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Section 1.1 Angle Measure and Arc Length

Section 1.1 Angle Measure and Arc Length. 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. Definition.

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Section 1.1 Angle Measure and Arc Length

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  1. 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.

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

  4. Standard Position A terminal ray C V initial ray

  5. Standard Position A 60 B C

  6. Standard Position F G E -120

  7. Standard Position A 80 B C

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

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

  10. Definition 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.

  11. Radian A B C 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. Definition The unit circle has its center at the origin with a radius of one unit.

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

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

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

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

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

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

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

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

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

  23. Quadrantal Angle A 90° B C

  24. Quadrantal Angle A 180° B C

  25. Quadrantal Angle A 270° C B

  26. Quadrantal Angle A C 360° B

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

  28. y t s (1,0)  (r,0) x

  29. s r = t 1 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: 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

  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

  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

  34. Homework: pp. 6-8

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

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

  37. Radians Degrees 9. 15 11. 15. 17. 31 7 p 12 p 6 7 ►A. Exercises Complete each line of the following table. /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. p 4

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

  40. p 8 4 p p 2 6 2 5 3 ►B. Exercises Find the missing angle measures, radius measures, and arc lengths in the table below. Give the angle measures in radians. Arc Length Radius Length Angle Measure (s) (r) (t) 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). s = rt = 8(/4) = 2

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

  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). s = rt /2= r(/6) r = 6/(/2) = 3

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

  45. d = (x2 – x1)2 + (y2 – y1)2 midpoint = ( , ) x1 + x2 2 y1 + y2 2 ►B. Exercises Determine the distance between each pair of points and also the midpoint of the segment joining them. 30. (3, -6), (1, 0)

  46. æ æ ö ö 3 + x 5 + y ç ç ÷ ÷ (9, -2) = , 2 2 è è ø ø ►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)

  47. 3 + x 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) 18 = 3 + x x = 15

  48. 5 + y -2 = 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) -4 = 5 + y y = -9

  49. ►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 = (15, -9)

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