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