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4.1 Angles and Their Measure

4.1 Angles and Their Measure. What is trigonometry?. The word “trigonometry” is Greek for triangle measurement. It was developed in ancient times to track the planets and stars. Trig was also used in navigation. 4.1 Vocabulary. Angle Initial Side Terminal Side Standard Position

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4.1 Angles and Their Measure

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  1. 4.1 Angles and Their Measure

  2. What is trigonometry? • The word “trigonometry” is Greek for triangle measurement. • It was developed in ancient times to track the planets and stars. • Trig was also used in navigation.

  3. 4.1 Vocabulary • Angle • Initial Side • Terminal Side • Standard Position • Counter-clockwise (CCW) • Clockwise (CW) • Coterminal Angles • QuadrantalAngle • Central Angle • Radian • Arc Length

  4. Angle, Initial Side, and Terminal Side • Def: An angle is a union of two rays. • Def: The side where the angle “begins” or opens up from is the initial side • Def: The side where the angle ends is the terminal side

  5. Standard Position • Def: An angle is in standard position when the vertex of the angle is at the origin and the initial side is on the positive x-axis

  6. Coterminal Angles • Def: Coterminal angles are angles in standard position whose terminal sides coincide (they have the same initial and terminal sides) • Counter-clockwise (CCW) angles have positive measurements • Clockwise (CW) angles have negative measurements • We denote the direction we move by drawing an arrow within the angle

  7. Quadrantal Angles Angles in the standard position where the terminal side lies on the x or y axis.

  8. Reference Angle • If angle A is in standard position, its reference angle (Ar) is the angle formed by the x-axis and the terminal side of angle A.

  9. Radians

  10. r arc length q r radius Def: A radian is the angle measure when a central angle of a circle intercepts an arc that has the same length as the radius of the circle. q= 1 radian

  11. r q r q = 1 radian What is q’s measure in radians when the arc length is equal to the radius?

  12. 2r q r q = 2 radians What is q’s measure in radians when the arc length is two times as long as the radius?

  13. 3r q r What is q’s measure in radians when the arc length is three times as long as the radius? q = 3 radians

  14. 3r 2r r q q q r r r

  15. What if the arc length is an entire revolution? q = 2pradians Then the arclength equals the circumference of the circle, or 2pr. 2p r q r

  16. Radian/Degree Equivalencies

  17. x x Conversion Factors Radians to Degrees: Degrees to Radians:

  18. Degrees and Radians Equivalency Circle

  19. Reference Angles of 30 Degrees

  20. 30 Degree Reference Angles in Radians

  21. Reference Angles of 45 Degrees

  22. 45 Degree Reference Angles in Radians

  23. Reference Angles of 60 Degrees

  24. 60 Degree Reference Angles in Radians

  25. Radians Review Questions • What is the definition of a radian? • Can you represent each degree measurement in radians? • Can you represent each radian measurement in degrees? • Can you identify the special angles if given in radian measurement?

  26. The End

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