7.1 Angles and Their Measure

1. 7.1 Angles and Their 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

2. Angles Trigonometry translated: _____________ of _____________ Angle Measure

3. Standard Position Vertex at origin The initial side of an angle in standard position is always located on the positivex-axis.

4. Positive and negative angles When sketching angles, always use an arrow to show direction.

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

6. Measuring Angles

7. Acute and Obtuse Angles Acute angles have measure between _____º and _____º. Obtuse angles have measure between ____º and _____º. Straight angles measure _______º.

8. Classifying Angles Angles are often classified according to the QUADRANTin which their terminal sides lie. Example: 50º is a ____ quadrant angle. 208º is a ____ quadrant angle. II I -75º is a _____ quadrant angle. III IV

9. 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.

10. Sketching Angles (Degrees) • Sketch in standard position. In which quadrant is  located? • Sketch in standard position. In which quadrant is  located?

11. Complementary and Supplementary Angles Complementary Angles Two positive angles are complementary if their sum is ______º Angles that measure 22º and 68º are complements. Supplementary Angles Two positive angles are supplementary if their sum is _______º Angles that measure 137º and 43º are supplements.

12. 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, for  in radians,

13. Radian Measure

14. Radian Measure

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

16. Conversions Between Degrees and Radians Example a) Convert from radians to degrees: b) Convert from radians to degrees: 3.8

17. Conversions Between Degrees and Radians Try it! c) Convert from degrees to radians (exact): d) Convert from radians to degrees:

18. Conversions Between Degrees and Radians Again! e) Convert from degrees to radians (to 3 decimal places): f) Convert from radians to degrees (to nearest tenth): 1 rad

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

20. Sketching Angles (Radians) • Sketch in standard position. In which quadrant is  located? • Sketch in standard position. In which quadrant is  located?

21. Degrees, minutes, and seconds • 1 minute (1’) = degree OR 1° = ______ minutes • 1 second (1”) = _____ minute = _____ degree OR 1’ = _____” • Example • Convert to decimal degrees:

22. Degrees, minutes, and seconds • Conversions between decimal degrees and degrees, minutes, • seconds can be easily accomplished using your TI graphing • calculator. • The ANGLE menu [ ] on your calculator has built-in features for converting between decimal degrees and DMS. APPS Note that the seconds (“) symbol is not in the ANGLE menu. Use  for “ symbol.

23. Practice • Using your TI graphing calculator, • Convert to decimal degrees to the nearest hundredth of a degree. • Convert 57.328° to an equivalent angle expressed to the nearest second.

24. End of Section 7.1