2. Copyright © 2009 Pearson Education, Inc. CHAPTER 6: The Trigonometric Functions 6.1The Trigonometric Functions of Acute Angles 6.2Applications of Right Triangles 6.3Trigonometric Functions of Any Angle 6.4Radians, Arc Length, and Angular Speed 6.5Circular Functions: Graphs and Properties 6.6Graphs of Transformed Sine and Cosine Functions

3. Copyright © 2009 Pearson Education, Inc. 6.5 Circular Functions: Graphs and Properties  Given the coordinates of a point on the unit circle, find its reflections across the x-axis, the y-axis, and the origin.  Determine the six trigonometric function values for a real number when the coordinates of the point on the unit circle determined by that real number are given.  Find the function values for any real number using a calculator.  Graph the six circular functions and state their properties.

4. Slide 6.5 - 4 Copyright © 2009 Pearson Education, Inc. Unit Circle We defined radian measure to be When r = 1, The arc length s on a unit circle is the same as the radian measure of the angle .

5. Slide 6.5 - 5 Copyright © 2009 Pearson Education, Inc. Basic Circular Functions For a real number s that determines a point (x, y) on the unit circle:

6. Slide 6.5 - 6 Copyright © 2009 Pearson Education, Inc. Reflections on a Unit Circle Let’s consider the radian measure π/3 and determine the coordinates of the point on the unit circle.

7. Slide 6.5 - 7 Copyright © 2009 Pearson Education, Inc. Reflections on a Unit Circle We have a 30º- 60º right triangle with hypotenuse 1 and side opposite 30º 1/2 the hypotenuse, or 1/2. This is the x-coordinate of the point. Let’s find the y-coordinate.

8. Slide 6.5 - 8 Copyright © 2009 Pearson Education, Inc. Example Each of the following points lies on the unit circle. Find their reflections across the x-axis, the y-axis, and the origin. Solution: a)a)

9. Slide 6.5 - 9 Copyright © 2009 Pearson Education, Inc. Example Solution continued

10. Slide 6.5 - 10 Copyright © 2009 Pearson Education, Inc. Example Solution continued

11. Slide 6.5 - 11 Copyright © 2009 Pearson Education, Inc. Find Function Values Knowing only a few points on the unit circle allows us to find trigonometric function values of frequently used numbers.

12. Slide 6.5 - 12 Copyright © 2009 Pearson Education, Inc. Example Find each of the following function values. Solution Locate the point on the unit circle determined by the rotation, and then find its coordinates using reflection if necessary.

13. Slide 6.5 - 13 Copyright © 2009 Pearson Education, Inc. Example Solution continued

14. Slide 6.5 - 14 Copyright © 2009 Pearson Education, Inc. Example Solution continued

15. Slide 6.5 - 15 Copyright © 2009 Pearson Education, Inc. Example Solution continued

16. Slide 6.5 - 16 Copyright © 2009 Pearson Education, Inc. Example Solution continued

17. Slide 6.5 - 17 Copyright © 2009 Pearson Education, Inc. Example Solution continued

18. Slide 6.5 - 18 Copyright © 2009 Pearson Education, Inc. Example Solution continued

19. Slide 6.5 - 19 Copyright © 2009 Pearson Education, Inc. Example Find each of the following function values of radian measures using a calculator. Round the answers to four decimal places. Solution: With the calculator in RADIAN mode:

20. Slide 6.5 - 20 Copyright © 2009 Pearson Education, Inc. Graph of Sine Function Make a table of values from the unit circle.

21. Slide 6.5 - 21 Copyright © 2009 Pearson Education, Inc.

22. Slide 6.5 - 22 Copyright © 2009 Pearson Education, Inc. Graph of Sine Function

23. Slide 6.5 - 23 Copyright © 2009 Pearson Education, Inc. Graph of Cosine Function Make a table of values from the unit circle.

24. Slide 6.5 - 24 Copyright © 2009 Pearson Education, Inc.

25. Slide 6.5 - 25 Copyright © 2009 Pearson Education, Inc. Graph of Cosine Function

26. Slide 6.5 - 26 Copyright © 2009 Pearson Education, Inc. Domain and Range of Sine and Cosine Functions The domain of the sine function and the cosine function is (–∞, ∞). The range of the sine function and the cosine function is [–1, 1].

27. Slide 6.5 - 27 Copyright © 2009 Pearson Education, Inc. Periodic Function A function with a repeating pattern is called periodic. The sine and cosine functions are periodic because they repeat themselves every 2π units. To see this another way, think of the part of the graph between 0 and 2π and note that the rest of the graph consists of copies of it. The sine and cosine functions each have a period of 2π. The period can be thought of as the length of the shortest recurring interval.

