Graphs of Sine, Cosine, Tangent, and Cotangent Functions: Periodic Behavior and Transformations

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Periodic Functions and Their Properties

Definition of Periodic Functions

A periodic function is a function that repeats its values in regular intervals or periods. Formally, a function f is periodic with period p if:

for all in the domain and for some positive number .

The most common examples in trigonometry are the sine and cosine functions, which have a period of .

Sine function:
Cosine function:

Key Terms: periodic function, period, sine wave (sinusoid), amplitude

Graph of the Sine Function

Behavior of Sine and Cosine Over One Period

As Increases from
$0\frac{\pi}{2}$	Increases from $0	Decreases from $1
to	Decreases from $1	Decreases from $0-1$
to	Decreases from $0-1$	Increases from to $0$
to	Increases from to $0$	Increases from $0

Reflection: The graph of the sine function has high points (maximums) and low points (minimums) at regular intervals.

Properties of the Sine Function

Domain:
Range:
The graph is continuous over its entire domain.
x-intercepts: , where is an integer.
Period:
The graph is symmetric with respect to the origin, so the sine function is an odd function:

Connection to the Unit Circle: The sine function relates to the y-coordinates of points on the unit circle corresponding to angle measures (arc lengths).

Graph of the Cosine Function

Properties of the Cosine Function

Domain:
Range:
The graph is continuous over its entire domain.
x-intercepts: , where is an integer.
Period:
The graph is symmetric with respect to the y-axis, so the cosine function is an even function:

Relationship to Sine: The graph of the cosine function is the graph of the sine function shifted to the left by units.

Amplitude and Period of Sine and Cosine Functions

Amplitude

The amplitude of or is .
The amplitude represents the maximum distance from the midline (usually the x-axis) to the peak of the wave.

Example: For , the amplitude is $2$.

Period

For or , the period is .
Increasing compresses the graph horizontally (more cycles per unit interval); decreasing stretches it.

Example: has period .

Guidelines for Sketching Graphs of Sine and Cosine Functions

Step-by-Step Method

Step 1: Find the period. For or , period is .
Step 2: Divide the interval of one period into four equal parts.
Step 3: Evaluate the function at each division point.
Step 4: Plot the points and connect them with a smooth, sinusoidal curve.
Step 5: Extend the graph over additional periods as needed.

Additional info: When is negative, the graph is reflected across the x-axis.

Transformations of Sine and Cosine Functions

Horizontal Translations (Phase Shifts)

The graph of is shifted to the right by units if , and to the left by units if .
For circular functions, this is called a phase shift.
In , the expression represents the horizontal translation.

Example: is the graph of shifted right by units.

Vertical Translations

The graph of is shifted up by units if , and down by units if .

Example: is the graph of shifted up by $3$ units.

General Form of Sine and Cosine Functions

Equation and Parameters

The general form: or
a: amplitude (vertical stretch/shrink)
b: affects period ()
c: vertical translation
d: horizontal translation (phase shift)

Additional info: These parameters allow modeling of real-world periodic phenomena, such as temperature or sound waves.

Modeling with Sine and Cosine Functions

Trigonometric Models

Periodic functions can model phenomena such as temperature, sound, and light.
Example: The average temperature at Mould Bay, Canada, can be modeled by .
Parameters are chosen to fit the amplitude, period, phase shift, and vertical translation to the data.

Graph of the Tangent and Cotangent Functions

Tangent Function

Domain: All real numbers except , where is an integer.
Range:
The graph has vertical asymptotes at .
x-intercepts:
Period:
No maximum or minimum values; the graph increases or decreases without bound between asymptotes.
The tangent function is odd:

Cotangent Function

Domain: All real numbers except , where is an integer.
Range:
The graph has vertical asymptotes at .
x-intercepts:
Period:
No maximum or minimum values; the graph increases or decreases without bound between asymptotes.
The cotangent function is odd:

Comparison: The tangent and cotangent functions are similar in shape but have their vertical asymptotes and x-intercepts at different locations.

Summary Table: Properties of Sine, Cosine, Tangent, and Cotangent Functions

Function	Domain	Range	Period	Symmetry	Vertical Asymptotes
				Odd	None
				Even	None
				Odd
				Odd

Applications and Modeling

Using Sine and Cosine Models

Periodic functions are used to model real-world phenomena such as temperature, sound, and light.
Parameters are adjusted to fit the amplitude, period, phase shift, and vertical translation to observed data.
Example: Modeling monthly temperatures with .

Additional info: Sine regression on graphing calculators can be used to fit trigonometric models to data.