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10.3: Extending the Trig Ratios

10.3: Extending the Trig Ratios. Expectation: G1.3.3: Determine the exact values of sine, cosine, and tangent for 0°, 30°, 45°, 60°, and their integer multiples and apply in various contexts.

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10.3: Extending the Trig Ratios

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  1. 10.3: Extending the Trig Ratios Expectation: G1.3.3: Determine the exact values of sine, cosine, and tangent for 0°, 30°, 45°, 60°, and their integer multiples and apply in various contexts. 10.3: Extending the Trig Ratios

  2. If the angles ∠X and ∠Y each measure between 0° and 90°, and if sin X = cos Y, what is the sum of the measures of the angles ∠X and ∠Y? • 30 • 45 • 60 • 90 • 135 10.3: Extending the Trig Ratios

  3. Angle of Rotation • An angle is an angle of rotation iff: • a. its vertex is the origin • b. one side is the positive x-axis • c. the other side is a rotation of the first side centered at the origin. 10.3: Extending the Trig Ratios

  4. Angles of Rotation θ 10.3: Extending the Trig Ratios

  5. Angles of Rotation θ 10.3: Extending the Trig Ratios

  6. Angles of Rotation θ 10.3: Extending the Trig Ratios

  7. Angles of Rotation θ 10.3: Extending the Trig Ratios

  8. Unit Circle • Defn: A circle is a unit circle iff: • a. its center is the origin (0,0). • b. its radius is 1. 10.3: Extending the Trig Ratios

  9. Unit Circle: x2 + y2 = 1 (0,1) (1,0) (-1,0) (0,-1) 10.3: Extending the Trig Ratios

  10. Who Cares? • We can use unit circles and trig to find coordinates of points on a unit circle. 10.3: Extending the Trig Ratios

  11. What are the coordinates of A? (0,1) A (1,0) (-1,0) 30° (0,-1) 10.3: Extending the Trig Ratios

  12. What are the coordinates of B? (0,1) B (-1,0) (1,0) 45° (0,-1) 10.3: Extending the Trig Ratios

  13. What are the coordinates of C? (0,1) (-1,0) (1,0) 60° C (0,-1) 10.3: Extending the Trig Ratios

  14. What are the coordinates of D? (0,1) (1,0) (-1,0) 60° D (0,-1) 10.3: Extending the Trig Ratios

  15. What is the angle of rotation for the hypotenuse below? (0,1) (.866, .5) A (-1,0) (1,0) (0,-1) 10.3: Extending the Trig Ratios

  16. ??????????? • What is the cos 30? • What is the sin 30? • Compare sin 30, cos 30 and the (x,y) coordinates of A. 10.3: Extending the Trig Ratios

  17. What is the angle of rotation for the hypotenuse below? (-.707, .707) (0,1) B (1,0) (-1,0) (0,-1) 10.3: Extending the Trig Ratios

  18. ??????????? What is the cos 135? What is the sin 135? Compare sin 135, cos 135 and the (x,y) coordinates of B. 10.3: Extending the Trig Ratios

  19. What is the angle of rotation for the hypotenuse below? (0,1) (1,0) (-1,0) (-.5, -.866) C (0,-1) 10.3: Extending the Trig Ratios

  20. ??????????? What is the cos 240? What is the sin 240? Compare sin 240, cos 240 and the (x,y) coordinates of C. 10.3: Extending the Trig Ratios

  21. What is the angle of rotation for the hypotenuse below? (0,1) (1,0) (-1,0) (.5, -.866) D (0,-1) 10.3: Extending the Trig Ratios

  22. ??????????? What is the cos 300? What is the sin 300? Compare sin 300, cos 300 and the (x,y) coordinates of D. 10.3: Extending the Trig Ratios

  23. Sine and Cosine on a Unit Circle • Defn: Let θ be a rotation angle. Then sin θ is the y-coordinate of the image of P(1,0) rotated θ about the origin and cos θ is the x-coordinate. P’= (cos θ, sin θ) 10.3: Extending the Trig Ratios

  24. What are sin (-60) and cos (-60)? What are the sin 440 and cos 440? 10.3: Extending the Trig Ratios

  25. Negative Angles • sin (-θ) = - sin θ • cos (- θ) = cos (θ) 10.3: Extending the Trig Ratios

  26. Angles Larger than 360° • If θ > 360, then: • sin θ = sin (θ - 360n) • cos θ = cos (θ - 360n) • where n is a whole number. 10.3: Extending the Trig Ratios

  27. Verify trig identity number 1: tan θ = sin θ cos θ 10.3: Extending the Trig Ratios

  28. Verify trig identity number 2: sin2 θ + cos2θ = 1 10.3: Extending the Trig Ratios

  29. Graphing Sine and Cosine For θ = 0, 30, 45, 60, 90, 120, 135, 150, 180, 210, 225, 240, 270, 300, 315, 330 and 360, determine sin θ and cos θ. It may be helpful to organize your data into a chart. Graph your data. 10.3: Extending the Trig Ratios

  30. 10.3: Extending the Trig Ratios

  31. A satellite orbits a planet at 1° per hour. Let the radius of the orbit equal 1 and determine the ordered pair coordinates of the satellite after 497 hours. Assume it starts at (1,0). 10.3: Extending the Trig Ratios

  32. Give 2 angles, θ, between 0 and 360 that have cos θ = .7071. 10.3: Extending the Trig Ratios

  33. Assignment • pages 652-653, • # 13-55 (odds) 10.3: Extending the Trig Ratios

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