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Radian Measure (3.1)

Radian Measure (3.1). JMerrill, 2007 Revised 2000. A Newer Kind of Angle Measurement: The Radian. 1 radian = the measure of the central angle of a circle that intercepts an arc equal in length to the radius of the circle.

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Radian Measure (3.1)

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  1. Radian Measure (3.1) JMerrill, 2007 Revised 2000

  2. A Newer Kind of Angle Measurement: The Radian • 1 radian = the measure of the central angle of a circle that intercepts an arc equal in length to the radius of the circle. The central angle is an angle that has its vertex at the center of a circle r r θ r

  3. The Radian 2 radians ≈ 114.6o 1 radian ≈ 57.3o 3 radians ≈ 171.9o 6 radians ≈ 343.8o 5 radians ≈ 286.5o 4 radians ≈ 229.2o

  4. Conversion Factor Between Radians and Degrees Radians can be expressed in decimal form or exact answers. The majority of the time, answers will be exact--left in terms of pi

  5. You Do • 196o = ? Radians (exact answer) • 1.35 radians = ? degrees

  6. Arc Length • In geometry, an arc length is represented by “s” • If any of these parts are unknown, use the formula s r θ r Where theta is in radians

  7. Arc Length • Example: A circle has a radius of 4 inches. Find the length of the arc intercepted by a central angle of 240o. • We will use s = rθ, but first we have to convert 240o to radians.

  8. Things You MUST Remember: • π radians = 180 degrees ( ½ revolution) • 2π radians = 360 degrees (1 revolution) • ¼ revolution = ? degrees = ? radians • 90 degrees π/2 radians

  9. Exact Angle Measurement • Angle measures that can be expressed evenly in degrees cannot be expressed evenly in radians, and vice versa. So, we use fractional multiples of π.

  10. Quadrant angles π 0o 180o 360o

  11. Special Angles & The Unit Circle P130

  12. Evaluating Trig Functions for Angles Using Radian Measure • Evaluate in exact terms • is equivalent to what degree? • So 60o

  13. You Do • Evaluate in exact terms

  14. Recall: Reference Angles Reference Angle: the smallest positive acute angle determined by the x-axis and the terminal side of θ ref angle ref angle ref angle ref angle

  15. Find Reference Angle 150° 30° 225° 45° 300° 60°

  16. Using Reference Angles a) sin 330° = = - sin 30° = - 1/2 b) cos 0° = = 1

  17. Using Reference Angles

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