1. Section 1.1 Angle Measure and Arc Length Section 1.1 Angle Measure and Arc Length

2. Objectives: 1.To draw an angle in standard position. 2.To convert between degree and radian measure. 3.To calculate arc length using angle measure and radius length. 4.To review the distance and midpoint formulas. Objectives: 1.To draw an angle in standard position. 2.To convert between degree and radian measure. 3.To calculate arc length using angle measure and radius length. 4.To review the distance and midpoint formulas.

3. An angle in standard position has the positive x-axis as its initial ray and the origin as its vertex. DefinitionDefinition

4. Standard Position initial ray terminal ray V V C C A A

5. B B C C A A 60  Standard Position

6. F F G G E E -120 

7. A A C C B B 80  Standard Position

8. Coterminal angles are angles in standard position with the same terminal ray. DefinitionDefinition

9. Coterminal Angles A A C C B B 440  A A C C B B 80 

10. A radian is the measure of an angle formed by two radii of a circle so that the intercepted arc has a length equal to the radius of the circle. DefinitionDefinition

11. Radian B B C C A A m  ABC = 1 radian  57.3 

12. Whenever an angle measure is given without degrees specified as the unit of measure, it is measured in radians.

13. The unit circle has its center at the origin with a radius of one unit. DefinitionDefinition

14. Unit Circle B B C C A A (1, 0)

15. Convert between degrees and radians by setting up a proportion. 180 π π = = deg m m rad n n In this proportion, substitute the known measure and solve for the other measure.

16. Change 60 degrees to radians. m deg n rad 180   = =

17. Change 60 degrees to radians. 60 deg n rad 180   = = n(180) = 60  n =  1.05 3 3  

18. Change 5.6 radians to degrees. m deg n rad 180   = =

19. Change 5.6 radians to degrees. m deg 5.6 rad 180   = = 5.6(180) = m  m =  320.9° 5.6(180)  

20. EXAMPLE 1 Find the measure of  in degrees and radians.    = ¼ of a circle ¼(360º) = 90º ¼ 2  =  = ¼ of a circle ¼(360º) = 90º ¼ 2  = 22 22

21. Practice Question: Find the measure of  in degrees.    = ¾ of a circle ¾(360º) = 270º  = ¾ of a circle ¾(360º) = 270º

22. A quadrantal angle is an angle whose terminal ray lies on one of the axes. DefinitionDefinition

23. Quadrantal Angle B B C C A A 90°

24. B B C C A A 180° Quadrantal Angle

25. B B C C A A 270° Quadrantal Angle

26. B B C C A A 360° Quadrantal Angle

27. Concentric circles have the same center but different radii. A central angle is an angle having its vertex at the center of a circle. DefinitionDefinition

28.   (r,0) (1,0) x y s s t t

29. From geometry it can be proved that the ratio of the arc length of two concentric circles is the same as the ratio of their radii: 1 1 r r t t s s = = Solving for s, s = rt, where s = arc length, r = radius, and t = angle measure ( in radians ).

30. In other words, the length of any arc equals the product of its radius and angle measure ( in radians )

31. Find the measure of the angle that cuts off an arc length of 8 in a circle with radius 2. s= rt 8= 2(t) t= 4 radians s= rt 8= 2(t) t= 4 radians

32. EXAMPLE 2 Find the measure of the angle that cuts off an arc length of 7 in a circle of radius 4. Answer s = rt 7 = 4t t = 7/4 = 1.75 radians Answer s = rt 7 = 4t t = 7/4 = 1.75 radians

33. Practice Question: Find the measure of the angle that cuts off an arc length of 8 in a circle of radius 5. Answer s = rt 8 = 5t t = 8/5 = 1.6 radians Answer s = rt 8 = 5t t = 8/5 = 1.6 radians

34. Homework: pp. 6-8 Homework: pp. 6-8

35. ►A. Exercises Sketch each angle in standard position. 3. 2  / 3 ►A. Exercises Sketch each angle in standard position. 3. 2  / 3 0 0   22 22 3232 3232

36. Sketch each angle in standard position. 290° Sketch each angle in standard position. 290° 0 0 180° 270°

37. ►A. Exercises Complete each line of the following table. ►A. Exercises Complete each line of the following table. RadiansDegrees 9.15 11. 15. 17.31 6 7  7 12   / 12 210 309 0.54

