In this section, we will study the following topics:

Transcription

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

2 6.1 Radian and Degree Measure Angles Trigonometry: measurement of triangles Section 4.1, Figure 4.1, Terminal and Initial Side of an Angle, pg. 248 Angle Measure Copyright Houghton Mifflin Company. All rights reserved. Digital Figures, 4 2 2

3 Standard Position Section 4.1, Figure 4.2, Standard Position of an Angle, pg Radian and Degree Measure Vertex at origin Copyright Houghton Mifflin Company. All rights reserved. Digital Figures, 4 3 The initial side of an angle in standard position is always located on the positive x-axis. 3

4 Positive and negative angles 6.1 Radian and Degree Measure Section 4.1, 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

5 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 Measuring Angles 6.1 Radian and Degree Measure Section 4.1, Figure 4.13, Common Degree Measures on the Unit Circle, pg Copyright Houghton Mifflin Company. All rights reserved. Digital Figures, 4 9 6

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8 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 8

9 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. 9

10 6.1 Radian and Degree Measure Coterminal Angles Angles that have the same initial and terminal sides are coterminal. Section 4.1, Figure 4.4, Coterminal Angles, pg. 248 Angles and are coterminal. Copyright Houghton Mifflin Company. All rights reserved. Digital Figures,

11 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 11

12 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.

13 6.1 Radian and Degree Measure Radian Measure A second way to measure angles is in radians. Definition of Radian: One radian is Section the measure 4.1, Figure of a 4.5, central Illustration angle of that intercepts arc s equal in length Arc to Length, the radius pg. r 249 of the circle. In general, s r Copyright Houghton Mifflin Company. All rights reserved. Digital Figures,

14 6.1 Radian and Degree Measure Radian Measure 2 radians corresponds to radians corresponds to radians corresponds to 90 2 Section 4.1, Figure 4.6, Illustration of Six Radian Lengths, pg Copyright Houghton Mifflin Company. All rights reserved. Digital Figures,

15 Section 4.1, Figure 4.7, Common Radian Angles, pg. 249 Radian Measure 6.1 Radian and Degree Measure 15

16 Conversions Between Degrees and Radians 6.1 Radian and Degree Measure 1. To convert degrees to radians, multiply degrees by 2. To convert radians to degrees, multiply radians by

17 Ex 5. Convert the degrees to radian measure. a) 60 b) 30 c) -54 d) -118 e) 45

18 Ex 6. Convert the radians to degrees. a) b) c) d)

19 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 coterminal with the angle = in standard position. 7 5

20 Degree and Radian Form of Special Angles

21 Class Work Convert from degrees to radians Convert from radians to degrees

22 Find one postive angle and one negative angle in standard position that are coterminal with the given angle

23 HW p odd, 37-41odd, 43-47odd 23