Trigonometry

Trigonometry is the branch of mathematics that studies the relationships between the angles and side lengths of triangles. Central to this field are the trigonometric functions—sine, cosine, and tangent—which establish precise ratios that allow for the calculation of unknown angles or sides when others are known, particularly within right-angled triangles. Its principles extend beyond simple geometric measurement, providing the foundational tools for describing and analyzing periodic waves and oscillations, making it indispensable in fields such as physics, engineering, and astronomy.

1. Foundations of Trigonometry
   1. Angles in the Plane
      1. Definition of an Angle
         1. Geometric Definition
         2. Algebraic Definition
         3. Ray Terminology
      2. Initial Side and Terminal Side
         1. Initial Side Properties
         2. Terminal Side Properties
         3. Angle Formation Process
      3. Standard Position
         1. Vertex at Origin
         2. Initial Side on Positive x-axis
         3. Counterclockwise Rotation Convention
      4. Positive and Negative Angles
         1. Counterclockwise as Positive
         2. Clockwise as Negative
         3. Angle Direction Conventions
      5. Coterminal Angles
         1. Definition of Coterminal Angles
         2. Finding Coterminal Angles
            1. Adding 360° (or 2π radians)
            2. Subtracting 360° (or 2π radians)
            3. General Formula for Coterminal Angles
         3. Properties of Coterminal Angles
            1. Equal Trigonometric Function Values
            2. Infinite Number of Coterminal Angles
      6. Quadrantal Angles
         1. Definition of Quadrantal Angles
         2. Identifying Quadrantal Angles
            1. 0°, 90°, 180°, 270°, 360°
            2. Radian Equivalents
         3. Properties of Quadrantal Angles
            1. Terminal Side on Coordinate Axes
            2. Special Trigonometric Values
   2. Measuring Angles
      1. Units of Angle Measurement
         1. Degrees
            1. Definition of Degree
            2. Circle Division into 360 Parts
            3. Historical Context
         2. Minutes and Seconds
            1. Definition of Minutes
            2. Definition of Seconds
            3. DMS Notation
            4. Converting Between Decimal Degrees and DMS
         3. Radians
            1. Definition of Radian
            2. Arc Length Equal to Radius
            3. Natural Unit for Calculus
      2. Conversion Between Degrees and Radians
         1. Fundamental Relationship
            1. 180° = π radians
            2. Conversion Factor Derivation
         2. Converting Degrees to Radians
            1. Multiplication by π/180
            2. Common Angle Conversions
         3. Converting Radians to Degrees
            1. Multiplication by 180/π
            2. Common Radian Conversions
      3. Comparing Degree and Radian Measures
         1. Advantages of Each System
         2. When to Use Each System
         3. Common Equivalent Angles
   3. Arcs and Sectors
      1. Definition of an Arc
         1. Portion of Circle Circumference
         2. Arc Endpoints
         3. Minor and Major Arcs
      2. Central Angle and Arc Relationship
         1. Central Angle Definition
         2. Proportional Relationship
         3. Arc Measure vs Arc Length
      3. Arc Length
         1. Concept of Arc Length
         2. Formula for Arc Length in Degrees
            1. s = (θ/360°) × 2πr
         3. Formula for Arc Length in Radians
            1. s = rθ
         4. Applications of Arc Length
            1. Real-world Distance Problems
            2. Circular Motion Applications
      4. Area of a Sector
         1. Definition of Sector
         2. Formula for Area of Sector in Degrees
            1. A = (θ/360°) × πr²
         3. Formula for Area of Sector in Radians
            1. A = (1/2)r²θ
         4. Applications of Sector Area
            1. Pizza Slice Problems
            2. Land Area Calculations
   4. Angular and Linear Speed
      1. Definition of Angular Speed
         1. Rate of Change of Angle
         2. Units of Angular Speed
         3. Constant vs Variable Angular Speed
      2. Definition of Linear Speed
         1. Rate of Change of Position
         2. Tangential Speed
         3. Units of Linear Speed
      3. Relationship Between Angular and Linear Speed
         1. Formula v = rω
         2. Derivation of Relationship
         3. Proportionality Concepts
      4. Applications of Angular and Linear Speed
         1. Wheel and Gear Problems
         2. Planetary Motion
         3. Mechanical Systems