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- 1. 6-3: Angles and Radian Measure © 2007 Roy L. Gover (www.mrgover.com) Learning Goals: •Define radians as another angle measure. •Extend the definition of an angle to include negative angles and angles greater than 180°
- 2. Definition Initial Side Terminal Side Verte x Angle A is in standard position A x y
- 3. Definition A If the terminal side moves counter- clockwise , angle A is positive x y
- 4. Definition A If the terminal side moves counter- clockwise , angle A is positive x y
- 5. Definition A If the terminal side moves counter- clockwise , angle A is positive x y
- 6. Definition A If the terminal side moves clockwise , angle A is negative x y
- 7. Definition A If the terminal side moves clockwise , angle A is negative x y
- 8. Definition A If the terminal side moves clockwise , angle A is negative x y
- 9. Definition A If the terminal side moves clockwise , angle A is negative x y
- 10. Definition A If the terminal side is on an axis, angle A is a quadrantel angle x y
- 11. Definition A If the terminal side is on an axis, angle A is a quadrantel angle x y
- 12. Definition A If the terminal side is on an axis, angle A is a quadrantel angle x y
- 13. Definition A If the terminal side is on an axis, angle A is a quadrantel angle x y Is this an angle?
- 14. Try This What kind of angle is this? (hint: where is the terminal side?) quadrantal angle
- 15. Try This What is the measure of this angle? a. -90° b. -45° c. -270° d. -360°
- 16. Try This What is the measure of this angle? a. -90° b. -45° c. -270° d. -360°
- 17. Try This What is the measure of this angle? a. 0° b. 45° c. 90° d. 120° e. 180°
- 18. Try This What is the measure of this angle? a. 0° b. 45° c. 90° d. 120° e. 180°
- 19. Try This What is the measure of this angle? a. -90° b. -45° c. -270° d. -360°
- 20. Try This What is the measure of this angle? a. -90° b. -45° c. -270° d. -360°
- 21. Try This What is the measure of this quadrantal angle? x y 0°
- 22. Try ThisIf a 143° angle is in standard position, determine the quadrant in which the terminal side lies. x y 2
- 23. Try ThisIf a 280° angle is in standard position, determine the quadrant in which the terminal side lies. x y 4
- 24. Important Idea There are two units of measure for angles: •degrees: used in geometry •radians: used in calculus In Precal, we use degrees and radians.
- 25. -1 1 -1 1 y x Radian: The length of the arc above the angle divided by the radius of the circle. Definition sr θ s r θ = , θ in radians (rads)
- 26. -1 1 -1 1 y x Definition s θ 1 s θ = , θ in radians (rads) Unit Circle: the circle with radius of 1 unit If r=1, =sθ 1
- 27. Definition The radian measure of an angle is the distance traveled around the unit circle. Since circumference of a circle is 2 r and r=1, the distance around the unit circle is 2 π π
- 28. Example Find the degree and radian measure of the angle in standard position formed by rotating the terminal side ½ of a circle in the positive direction. Leave your radian answer in terms of .π
- 29. Example Find the degree and radian measure of the angle in standard position formed by rotating the terminal side 5/6 of a circle in the negative direction. Leave your radian answer in terms of .π
- 30. Try This Find the degree and radian measure of the angle in standard position formed by rotating the terminal side 2/3 of a circle in the positive direction. Leave your radian answer in terms of .π
- 31. Solution 2 360 240 3 ° = °g 2 4 2 3 3 π π =g radians
- 33. Example 45 (degrees) 4 π radians ≈ .785 radians
- 34. Example 90 (degrees) 2 π radians ≈ 1.57 radians
- 37. Try This ≈ rads 270 rads 4.71 1 3 1 or 2 2 π π Do you see a pattern?
- 38. Important Idea Radian measure allows the expansion of trig functions to model real-world phenomena where independent variables represent distance or time and not just an angle measure in degrees.
- 39. Important Idea If a circle contains 360° or 2π radians, how many radians are in 180° • Use to change rads to degrees 180° π rads • Use to change degrees to rads π rads 180°
- 40. Example Change 30° to radian measure in terms of π.
- 41. Try This Change 120° to radian measure in terms of π. 2 rads 3 π
- 42. Try This Change 240° to radian measure in terms of π. 4 rads 3 π
- 43. Example Change radians to degree measure. 3 4 π
- 44. Example Change 2.356 radians to degree measure. (hint: radians are not always stated in terms of π.)
- 45. Try This Change radians to degree measure. 157.5° 7 8 π
- 46. Try This Change -3.5 radians to degree measure to the nearest tenth. -200.5°
- 50. Definition 0 2 π π The quadrantal angles in radians 2π The terminal side is on an axis.
- 51. Definition Coterminal Angles: Angles that have the same terminal side. Important Idea In precal, angles can be larger than 360° or 2 radians. π
- 52. Example Find positive angles and negative angles that are coterminal with 30°.
- 53. Important Idea To find coterminal angles, simply add or subtract either 360° or 2 radians to the given angle or any angle that is already coterminal to the given angle. π
- 54. Analysis 30° and 390° have the same terminal side, therefore, the angles are coterminal 30° x y x y 390°
- 55. Analysis 30° and 750° have the same terminal side, therefore, the angles are coterminal 30° x y x y 750°
- 56. Analysis 30° and 1110° have the same terminal side, therefore, the angles are coterminal 30° x y x y 1110°
- 57. Analysis 30° and -330° have the same terminal side, therefore, the angles are coterminal 30° x y x y -330°
- 58. How many angles can you find that are coterminal with a 30° angle and how do you find them? With Mr. Gover
- 59. Try This Find 3 angles coterminal with 60° 420°,780° and -300°
- 60. Try This Find one positive angle and two negative angle coterminal with radians.3 2 π and7 2 π 2 π− 5 2 π−,
- 61. Try This Find two positive angle and one negative angle coterminal with radians.5 6 π− and7 6 π 19 6 π 17 6 π−,
- 62. Lesson Close In your own words and without looking at your notes, write a definition for: •Coterminal angle •Radian