Trigonometry

4.

Graphs of Trigonometric Functions

4.1.

Periodic Functions

4.1.1.

Definition of Periodicity

4.1.1.1.

Repeating Pattern

4.1.1.2.

Fundamental Period

4.1.2.

Period and Amplitude

4.1.2.1.

Period Definition

4.1.2.1.1.

Horizontal Length of One Cycle

4.1.2.1.2.

Calculating Period from Function

4.1.2.2.

Amplitude Definition

4.1.2.2.1.

Vertical Distance from Midline

4.1.2.2.2.

Calculating Amplitude

4.1.3.

4.1.3.1.

Definition as Reciprocal of Period

4.1.3.2.

Relationship to Period

4.1.3.3.

Applications in Wave Motion

4.2.

Graph of Sine Function

4.2.1.

Basic Sine Function y = sin x

4.2.1.1.

Domain and Range

4.2.1.2.

4.2.1.3.

4.2.2.

4.2.2.1.

Zeros at Multiples of π

4.2.2.2.

Maximum Values at π/2 + 2πn

4.2.2.3.

Minimum Values at 3π/2 + 2πn

4.2.2.4.

y-intercept at Origin

4.2.3.

Sketching the Basic Sine Graph

4.2.3.1.

Five Key Points Method

4.2.3.2.

Smooth Curve Connection

4.3.

Graph of Cosine Function

4.3.1.

Basic Cosine Function y = cos x

4.3.1.1.

Domain and Range

4.3.1.2.

4.3.1.3.

4.3.2.

4.3.2.1.

Zeros at π/2 + πn

4.3.2.2.

Maximum Values at Multiples of 2π

4.3.2.3.

Minimum Values at π + 2πn

4.3.2.4.

y-intercept at (0, 1)

4.3.3.

Sketching the Basic Cosine Graph

4.3.3.1.

Five Key Points Method

4.3.3.2.

Relationship to Sine Graph

4.4.

Transformations of Sine and Cosine

4.4.1.

Amplitude Changes

4.4.1.1.

Vertical Stretch: y = A sin x

4.4.1.2.

Vertical Compression: y = A sin x (|A| < 1)

4.4.1.3.

Reflection: y = -A sin x

4.4.2.

4.4.2.1.

Horizontal Compression: y = sin(Bx), B > 1

4.4.2.2.

Horizontal Stretch: y = sin(Bx), 0 < B < 1

4.4.2.3.

Period Formula: 2π/|B|

4.4.3.

Phase Shift (Horizontal Shift)

4.4.3.1.

Left Shift: y = sin(x + C), C > 0

4.4.3.2.

Right Shift: y = sin(x - C), C > 0

4.4.3.3.

Phase Shift from y = sin(B(x - C))

4.4.4.

4.4.4.1.

Upward Shift: y = sin x + D, D > 0

4.4.4.2.

Downward Shift: y = sin x + D, D < 0

4.4.4.3.

Midline Changes

4.4.5.

General Form: y = A sin(B(x - C)) + D

4.4.5.1.

Identifying All Parameters

4.4.5.2.

Order of Transformations

4.4.5.3.

Graphing Strategy

4.5.

Graph of Tangent Function

4.5.1.

Basic Tangent Function y = tan x

4.5.1.1.

Domain: x ≠ π/2 + πn

4.5.1.2.

Range: All Real Numbers

4.5.1.3.

4.5.2.

4.5.2.1.

Zeros at Multiples of π

4.5.2.2.

Vertical Asymptotes at π/2 + πn

4.5.2.3.

Increasing on Each Interval

4.5.3.

Sketching the Tangent Graph

4.5.3.1.

Asymptote Identification

4.5.3.2.

Key Points Between Asymptotes

4.5.3.3.

Repeating Pattern

4.6.

Graph of Cotangent Function

4.6.1.

Basic Cotangent Function y = cot x

4.6.1.1.

Domain: x ≠ πn

4.6.1.2.

Range: All Real Numbers

4.6.1.3.

4.6.2.

4.6.2.1.

Zeros at π/2 + πn

4.6.2.2.

Vertical Asymptotes at Multiples of π

4.6.2.3.

Decreasing on Each Interval

4.6.3.

Sketching the Cotangent Graph

4.7.

Graphs of Secant and Cosecant Functions

4.7.1.

Secant Function y = sec x

4.7.1.1.

Relationship to Cosine

4.7.1.2.

Domain: x ≠ π/2 + πn

4.7.1.3.

Range: (-∞, -1] ∪ [1, ∞)

4.7.1.4.

Vertical Asymptotes

4.7.1.5.

U-shaped Branches

4.7.2.

Cosecant Function y = csc x

4.7.2.1.

Relationship to Sine

4.7.2.2.

Domain: x ≠ πn

4.7.2.3.

Range: (-∞, -1] ∪ [1, ∞)

4.7.2.4.

Vertical Asymptotes

4.7.2.5.

U-shaped Branches

4.7.3.

Graphing Strategy for Reciprocal Functions

4.7.3.1.

Graph Base Function First

4.7.3.2.

Identify Asymptotes

4.7.3.3.

Sketch Reciprocal Curves

4.8.

Modeling with Sinusoidal Functions

4.8.1.

Identifying Periodic Phenomena

4.8.1.1.

4.8.1.2.

Mechanical Oscillations

4.8.1.3.

Economic Patterns

4.8.2.

Constructing Sinusoidal Models

4.8.2.1.

Determining Amplitude

4.8.2.2.

4.8.2.3.

Identifying Phase Shift

4.8.2.4.

Determining Vertical Shift

4.8.3.

Interpreting Parameters in Context

4.8.3.1.

Physical Meaning of Each Parameter

4.8.3.2.

Units and Scale Considerations

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3. Trigonometric Functions of Any Angle

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5. Trigonometric Identities