Section 4.1 Graphs of Sine and Cosine

1. Chapter 4Graphs of the Circular Function Section 4.1 Graphs of Sine and Cosine Section 4.2 Translations of Sin and Cos Section 4.3 Other Circular Functions

2. Section 4.1 Graphs of Sin & Cos • Identify Periodic Functions • Graph the Sine Function • Graph the Cosine Function • Identify Amplitude and Period • Use a Trigonometric Model

3. Periodic Functions A periodic function is a function f such that: f(x) = f(x + np) for every real number x in the domain of f, every integer n, and some positive real number p. The smallest possible value of p is the period of the function.

4. Graph of the Sine Function POSEIDON/TOPEX Imagery

5. Graph of the Sine Function Characteristics of the Sine Function. Domain: (-ë, ë) Range: [-1, 1] Over the interval [0, é/2] 0 æ 1 Over the interval [é/2, é] 1 æ 0 Over the interval [é, 3é/2] 0 æ -1 Over the interval [3é/2, 2é] -1æ 0 The graph is continuous over its entire domain and symmetric with repeat to the origin. x-intercepts:né Period: 2é

6. Graph of the Cosine Function Characteristics of the Cosine Function. Domain: (-ë, ë) Range: [-1, 1] Over the interval [0, é/2] 1 æ 0 Over the interval [é/2, é] 0 æ-1 Over the interval [é, 3é/2] -1æ 0 Over the interval [3é/2, 2é] 0æ 1 The graph is continuous over its entire domain and symmetric with repeat to the origin. x-intercepts: é/2 + né Period: 2é

7. Amplitude of Sine and Cosine Functions Example with a sound wave The graph of y= a sin x or y = a cos x, with a å 0, will have the same shape as the graph of y = sin x or y= cos x, respectively, except with the range [-|a|, |a|]. |a| is called the amplitude.

8. Period of Sine and Cosine Functions For b> 0, the graph of y = sin bx will look like that of y = sin x, but with a period of 2é/b. Also the graph of y = cos bx will look like that of y = cos x, but with a period of 2é/b.

9. Guidelines for Sketching Graphs of Sine and Cosine • Find the period • Divide the interval into four equal parts • Evaluate the function for each of the five x-values resulting from step 2. • Plot the points and join them with a sinusoidal curve. • Draw additional cycles on the right and left as needed.

10. Section 4.2 Translations of the Graphs of Sin and Cos • Understand Horizontal Translations • Understand Vertical Translations • Understand Combinations of Translations • Determine a Trigonometric Maodel using Curve Fitting

11. Horizontal Translations • A horizontal translation is called a phase shift when dealing with circular functions. In the function y = f(x-d),the expression (x-d) is called the argument with a shift of d units to the right if d >0 and |d| units to the left if d<0.

13. Vertical Translations The graph of a function of the form y = c + f(x) is translated vertically as compared to the graph of y = f(x) with a shift of c units upif c >0 and |c| units down if c<0.

15. Combinations of Translations The graph of a function of the form y = c + f(x - d) has both a horizontal and a vertical shift. To graph the function it doesn’t matter which one you look at first.

17. Determining a Trig ModelUsing Curve Fitting • http://mathdemos.gcsu.edu/mathdemos/sinusoidapp/sinusoidapp.html

18. Section 4.3 Graphs of the Other Circular Functions • Graph the Cosecant • Graph the Secant • Graph the Tangent • Graph the Cotangent • Understand Addition of Ordinates

19. Sine Graph

20. Cosine Graph

21. Cosecant Graph

22. Secant Graph

23. Tangent Graph

24. Cotangent Graph

25. Addition of Ordinates • New functions can be formed by combining other functions. Example: y = sin x + cos x • Since the y coordinate is called the ordinate Addition of ordinates means we add to get the y coordinate (x, sin x + cos x) • On the graphing calculator we use Y1= sin x and Y2= cos x with Y3= Y1 + Y2