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

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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

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

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Graph of the Sine Function POSEIDON/TOPEX Imagery

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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

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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

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Amplitude of Sine and Cosine Functions 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|]. amplitude |a| is called the amplitude. Example with a sound wave

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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.

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Guidelines for Sketching Graphs of Sine and Cosine 1.Find the period 2.Divide the interval into four equal parts 3.Evaluate the function for each of the five x-values resulting from step 2. 4.Plot the points and join them with a sinusoidal curve. 5.Draw additional cycles on the right and left as needed.

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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

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Horizontal Translations phase shift y = f (x-d) (x-d) argumentshift of d unitsright if d >0 |d| units to the left if d 0 and |d| units to the left if d<0.

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Vertical Translations The graph of a function of the form y = c + f (x) y = f (x)shift of c units upif c >0 |c| units down if c 0 and |c| units down if c<0.

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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.

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Determining a Trig Model Using Curve Fitting http://mathdemos.gcsu.edu/mathdemos/si nusoidapp/sinusoidapp.htmlhttp://mathdemos.gcsu.edu/mathdemos/si nusoidapp/sinusoidapp.html

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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

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Sine Graph

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Cosine Graph

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Cosecant Graph

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Secant Graph

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Tangent Graph

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Cotangent Graph

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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 Y 1 = sin x and Y 2 = cos x with Y 3 = Y 1 + Y 2

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