree, Minutes, Seconds Portions of a degree are measured in minutes and seconds. One minute is 1/60 of a degree. 1 1' 60 60' 1 or One second is 1/60 of a minute. 1 1 1" 60 3600 or 60" 1' 12° 42’ 38” represents 12 degrees, 42 minutes, 38 seconds. Copyright © 2005 Pearson Education, Inc. Slide 1-11 Calculations Perform the calculation. 27 34' 26 52' Perform the calculation. 72 15 18' 27 34' 26 52' 53 86' Write 72 as 71 60' 71 60 Since 86 = 60 + 26, the sum is written 53 15 18' 1 26' 56 42' 54 26' Copyright © 2005 Pearson Education, Inc. Slide 1-12 Conversions Converting between decimal degrees and degrees, minutes & seconds. Convert 74 12' 18" 12 18 74 12' 18" 74 60 3600 74 .2 .005 74.205 Convert 36.624 34.624 34 .624 34 .624(60') 34 37.44' 34 37 ' .44' 34 37 ' .44(60") 34 37 ' 26.4" 34 37 ' 26.4" Copyright © 2005 Pearson Education, Inc. Slide 1-13 Warm up Convert 74° 8’ 14” to decimal degrees. Convert 34.817° to degrees, minutes, and seconds. Copyright © 2005 Pearson Education, Inc. Slide 1-14 Section1.1- Angles Objective: SWBAT learn the basic terminology of angles and their degree measures. In addition students will be able to determine an angle in standard position with coterminal angles. Copyright © 2005 Pearson Education, Inc. 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. Angles in standard position having their terminal sides along the x-axis or y-axis, such as angles with measures 90, 180, 270, and so on, are called quadrantal angles. Copyright © 2005 Pearson Education, Inc. Slide 1-16 Coterminal Angles A complete rotation of a ray results in an angle measuring 360. By continuing the rotation, angles of measure larger than 360 can be produced. Such angles are called coterminal angles. Copyright © 2005 Pearson Education, Inc. Slide 1-17 Coterminal Angles Find the angles of smallest possible positive measure coterminal with each angle. a) 1115 b) 187 Add or subtract 360 as may times as needed to obtain an angle with measure greater than 0 but less than 360. o o o a) 1115 3(360 ) 35 b) 187 + 360 = 173 Copyright © 2005 Pearson Education, Inc. Slide 1-18 Homework Worksheet left side Copyright © 2005 Pearson Education, Inc. Slide 1-19