An Introduction to Trigonometry

I. Basic Concepts

• Trigonometric functions are based on the unit circle (radius r=1).
• Angle Measurements:
  • Degree Measure: Circle divided into 360 parts; right angle = 90°.
  • Radian Measure: Arc length; one revolution (360°) = 2π radians.
  • Conversion: 180° = π radians.
  • Example conversions:
    • 270° = 3π/2 radians.
    • 7π/5 radians = 252°.
• Trigonometric Functions:
  • Sine (sin): y = sin(θ)
  • Cosine (cos): x = cos(θ)
  • Others derived: Tangent (tan), Secant (sec), Cosecant (csc), Cotangent (cot).

II. Calculating Trigonometric Functions of Special Angles

• Special Angles and their Values:
  • 0°, 30° (π/6), 45° (π/4), 60° (π/3), 90° (π/2).
  • Example values: sin(π/4)=√2/2, cos(π/4)=√2/2.
• Use these values to calculate other functions like tan, csc.
• Important Note: Some functions are undefined at certain angles, e.g., tan(π/2).

III. Other Angles

• Techniques for finding functions for angles based on special angles, e.g., -π/3, 7π/4.
• Mirror image concept for positive/negative angles on unit circle.
• Examples provided for practice.

IV. Using the Calculator

• Calculators provide decimal approximations, not exact values.
• Ensure correct setting: degrees or radians.
• Practice exercises for calculator use.

V. Right-Triangle Applications

• Concept of Similar Triangles: Used to apply trigonometric functions to any right triangle.
• Key Formulas:
  • sin(θ) = opposite / hypotenuse
  • cos(θ) = adjacent / hypotenuse
  • tan(θ) = opposite / adjacent
• Examples include calculating structure heights and distances.

VI. Simple Trigonometric Equations

• Solving for angles given trigonometric values, e.g., sin(x)=1/2.
• Infinite solutions possible using angle periodicity (2πk rule).

VII. Trigonometric Identities

• Key Identities:
  • cos(-θ) = cos(θ), sin(-θ) = -sin(θ).
  • Pythagorean identity: sin²(θ) + cos²(θ) = 1.
  • Sum and difference formulas: sin(A+B), cos(A+B).
• Derivation of other formulas from basic identities.

VIII. Graphing Trigonometric Functions

• Graphing y = sin(x), y = cos(x), y = tan(x).
• Understanding periodicity and transformations (shifts and scalings).
• Examples of graph transformations provided.

Exercises and Solutions

• Exercises throughout for practice in calculations and applications.
• Solutions provided for self-assessment.

These notes cover the fundamental concepts of trigonometry, calculations of specific angles, right-triangle applications, solving equations, identities, and graphing functions, providing a comprehensive overview suitable for study and reference.