Lecture on Trigonometry: Unit Circle and Trig Graphs

Overview

• Continuing with Module Three
  • Covers third homework and quiz
  • Focus on unit circle and trig graphs

Unit Circle Recap

• Definition: Essential for determining trig functions of special angles
• Angles: 0 to 360 degrees or 0 to 2π in radians
• Special Angles: 30, 45, 60 degrees, etc.
• Cosine and Sine:
  • Cosine of angle = x-coordinate of point
  • Sine of angle = y-coordinate of point
  • Example: Cosine of π/6 is √3/2; Sine of π/6 is 1/2
• Quadrants: Use unit circle to find functions in different quadrants

Introduction to Trig Graphs

• Purpose: Graphically represent sine and cosine functions
• Creating Tables:
  • Start with an angle (θ)
  • Determine cosine and sine for key angles: 0, π/2, π, 3π/2, 2π, etc.
• Graphing:
  • Use values from unit circle
  • Create graphs with angles as x-axis and trig values as y-axis

Graphing Cosine and Sine Functions

• Standard Functions:
  • Cosine starts high (1) at angle 0
  • Sine starts at 0 at angle 0
• Plotting Points for cosine and sine from the unit circle

Variations in Trig Graphs

• Amplitude and Period:
  • y = a * cos(ωx) or y = a * sin(ωx)
  • Amplitude (|a|): From middle to highest/lowest point
  • Period (T): Distance before graph repeats, calculated as 2π/ω
• Graph Adjustments:
  • Vertical Shift: Adjust graph up/down after plotting
  • Negative Coefficients: Reflect graph across the x-axis

Example Problems

• Equation to Graph Conversion:
  • Identify amplitude and period
  • Plot the basic graph, apply vertical shifts
• Graph to Equation Conversion:
  • Determine if function is cosine or sine by starting point
  • Calculate amplitude and period to determine equation

Key Takeaways

• Memorize unit circle values for quick reference
• Understand cosine and sine graph characteristics
• Use amplitude and period formulas to adjust and interpret graphs
• Practice converting between equations and graphs for both sine and cosine functions

This lecture focused on understanding the use of the unit circle for determining trigonometric functions and graphically representing these functions using sine and cosine graphs, emphasizing the calculation of amplitude and period for various transformations.