38. ►B. Exercises Give three angle measures in radians which are coterminal with each of the following. Include at least one positive and one negative angle measure. 21. ►B. Exercises Give three angle measures in radians which are coterminal with each of the following. Include at least one positive and one negative angle measure. 21. 4 4  

39. ►B. Exercises 23.How many radians are in one degree? ►B. Exercises 23.How many radians are in one degree? n rad (180) =  deg 1 deg n rad 180   = = n = 180    0.01745

40. ►B. Exercises Find the missing angle measures, radius measures, and arc lengths in the table below. Give the angle measures in radians. ►B. Exercises Find the missing angle measures, radius measures, and arc lengths in the table below. Give the angle measures in radians. Arc LengthRadius LengthAngle Measure (s)(r)(t) 8 5 4  2  6  3 22 25. 27. 29. 25. 27. 29.

41. Find the missing angle measures, radius measures, and arc lengths in the table below. Give the angle measures in radians. 25.Given r = 8 and t =  / 4, use s = rt to find the arc length (s). Find the missing angle measures, radius measures, and arc lengths in the table below. Give the angle measures in radians. 25.Given r = 8 and t =  / 4, use s = rt to find the arc length (s). s =rt =8(  / 4 ) =2  s =rt =8(  / 4 ) =2=2

42. Arc LengthRadius LengthAngle Measure (s)(r)(t) 8 5 4  2  6  3 22 22 22 ►B. Exercises Find the missing angle measures, radius measures, and arc lengths in the table below. Give the angle measures in radians. ►B. Exercises Find the missing angle measures, radius measures, and arc lengths in the table below. Give the angle measures in radians. 25. 27. 29. 25. 27. 29.

43. Find the missing angle measures, radius measures, and arc lengths in the table below. Give the angle measures in radians. 27.Given s =  / 2 and t =  / 6, use s = rt to find the radius length (r). Find the missing angle measures, radius measures, and arc lengths in the table below. Give the angle measures in radians. 27.Given s =  / 2 and t =  / 6, use s = rt to find the radius length (r). s =rt  / 2 =r(  / 6 ) r = 6 /  (  / 2 ) = 3 s =rt  / 2 =r(  / 6 ) r = 6 /  (  / 2 ) = 3

44. Arc LengthRadius LengthAngle Measure (s)(r)(t) 8 5 4  2  6  3 22 22 22 3 3 ►B. Exercises Find the missing angle measures, radius measures, and arc lengths in the table below. Give the angle measures in radians. ►B. Exercises Find the missing angle measures, radius measures, and arc lengths in the table below. Give the angle measures in radians. 25. 27. 29. 25. 27. 29.

45. ►B. Exercises Determine the distance between each pair of points and also the midpoint of the segment joining them. 30.(3, -6), (1, 0) ►B. Exercises Determine the distance between each pair of points and also the midpoint of the segment joining them. 30.(3, -6), (1, 0) d = (x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 midpoint = (, ) x 1 + x 2 2 x 1 + x 2 2 y 1 + y 2 2 y 1 + y 2 2

46. ►B. Exercises If M is the midpoint of AB, find B. Then use distances to justify your answers. 35.A(3, 5), M(9, -2) ►B. Exercises If M is the midpoint of AB, find B. Then use distances to justify your answers. 35.A(3, 5), M(9, -2)             (9, -2) = 2 2 5 + y,, 2 2 3 + x

47. 18 = 3 + x 2 2 3 + x 9 = x = 15 ►B. Exercises If M is the midpoint of AB, find B. Then use distances to justify your answers. 35.A(3, 5), M(9, -2) ►B. Exercises If M is the midpoint of AB, find B. Then use distances to justify your answers. 35.A(3, 5), M(9, -2)

48. -4 = 5 + y 2 2 5 + y -2 = y = -9 ►B. Exercises If M is the midpoint of AB, find B. Then use distances to justify your answers. 35.A(3, 5), M(9, -2) ►B. Exercises If M is the midpoint of AB, find B. Then use distances to justify your answers. 35.A(3, 5), M(9, -2)

49. B = (15, -9) ►B. Exercises If M is the midpoint of AB, find B. Then use distances to justify your answers. 35.A(3, 5), M(9, -2) ►B. Exercises If M is the midpoint of AB, find B. Then use distances to justify your answers. 35.A(3, 5), M(9, -2)