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Trigonometric Ratios in the Unit Circle

Trigonometric Ratios in the Unit Circle. Warm-up (2 m). Sketch the following radian measures:. Trigonometric Ratios in the Unit Circle. The unit circle has a radius of 1. Quadrant II. Quadrant I. x is y is. x is y is. Quadrant III. Quadrant IV. x is y is. x is y is.

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Trigonometric Ratios in the Unit Circle

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  1. Trigonometric Ratios in the Unit Circle

  2. Warm-up (2 m) • Sketch the following radian measures:

  3. Trigonometric Ratios in the Unit Circle • The unit circle has a radius of 1

  4. Quadrant II Quadrant I x is y is x is y is Quadrant III Quadrant IV x is y is x is y is

  5. “All Students Take Calculus” S A sine is positive all ratios are positive cosecant is positive T C cosine is positive tangent is positive cotangent is positive secant is positive

  6. Example:

  7. Example:

  8. Your Turn: • Complete problems 1 - 3

  9. Sketching Negative Radians and/or Multiple Revolutions • Whenever the angle is less than 0 or more than 2 pi, solve for the coterminal angle between 0 and 2 pi • Sketch the coterminalangle

  10. Example #3:

  11. Example #4:

  12. Your Turn: • Complete practice problems 4 – 7

  13. Reminder: Special Right Triangles 30° – 60° – 90° 45° – 45° – 90° 45° 60° 30° 45°

  14. Investigation! • Fit the paper triangles onto the picture below. The side with the * must be on the x-axis. Use the paper triangles to determine the coordinates of the three points.

  15. Special Right Triangles & the Unit Circle

  16. Special Right Triangles & the Unit Circle: 30°- 60°

  17. 30°- 60°

  18. 45° or

  19. 45° or

  20. Summarizing Questions • In which quadrants is tangent positive? Why? • In which quadrants is cosecant negative? Why? • How do I sketch negative angles? • How can I sketch angles with multiple revolutions? • What are some ways of remembering the radian measures of the Unit Circle? • How do we get the coordinates for π/6, π/4, and π/3?

  21. Example #5

  22. Example #6

  23. Your Turn: • Use your unit circle to solve for the exact values of sine, cosine, and tangent of problems 8 – 11. Rationalize the denominator if necessary.

  24. Reference Angles • Reference angles make it easier to find exact values of trig functions in the unit circle • Measure an angle’s distance from the x-axis

  25. Reference Angles, cont. • Always • Coterminal • Acute (less than ) • Have one side on the x-axis

  26. Solving for Reference Angles • Step 1: Calculate the coterminal angle if necessary (Remember, coterminal angles are positive and less than 2π.) • Step 2: Sketch either the given angle (if less than 2π) or the coterminal angle (if greater than 2π) • Step 3: Determine the angle’s distance from the x-axis (It is almost always pi/denominator!!!) This is the reference angle!!!!

  27. Example #7:

  28. Example #8:

  29. Example #9:

  30. Your Turn:

  31. Your Turn:

  32. Your Turn:

  33. Your Turn:

  34. Your Turn:

  35. Solving for Exact Trig Values • Step 1: Solve for the coterminal angle between 0 and 2π if necessary • Step 2: Solve for the reference angle (Note the quadrant) • Step 3: Identify the correct coordinates of the angle(Make sure the signs of the coordinates match the quadrant!) • Step 4: Solve for the correct trig ratio (Rationalize the denominator if necessary)

  36. Example #10: CoterminalAngle: Reference Angle:

  37. Example #10: Coordinates: Tangent: Sine: Cosine:

  38. Example #11: CoterminalAngle: Reference Angle:

  39. Example #11: Coordinates: Tangent: Sine: Cosine:

  40. Example #12: CoterminalAngle: Reference Angle:

  41. Example #12: Coordinates: Tangent: Sine: Cosine:

  42. Your Turn: • Complete problems 12 – 18.

  43. Exit Ticket • Solve for the exact values of the following: 1. 2. 3.

  44. Summarizing Questions How do we get the coordinates for using the 45° – 45° – 90° triangle? Why are the coordinates of negative? What are the sine, cosine, and tangent of ? What is a reference angle?

  45. Exit Ticket – “The Important Thing” • On a sheet of paper (with your name!) complete the sentence below: Three important ideas/things from today’s lesson are ________, ________, and ________, but the most important thing I learned today was ________.

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