28. Slide 6.5 - 28 Copyright © 2009 Pearson Education, Inc. Periodic Function A function f is said to be periodic if there exists a positive constant p such that for all s in the domain of f. The smallest such positive number p is called the period of the function.

29. Slide 6.5 - 29 Copyright © 2009 Pearson Education, Inc. Amplitude The amplitude of a periodic function is defined to be one half the distance between its maximum and minimum function values. It is always positive. Both the graphs and the unit circle verify that the maximum value of the sine and cosine functions is 1, whereas the minimum value of each is –1.

30. Slide 6.5 - 30 Copyright © 2009 Pearson Education, Inc. Amplitude of the Sine Function the amplitude of the sine function

31. Slide 6.5 - 31 Copyright © 2009 Pearson Education, Inc. Amplitude of the Cosine Function the amplitude of the cosine function

32. Slide 6.5 - 32 Copyright © 2009 Pearson Education, Inc. Odd and Even Consider any real number s and its opposite, –s. These numbers determine points T and T 1.

33. Slide 6.5 - 33 Copyright © 2009 Pearson Education, Inc. Odd and Even Because their second coordinates are opposites of each other, we know that for any number s, Because their first coordinates are opposites of each other, we know that for any number s, The sine function is odd. The cosine function is even.

34. Slide 6.5 - 34 Copyright © 2009 Pearson Education, Inc. Graph of the Tangent Function Instead of a table, let’s begin with the definition and a few points on the unit circle.

35. Slide 6.5 - 35 Copyright © 2009 Pearson Education, Inc. Graph of the Tangent Function Tangent function is not defined when x, the first coordinate, is 0; that is, when cos s = 0: Draw vertical asymptotes at these locations.

36. Slide 6.5 - 36 Copyright © 2009 Pearson Education, Inc. Graph of the Tangent Function Note: Add these ordered pairs to the graph. Use a calculator to add some other points in (–π/2, π/2).

37. Slide 6.5 - 37 Copyright © 2009 Pearson Education, Inc. Graph of the Tangent Function Now we can complete the graph.

38. Slide 6.5 - 38 Copyright © 2009 Pearson Education, Inc. Graph of the Tangent Function From the graph, we see that: Period is π. There is no amplitude (no maximum or minimum values). Domain is the set of all real numbers except (π/2) + kπ, where k is an integer. Range is the set of all real numbers.

39. Slide 6.5 - 39 Copyright © 2009 Pearson Education, Inc. Graph of the Cotangent Function The cotangent function (cot s = cos s/sin s) is not defined when y, the second coordinate, is 0; that is, it is not defined for any number s whose sine is 0. Cotangent is not defined for s = 0, ±2π, ±3π, … The graph of the cotangent function is on the next slide.

40. Slide 6.5 - 40 Copyright © 2009 Pearson Education, Inc. Graph of the Cotangent Function

41. Slide 6.5 - 41 Copyright © 2009 Pearson Education, Inc. Graph of the Cotangent Function From the graph, we see that: Period is π. There is no amplitude (no maximum or minimum values). Domain is the set of all real numbers except kπ, where k is an integer. Range is the set of all real numbers.

42. Slide 6.5 - 42 Copyright © 2009 Pearson Education, Inc. Graph of the Cosecant Function The cosecant and sine functions are reciprocals. The graph of the cosecant function can be constructed by finding the reciprocals of the values of the sine function. The cosecant function is not defined for those values of s whose sine is 0. The graph of the cosecant function is on the next slide with the graph of the sine function in gray for reference.

43. Slide 6.5 - 43 Copyright © 2009 Pearson Education, Inc. Graph of the Cosecant Function

44. Slide 6.5 - 44 Copyright © 2009 Pearson Education, Inc. Graph of the Cosecant Function From the graph, we see that: Period is 2π. There is no amplitude (no maximum or minimum values). Domain is the set of all real numbers except kπ, where k is an integer. Range is (–∞, –1] U [1, ∞).

45. Slide 6.5 - 45 Copyright © 2009 Pearson Education, Inc. Graph of the Secant Function The secant and cosine functions are reciprocals. The graph of the secant function can be constructed by finding the reciprocals of the values of the cosine function. The secant function is not defined for those values of s whose cosine is 0. The graph of the secant function is on the next slide with the graph of the cosine function in gray for reference.

46. Slide 6.5 - 46 Copyright © 2009 Pearson Education, Inc. Graph of the Cosecant Function

47. Slide 6.5 - 47 Copyright © 2009 Pearson Education, Inc. Graph of the Secant Function From the graph, we see that: Period is 2π. There is no amplitude (no maximum or minimum values). Domain is the set of all real numbers except kπ, where k is an integer. Range is (–∞, –1] U [1, ∞).