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Angle Measurement and Conversion: Degrees and Radians

Learn how to measure angles in degrees and radians, convert between different units, and understand the concept of coterminal angles. Explore the relationship between central angles, arc lengths, and radii of a circle.

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Angle Measurement and Conversion: Degrees and Radians

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  1. Lesson 7-1 Angles, Arcs, and Sectors

  2. Objective:

  3. To find the measure of an angle in either degrees or radians and to find coterminal angles. Objective:

  4. An angle is made up of two rays:

  5. An angle is made up of two rays: Initial Ray

  6. An angle is made up of two rays: Terminal Ray Initial Ray

  7. An angle is made up of two rays: Terminal Ray θ Initial Ray The angle measure (or opening between the two rays) we define as θ.

  8. An angle is made up of two rays: Terminal Ray θ Initial Ray Our common unit for measuring smaller angles is the degree.

  9. We all know there is 180° in a triangle and we also should all know by now there is 360° in a circle.

  10. A circle could just represent one complete revolution about a point.

  11. A circle could just represent one complete revolution about a point. So therefore in a revolution we say there is 360°.

  12. For angles that are very large, we may measure them in revolutionsinstead of degrees.

  13. For angles that are very large, we may measure them in revolutions instead of degrees. So, 5 revolutions= 5(360) = 1800°.

  14. Now, degrees can be broken down into smaller units.

  15. Now, degrees can be broken down into smaller units. For instance every degree has 60 minutes in its unit or 1° = 60 minutes.

  16. Just like in a clock each minute can be broken down into smaller units called seconds.

  17. How many seconds do you think are in 1 minute (denoted 1’)?

  18. How many seconds do you think are in 1 minute (denoted 1’)? 60 seconds – or 60” = 1’

  19. A measurement that looks like this  25° 20’ 6” is read as 25 degrees, 20 minutes, 6 seconds.

  20. Convert this measurement 25° 20’ 6”.

  21. Convert this measurement 25° 20’ 6”.

  22. Convert this measurement 25° 20’ 6”.

  23. Convert this measurement 12.3°.

  24. Convert this measurement 12.3°.

  25. Another unit of measure for angle is called the radian:

  26. Another unit of measure for angle is called the radian:

  27. Another unit of measure for angle is called the radian: Central Angle

  28. Another unit of measure for angle is called the radian: Central Angle The vertex of the angle is located at the center of a circle.

  29. Another unit of measure for angle is called the radian: Central Angle The rays of the angle are both radii of the circle.

  30. Another unit of measure for angle is called the radian: Central Angle Traditionally call a central angle’s measure θ.

  31. Another unit of measure for angle is called the radian: The piece of the circle that is located between the two rays of the circle is called thearc length of the central angle.

  32. Another unit of measure for angle is called the radian: In general, the radian measure of a central angle is the number of radiusunits in the length of the intercepted arc of the circle between the two rays.

  33. Another unit of measure for angle is called the radian: Arc length is always denoted as s.

  34. Another unit of measure for angle is called the radian: Radius is always denoted as r.

  35. Another unit of measure for angle is called the radian: The measure of the central angle is always θ.

  36. Another unit of measure for angle is called the radian: This formula is always true  s = rθ so,

  37. One important conversion from radians to degrees is

  38. One important conversion from radians to degrees is

  39. From this we cancome up with these two equivalences:

  40. From this we cancome up with these two equivalences:

  41. From this we cancome up with these two equivalences:

  42. Convert 196° to radians (to the nearest hundredth).

  43. Convert 1.35 radians to decimal degrees (to the nearest tenth) and to degrees and minutes (to the nearest ten minutes).

  44. Angle measures are often given in terms of π.

  45. Angle measures are often given in terms of π. Some common angles and their conversion in terms of radians:

  46. Angle measures are often given in terms of π. Some common angles and their conversion in terms of radians:

  47. Angle measures are often given in terms of π. Some common angles and their conversion in terms of radians:

  48. Angle measures are often given in terms of π. Some common angles and their conversion in terms of radians:

  49. If an angle is in standard position,then the vertex is located at the origin.

  50. The initial ray is located on the positive x-axis.

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