Transcript Ch. 6-1
6.1 Radian and Degree Measure In this section, we will study the following topics: Terminology used to describe angles Degree measure of an angle Radian measure of an angle Converting between radian and degree measure Find coterminal angles 1 6.1 Radian and Degree Measure Angles Trigonometry: measurement of triangles Section 4.1, Figure 4.1, Terminal and Angle Measure Initial Side of an Angle , pg. 248 Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–2 2 6.1 Radian and Degree Measure Standard Position Section 4.1, Figure 4.2, Standard Position of an Angle, pg. 248 Vertex at origin Copyright © Houghton Mifflin Company. All rights reserved. The initial side of an angle in standard position is always located on the positive x-axis. Digital Figures, 4–3 3 6.1 Radian and Degree Measure Positive and Section negative4.1, angles Figure 4.3, Positive and Negative Angles, pg. 248 When sketching angles, always use an arrow to show direction. Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–4 4 6.1 Radian and Degree Measure Measuring Angles The measure of an angle is determined by the amount of rotation from the initial side to the terminal side. There are two common ways to measure angles, in degrees and in radians. We’ll start with degrees, denoted by the symbol º. 1 One degree (1º) is equivalent to a rotation of 360 revolution. of one 5 6.1 Radian and Degree Measure Section 4.1, Figure 4.13, Common Degree Measures on the Unit Circle, pg. 251 Measuring Angles 1 360 Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–9 6 6.1 Radian and Degree Measure Classifying Angles Angles are often classified according to the quadrant in which their terminal sides lie. Ex1: Name the quadrant in which each angle lies. 50º Quadrant 1 208º Quadrant 3 II I -75º Quadrant 4 III IV 7 6.1 Radian and Degree Measure Classifying Angles Standard position angles that have their terminal side on one of the axes are called quadrantal angles. For example, 0º, 90º, 180º, 270º, 360º, … are quadrantal angles. 8 6.1 Radian and Degree Measure Coterminal Angles 4.1, Figure 4.4, Coterminal Angles thatSection have the same initial and terminal sides are coterminal. Angles, pg. 248 Angles and are coterminal. Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–5 9 6.1 Radian and Degree Measure Example of Finding Coterminal Angles You can find an angle that is coterminal to a given angle by adding or subtracting multiples of 360º. Ex 2: Find one positive and one negative angle that are coterminal to 112º. For a positive coterminal angle, add 360º : 112º + 360º = 472º For a negative coterminal angle, subtract 360º: 112º - 360º = -248º 10 Ex 3. Find one positive and one negative angle that is coterminal with the angle = 30° in standard position. Ex 4. Find one positive and one negative angle that is coterminal with the angle = 272 in standard position. 6.1 Radian and Degree Measure Radian Measure A second way to measure angles is in radians. Definition of Radian: 4.1, Figure Illustration One radian isSection the measure of a 4.5, central angle ofthat intercepts pg. r249 arc s equal in lengthArc to Length, the radius of the circle. In general, s r Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 4–6 12 6.1 Radian and Degree Measure Radian Measure 2 6.28 3.14 2 radians corresponds to 360 radians corresponds to 180 2 radians corresponds 904.6, Illustration of Section 4.1,to Figure Six Radian Lengths, pg. 249 Copyright © Houghton Mifflin Company. All rights reserved. 2 1.57 Digital Figures, 4–7 13 6.1 Radian and Degree Measure Common Section 4.1, Figure 4.7, Radian Measure Radian Angles, pg. 249 14 6.1 Radian and Degree Measure Conversions Between Degrees and Radians 1. 2. To convert degrees to radians, multiply degrees by To convert radians to degrees, multiply radians by 180 180 15 Ex 5. Convert the degrees to radian measure. a) 60 b) 30 c) -54 d) -118 e) 45 Ex 6. Convert the radians to degrees. a) 6 b) 2 c) 11 18 d) 9 Ex 7. Find one positive and one negative angle that is coterminal with the angle = in standard position. 3 Ex 8. Find one positive and one negative angle that is 7 coterminal with the angle = in standard position. 5 Degree and Radian Form of “Special” Angles 90 ° 120 ° 60 ° 135 ° 45 ° 150 ° 30 ° 0° 180 ° 360 ° 210 ° 330 ° 225 ° 315 ° 240 ° 300 ° 270 ° 19 Class Work Convert from degrees to radians. 1. 54 2. -300 Convert from radians to degrees. 3. 11 3 4. 13 12 Find one postive angle and one negative angle in standard position that are coterminal with the given angle. 5. 135 11 6. 6 HW p474 1-29 odd, 37-41odd, 43-47odd 22