section 4.1 graphs of sine and cosine
DESCRIPTION
Chapter 4 Graphs 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. Section 4.1 Graphs of Sin & Cos. Identify Periodic Functions Graph the Sine Function Graph the Cosine Function - PowerPoint PPT PresentationTRANSCRIPT
Section 4.1 Graphs of Sine and Cosine
Section 4.2 Translations of Sin and Cos
Section 4.3 Other Circular Functions
Chapter 4Graphs of the Circular Function
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
Periodic FunctionsA periodic function is a function ff such that:
f(x) = f(x + np)f(x) = f(x + np)for every real number xx in the domain of ff, every integer nn, and some positive real number pp.
The smallest possible value of pp is the periodperiod of the function.
Graph of the Sine Function
POSEIDON/TOPEX Imagery
Graph of the Sine Function
Characteristics of the Sine Function.Domain: (-Domain: (-ë, ëë, ë))Range: [-1, 1]Range: [-1, 1]
Over the interval [0, Over the interval [0, é/2] 0 æ 1é/2] 0 æ 1Over the interval [Over the interval [é/2, é] 1 æ 0é/2, é] 1 æ 0Over the interval [Over the interval [éé, 3, 3é/2] 0 æ -1é/2] 0 æ -1Over the interval [3Over the interval [3é/2, 2é] -1æ 0é/2, 2é] -1æ 0
The graph is continuous over its entire domain and symmetric with repeat to the origin.
x-intercepts:nx-intercepts:néé Period: 2é Period: 2é
Graph of the Cosine Function
Characteristics of the Cosine Function.Domain: (-Domain: (-ë, ëë, ë))Range: [-1, 1]Range: [-1, 1]
Over the interval [0, Over the interval [0, é/2] 1 æ 0é/2] 1 æ 0Over the interval [Over the interval [é/2, é] 0 æ-1é/2, é] 0 æ-1Over the interval [Over the interval [éé, 3, 3é/2] -1æ 0é/2] -1æ 0Over the interval [3Over the interval [3é/2, 2é] 0æ 1é/2, 2é] 0æ 1
The graph is continuous over its entire domain and symmetric with repeat to the origin.
x-intercepts: x-intercepts: é/2 + é/2 + nnéé Period: 2éPeriod: 2é
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|].
|a| is called the amplitudeamplitude.
Example with a sound wave
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.
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.
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
Horizontal Translations
• A horizontal translation is called a phase phase shiftshift when dealing with circular functions. In the function y = y = ff(x-d)(x-d), the expression (x-d) (x-d) is called the argumentargument with a shift of shift of d unitsd units to the right if d >0 right if d >0 and |d| units to |d| units to the left if d<0the left if d<0.
Vertical Translations
The graph of a function of the form
y = c + y = c + ff(x)(x)
is translated vertically as compared to the graph of y = y = ff(x)(x) with a shift of c units upshift of c units up if if c >0 c >0 and |c| units down if c<0|c| units down if c<0.
Combinations of Translations
The graph of a function of the form
y = c + y = c + ff(x - d)(x - d)
has both a horizontal and a vertical shift. To graph the function it doesn’t matter which one you look at first.
Determining a Trig ModelUsing Curve Fitting
• http://mathdemos.gcsu.edu/mathdemos/sinusoidapp/sinusoidapp.html
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
Sine Graph
Cosine Graph
Cosecant Graph
Secant Graph
Tangent Graph
Cotangent Graph
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