Graphs and Properties of Trigonometric Functions

Graphing Sine and Cosine Functions

The sine and cosine functions are fundamental periodic functions in trigonometry. Their general forms are:

• Sine:

• Cosine:

Key properties:

• Amplitude: (the maximum displacement from the midline)

• Period: (the length of one complete cycle)

• Phase Shift: (horizontal shift of the graph)

• Vertical Shift: (moves the graph up or down)

• x-coordinates of quarter points: Start at phase shift, add period 4

• y-coordinates of quarter points:

Graphing Tangent Functions

The tangent function has a different periodicity and features vertical asymptotes:

• General form:

• Period:

• Vertical Asymptotes: Occur at the endpoints of the principal cycle

• x-coordinates of center, left & right halfway points: Start at left vertical asymptote, add period 4

• y-coordinates of center, left & right halfway points:

• Tangent values: Left: , Center: $0

Graphs of Trigonometric Functions

Each trigonometric function has a distinct graph, characterized by its period, amplitude, and symmetry.

Even and Odd Properties of Trigonometric Functions

Trigonometric functions can be classified as even or odd, which affects their symmetry:

• Odd Functions: , , ,

• Even Functions: ,

Inverse Trigonometric Function Intervals

Inverse trigonometric functions are defined on restricted intervals to ensure they are one-to-one:

• : values , angles

• : values , angles

• : values , angles

Evaluating Inverse Trigonometric Functions

Inverse trigonometric functions return the angle whose trigonometric value is given. Their geometric interpretation is shown below:

Special Values of Trigonometric Functions

Trigonometric functions have well-known values at special angles. These are often summarized in a table:

Example Problems

Example 1: Sketching

Sketch the graph of and identify its properties: amplitude , period , phase shift , vertical shift . The graph is symmetric about the origin (odd function).

Example 2: Using Odd Function Property

Use the fact that is an odd function to determine which expression is equivalent to .

Example 3: Sketching

Sketch the graph of and identify its properties: period , vertical asymptotes at , odd function.

Example 4: Identifying Trigonometric Function from Graph

Given a graph of one cycle of a trigonometric function, determine the equation of the function in the form or .

Example 5: Determining Period and Quarter Points

Determine the period of and sketch its graph. The period is .

Example 6: Amplitude, Range, Period, and Phase Shift

Determine the amplitude, range, period, and phase shift for a given function and sketch the graph using quarter points.

Example 7: Tangent Function Principal Cycle

For a tangent function, determine the interval for the principal cycle, period, equations of vertical asymptotes, and coordinates of center and halfway points. Sketch the graph.

Example 8: Range of a Sine Function

Determine the range of . The range is .

Example 9: Exact Values of Trigonometric Expressions

Find the exact value of trigonometric expressions at special angles using the table above.

Summary Table: Trigonometric Function Properties

Additional info: Academic context and explanations have been expanded for clarity and completeness. Only directly relevant images and tables have been included to reinforce the study material.