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Trigonometric Ratios in the Unit Circle. Warm-up (2 m) 1. Sketch the following radian measures:

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Presentation on theme: "Trigonometric Ratios in the Unit Circle. Warm-up (2 m) 1. Sketch the following radian measures:"— Presentation transcript:

1 Trigonometric Ratios in the Unit Circle

2 Warm-up (2 m) 1. Sketch the following radian measures:

3

4 Trigonometric Ratios in the Unit Circle The unit circle has a radius of 1

5 x is y is x is y is x is y is x is y is Quadrant IQuadrant II Quadrant IIIQuadrant IV

6 “All Students Take Calculus” AS CT all ratios are positive sine is positive tangent is positive cosine is positive cosecant is positive cotangent is positive secant is positive

7 Example: Trigonometric Ratio Sine Cosine Tangent

8 Example: Trigonometric Ratio Sine Cosine Tangent

9 Your Turn: Complete problems 1 - 3

10 Sketching Negative Radians and/or Multiple Revolutions 1. Whenever the angle is less than 0 or more than 2 pi, solve for the coterminal angle between 0 and 2 pi 2. Sketch the coterminal angle

11 Example #3: Trigonometric Ratio Sine Cosine Tangent

12 Example #4: Trigonometric Ratio Sine Cosine Tangent

13 Your Turn: Complete practice problems 4 – 7

14 Reminder: Special Right Triangles 30° 60° 45° 30° – 60° – 90°45° – 45° – 90°

15 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.

16 Special Right Triangles & the Unit Circle

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

18 30°- 60°

19 45° or

20

21

22 Summarizing Questions 1. In which quadrants is tangent positive? Why? 2. In which quadrants is cosecant negative? Why? 3. How do I sketch negative angles? 4. How can I sketch angles with multiple revolutions? 5. What are some ways of remembering the radian measures of the Unit Circle? 6. How do we get the coordinates for π/6, π/4, and π/3?

23 Example #5

24 Example #6

25 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.

26 8. Sine Cosine Tangent 9. Sine Cosine Tangent

27 10. Sine Cosine Tangent 11. Sine Cosine Tangent

28 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

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

30 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!!!!

31 Example #7:

32 Example #8:

33 Example #9:

34 Your Turn:

35

36

37

38

39 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)

40 Example #10: Reference Angle: Coterminal Angle:

41 Example #10: Coordinates: Sine: Tangent: Cosine:

42 Example #11: Reference Angle: Coterminal Angle:

43 Example #11: Coordinates: Sine: Tangent: Cosine:

44 Example #12: Reference Angle: Coterminal Angle:

45 Example #12: Coordinates: Sine: Tangent: Cosine:

46 Your Turn: Complete problems 12 – 18.

47 Exit Ticket Solve for the exact values of the following:

48 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?

49 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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