Advanced Trigonometric Functions and Graph Writing

This section delves into the techniques for writing equations from graphs and understanding more complex trigonometric functions including tangent, cotangent, secant, and cosecant.

Definition: The process of writing equations involves finding amplitude (A), vertical shift (D), period factor (B), and phase shift (C).

Example: For finding amplitude: Calculate half the distance between maximum and minimum points of the graph.

Highlight: Special characteristics of advanced functions:

• Tangent: Period π, asymptotes at x = π/2 + πk
• Cotangent: Period π, asymptotes at x = πk
• Secant: Period 2π, range $-∞,-1] ∪ [1,∞$
• Cosecant: Period 2π, range $-∞,-1] ∪ [1,∞$

Vocabulary: Asymptotes are lines that a graph approaches but never touches, crucial in understanding tangent, cotangent, secant, and cosecant functions.

The Unit Circle and Basic Trigonometric Functions

This section covers the fundamental concepts of the unit circle and the graphing of sine and cosine functions. The unit circle is presented with its key points and angles, followed by detailed instructions for graphing basic trigonometric functions.

Definition: The unit circle is a circle with radius 1 centered at the origin, used to define trigonometric functions.

Example: To find values quickly: First draw planes with values, determine the quadrant, and for special angles like π/3 or π/6, use 30-60-90 triangle relationships.

Highlight: Sine and cosine functions have specific characteristics:

• Sine: Domain (-∞,∞), range $-1,1$, period 2π
• Cosine: Domain (-∞,∞), range $-1,1$, period 2π, even function

Vocabulary: Transformations are expressed as g$x$ = ±Af$B(x-C)$ + D, where:

• A affects amplitude
• B affects period
• C creates phase shift
• D creates vertical shift