1. 13.2 General Angles and Radian Measure

2. 1. I want the exact value (not the decimal value from the calculator) 1. I want the exact value (not the decimal value from the calculator)2.3. Quiz 13-1 25º ABC x 7

3. What you’ll learn about  The Problem of Angular Measure  Degrees and Radians  Arc Length  Area of a Sector … and why Angles are the domain elements of the trigonometric functions.

4. Why 360 º ? The idea of dividing a circle into 360 equal pieces dates back to the sexagesimal (60-based) counting system of the ancient Sumarians. Early astronomical calculations linked the sexagesimal system to circles.

5. Angles vs. lengths If I lengthen the sides of the angle, does the measure of the angle change? NO A Notice that the length of the arc depends upon how far the arc is from the vertex of the angle. 37°

6. Radians vs. Degrees How far you are from the vertex of the angle does not change the degree measure of the angle. A Another way of measuring the angle that takes into account the change in the arc length will be useful. 37°

7. Degrees: The measure of an angle was a portion of 360° (the angle measure of a circle). Vocabulary Radians: the measure of an angle that takes into account the lengths of the sides of the angle.

8. Circumference: The distance all the way around a circle. a circle.Vocabulary What is the circumference of a circle whose radius equals 1 (inch, meter, yard, mile, etc.) ? radius equals 1 (inch, meter, yard, mile, etc.) ? Radians: the measure of an angle that takes into account the lengths of the sides of the angle.

9. Degrees: The measure of an angle as a portion of 360° (the angle measure of a circle). Vocabulary Radians: the measure of an angle as a portion of the circumference of a circle.

10. Vocabulary Radian measure: the ratio of the arc length to the radius of the circle: the radius of the circle: Radians = inches/inches

11. A circle has a radius of 5 inches. What is the radian measure of an angle that is ½ of the circle? Radian measure: the ratio of the arc length to the radius of the circle: the radius of the circle: Radian measure of the arc = pi radians

12. Your Turn: 1. A circle has a radius of 2 mm, what is the circumference of the circle? (leave ‘π’ in your answer) circumference of the circle? (leave ‘π’ in your answer) 2. An arc subtends 1/3 of a circle that has a radius of 3 inches. What is the radian measure of the arc? (leave ‘π’ in your answer)

13. Your Turn: 4. An angle subtends 1/9 of a circle. The circle has a radius of 10 inches. What is the radian measure of the angle? 3. A circle has a circumference of 10π inches. What is the radius of the circle?

14. Degree-Radian Conversion Degree-Radian Conversion 180° = π radians These are “conversion factors” “conversion factors” When you multiply a number by one of these factors, it converts the units. of these factors, it converts the units.

15. Converting from Degrees to Radian Measure 140 ° Converting from Radian Measure to Degrees

16. Your Turn: Convert between radians and degrees. 5.6.

17. Arc Length Radian = Greek letter “theta” Theta is often used as a variable to denote the measure of an angle. the measure of an angle. “arc length = radius * angle measure (in radians)” ‘s’ is the measure of the arc. Theta is in radian measure not degrees!!

18. Arc Length “arc length = radius * angle measure (in radians)” A circle has a radius of 8 inches. The radian measure of an angle in the circle is measure of an angle in the circle is What is the length of the arc?

19. Arc Length “arc length = radius * angle measure (in radians)” r = 5 inches Arc length = ?

20. 5. Your turn: r = 22 inches Arc length = ?

21. 6. Your turn: Arc length = 7π inches Angle measure = ?

22. 7. Your turn: Angle measure = 3π radians Arc length = 12π inches Radius = ?

23. Formula for Sector Area 8 ft What is the area of a 20° sector of a circle whose radius is 2 ft? circle whose radius is 2 ft? 20º Gotcha!!! This formula uses angle measure in radians.

24. Your turn: 10 ft 9. What is the area of a 40° sector of a circle whose radius is 2 ft? 40º Gotcha!!! This formula uses angle measure in radians.

25. Your turn: 6 ft 11. What is the area of a 120º sector of a circle whose radius is 6 ft? of a circle whose radius is 6 ft? 120º Be careful of the radian/degree gotch.

