1. AAT-A 4/25/14 Obj: SWBAT convert from degrees to radians and vice versa. Agenda Bell Ringer: Inquiry: Angle measure HW Requests: Comments on ACT Turn in Break Packets Tabled: Skills Practice WS Angle of Elevation and Depression Chapter 8 Take Home Test Homework: Read Section 13.2 #19-49 odds Announcements: Larie, Midterm???

2. 2 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

3. Angle- formed by rotating a ray about its endpoint (vertex) Initial Side Starting position Terminal Side Ending position Standard Position Initial side on positive x-axis and the vertex is on the origin

4. An angle describes the amount and direction of rotation 120°–210° Positive Angle- rotates counter-clockwise (CCW) Negative Angle- rotates clockwise (CW) When sketching angles, always use an arrow to show direction.

5. 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. One degree (1º) is equivalent to a rotation of of one revolution.

6. 6 6.1 Radian and Degree Measure Measuring Angles

7. 7 In general, for  in radians, A second way to measure angles is in radians. Radian Measure Definition of Radian: One radian is the measure of a central angle  that intercepts arc s equal in length to the radius r of the circle.

8. 8 Radian Measure

9. 9 Conversions Between Degrees and Radians 1. To convert degrees to radians, multiply degrees by 2. To convert radians to degrees, multiply radians by Example Convert from degrees to radians: 210º

10. Convert from degrees to radians. 1. 54  2. -300  Convert from radians to degrees. 3. 4.

11. 11 6.1 Radian and Degree Measure Coterminal Angles Angles that have the same initial and terminal sides are coterminal. Angles  and  are coterminal.

12. 12 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

13. Coterminal Angles: Two angles with the same initial and terminal sides Find a positive coterminal angle to 20º Find a negative coterminal angle to 20º Types of questions you will be asked: Identify a) ALL angles coterminal with 45º, then b) find one positive coterminal angle and one negative coterminal angle. a) 45º + 360k (where k is any given integer). b) Some possible answers are 405º, 765º, - 315º, - 675º To find a coterminal angle add or subtract multiples of 360º if in degrees or 2π if in radians

14. Ex 5. Convert the degrees to radian measure. a) 60 b) 30  c) d) -54  e) -118  f) g) 45  Class Work a) b) c) d)

15. 15 Class Work Time permitting Pg 712 #1, 2, 4-16 HW: Read Section 13.2 #19-49 odds

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

17. 17 Angles are often classified according to the quadrant in which their terminal sides lie. Ex1: Name the quadrant in which each angle lies. 50º 208º II I -75º III IV 6.1 Radian and Degree Measure Classifying Angles Quadrant 1 Quadrant 3 Quadrant 4

18. 18 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.

19. 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.

20. 20 6.1 Radian and Degree Measure Radian Measure A second way to measure angles is in radians. Definition of Radian: One radian is the measure of a central angle  that intercepts arc s equal in length to the radius r of the circle. In general,

21. 21 6.1 Radian and Degree Measure Radian Measure

22. 22 6.1 Radian and Degree Measure Radian Measure

23. 23 6.1 Radian and Degree Measure Conversions Between Degrees and Radians 1. To convert degrees to radians, multiply degrees by 2. To convert radians to degrees, multiply radians by

24. Ex 7. Find one positive and one negative angle that is coterminal with the angle  = in standard position. Ex 8. Find one positive and one negative angle that is coterminal with the angle  = in standard position.

25. 25 0°  360 °  30 °  45 °  60 °  330 °  315 °  300 °   120 °  135 °  150 °  240 °  225 °  210 °  180 ° 90 °  270 °   Degree and Radian Form of “Special” Angles

26. Find one postive angle and one negative angle in standard position that are coterminal with the given angle. 5. 135  6.