Trigonometry Background for Visualizing Trigonometric Functions

Introduction

• Presenter: Dennis Davis
• Purpose: Provide trigonometry background for understanding six trigonometric functions.
• Note: Not a comprehensive course, but covers five important concepts.

Basics of Trigonometry

• Meaning: Trigonometry ("Trig") means triangle measurement, but focuses on angles.
• Angle Representation: Often denoted by Greek letters, commonly θ (theta).

Imaginary Triangle and Trig Functions

• Creating a Triangle: Use angle θ and a paper cutter analogy.
  • Sides:
    • Opposite (OPP): Across from θ
    • Hypotenuse (HYP): Longest side, opposite the right angle
    • Adjacent (ADJ): Next to θ, aligned with the guide
• Six Trig Functions: Ratios of triangle side lengths.
  • Functions: Sine, Cosine, Tangent, Cotangent, Secant, Cosecant
• SOHCAHTOA Mnemonic:
  • Sine (SOH): Opposite / Hypotenuse
  • Cosine (CAH): Adjacent / Hypotenuse
  • Tangent (TOA): Opposite / Adjacent

Reciprocal Trigonometric Functions

• Cosecant: 1/Sine = Hypotenuse/Opposite
• Secant: 1/Cosine = Hypotenuse/Adjacent
• Cotangent: 1/Tangent = Adjacent/Opposite
• Pairing: Each reciprocal pair has one function starting with "co".

Properties of Trig Functions

1. Properties of Angles: Functions of angles rather than triangles.
2. Unitless Ratios: Ratios of lengths, units cancel out.
3. Notation: Trig functions written without parentheses.
4. Square Functions: Write exponent between function abbreviation and angle.

Coordinate Plane and Quadrants

• Axes: X-axis (horizontal), Y-axis (vertical)
• Quadrants: Divided into four sections with different sign combinations for coordinates.

Angles Measurement

• Degrees: Familiar unit, e.g., 0, 90, 180, 270.
• Radians: More common in math/science.
  • Conversion: 2π radians = 360 degrees
  • Common Angles: Expressed in fractions of π

Pythagorean Theorem

• Right Triangles: a² + b² = c² (c is hypotenuse)
• Example: 3-4-5 triangle

Unit Circle

• Definition: Circle with radius 1, centered at origin of the xy-plane.
• Sine & Cosine: Directly derived from unit circle coordinates.
  • Sine = y-coordinate, Cosine = x-coordinate
  • Tangent = Sine/Cosine

Memorization Tips

• Focus on understanding rather than memorizing.
• Use visual aids and mnemonics.
• Practice deriving trigonometric values through understanding of patterns.

Conclusion

• Objective: Lay foundation for understanding graphical representation of trig functions.
• Encouragement: View other series for deeper insights.
• Advice: Avoid memorization, use logic and understanding.

Additional Resources

• Consider making flashcards for angles and their trigonometric values in both degrees and radians.