26. Your turn: 14 inch pizza (diameter) Slice is 1/8 of the pizza 10. Find the area of a slice of pizza.

27. Your Turn: 11. What is the area of a 45° sector of a circle whose radius = 12 inches? of a circle whose radius = 12 inches? 12. What is the radius of a circle with a 100° sector whose sector area is whose sector area is 13. The radius of a circle is 20 inches. What is the sector angle? What is the sector angle?

28. End of core required knowledge.

29. Initial Side, Terminal Side beginning position of the ray final position of the ray Vertex α, β, θ = the measure of the angle

30. Vocabulary Standard Position An angle with one ray along the x-axis and the other ray rotated counter-clockwise from the first ray. ray rotated counter-clockwise from the first ray. Terminal Side In Trigonometry, we use a circle with the vertex of the angle at vertex of the angle at the center of the circle. the center of the circle.

31. Vocabulary Standard Position An angle with one ray along the positive x-axis and the other ray rotated counter- clockwise from the first ray. Initial Side Terminal Side In Trigonometry, we sometimes use a circle sometimes use a circle with the vertex of the with the vertex of the angle at the center of angle at the center of the circle. the circle. Reference Angle: the measure of the acute angle with the x-axis. Reference Angle: the acute angle between the terminal side and the x-axis. between the terminal side and the x-axis. 0 90 180 270

32. Vocabulary Standard Position An angle with one ray along the positive x-axis and the other ray rotated counter- clockwise from the first ray. Initial Side Terminal Side Reference Angle: the measure the measure of the acute angle with the x-axis.

33. Angle measures 0º0º0º0º 90º 180º 270º Draw the standard position angle with the given measure. 220º 40º What is the reference angle for this standard position angle?

34. Your turn: Draw an angle in standard position that has a measure of: 15. 135º 17. 290º 16. What is the reference angle for 135° ? 18. What is the reference angle for 290° ?

35. Co-terminal Angles 45º What is the difference in position on the unit circle position on the unit circle if terminal side stops at if terminal side stops at 45º or goes all the way 45º or goes all the way around and stops at 405º ? around and stops at 405º ?

36. Co-terminal Angles What is the difference in position on the unit circle position on the unit circle if terminal side stops at if terminal side stops at 45º or goes all the way 45º or goes all the way around and stops at 405º ? around and stops at 405º ? 45º There is no difference !! Although the angular measure is different they measure is different they are co-terminal angles. are co-terminal angles. 405º

37. Finding Co-terminal Angles Find a positive and a negative angle that are co-terminal with 45°. co-terminal with 45°. We’ve already found one positive co-terminal one positive co-terminal angle with 45° (405°). angle with 45° (405°). Can you find another? 405° + 360° = 765° Negative angle: 45°- 360° = -315° 45º

38. Find a positive co-terminal angle with 60° Find a negative co-terminal angle with 140° 60° + 360° = 420° 140° - 360° = -220°

39. Your Turn: 19. 21. Find a positive co-terminal angle with 120° Find a negative co-terminal angle with 270° 20. What is the reference angle for these two angles? 22. What is the reference angle for these two angles?

40. Finding Co-terminal Angles Find a positive and negative co-terminal angle with: Notice the angle measure is now in radians. Go to then go around the circle one complete revolution. How many radians does it take to go How many radians does it take to go all the way around the circle? all the way around the circle?

41. Your Turn: 23. Find a negative co-terminal angle with 24. Find a positive co-terminal angle with

42. HOMEWORK Section 13-2: page 862, even problems: 4-12, 16-38, even problems: 4-12, 16-38, and 62 (remember that you can’t have a radical in the denominator).

43. Your Turn: Convert between radians and degrees. 15.16.

44. Your Turn: 17. The radius of a circle is 5 inches. What is its circumference? 18. The interior angle of a circle is radians. The circle has a radius of 10 inches. What is the length of the arc has a radius of 10 inches. What is the length of the arc subtended by the angle? subtended by the angle? 19. What is the area of a 25° sector of a circles whose radius = 10 inches?

45. Sector Area 1. Find the area of the circle. 2.Find the circle fraction  3. Sector Area = (circle fraction)*(circle Area)

46. Sector Area 10 ft Area of a circle: What fraction of the circle is a 30º sector? is a 30º sector? 30º Sector Area = (fraction of circle)*(Area of circle ) Sector Area =

47. Sector Area 8 ft Area of a circle: What fraction of the circle is a 20º sector? is a 20º sector? 20º Sector Area = (fraction of circle)*(Area of circle ) Sector Area =