Trigonometry - In-depth Notes on Trigonometric Functions and Applications

Unit 4: Trigonometric Functions

Introduction
  • Trigonometry: Derived from Greek word ‘trigon’ (triangle) and ‘metron’ (measuring), it involves measuring the sides of triangles.
  • Applications: Initially used by sea captains, surveyors, and engineers. Now applied in areas like seismology, circuit design, physics, and music.
  • Focus on trigonometric identities and their applications, great for mechanics and understanding waves.
4.1 Radian Measure of Angle
  • Angle Definition: A measure of rotation of a ray about its initial point;
    • Initial Side: Original position of the ray
    • Terminal Side: Position after rotation
    • Vertex: Point of rotation
  • Positive vs Negative Angles:
    • Positive: Anticlockwise rotation
    • Negative: Clockwise rotation
Angles in Standard Position
  1. Vertex at the origin.
  2. Initial side on the positive x-axis.
  3. Examples:
    • 200°, 1125°, -900° conversions.
Measurements of Angles

  1. Degree Measure: 1 degree (°) is 1/360 of a complete revolution.

    • 1° = 60', 1' = 60" (minutes and seconds).
    • Example conversions: 360°, 180°, -30°, etc.

  2. Radian Measure:

    • An angle subtended at the center of a circle by an arc of equal length to the radius.
    • $2 heta = 2 rac{ ext{circumference}}{r} = rac{l}{r}$; $l = r heta$ where $l$ is arc length.
    • $360° = 2 heta
      ightarrow heta = rac{ ext{radian}}{ ext{angle}}$ conversions: Table of common angles provided.
4.2 Basic Trigonometric Functions
  • Definitions of Trigonometric Functions for an angle in a right-angled triangle:
    • Sine (sin): Ratio of opposite side to hypotenuse.
    • Cosine (cos): Ratio of adjacent side to hypotenuse.
    • Tangent (tan): Ratio of opposite side to adjacent side.
Pythagorean Identities
  • Basic Identities:
    1. $sin^2( heta) + cos^2( heta) = 1$
    2. $1 + tan^2( heta) = sec^2( heta)$
    3. $1 + cot^2( heta) = csc^2( heta)$
Angle Relationships
  • Complementary Angles:
    • Sum to 90°; example: 60° and 30° (complement of 30° is 60°).
  • Supplementary Angles:
    • Sum to 180°; example: 90° and 90° (supplement of 30° is 150°).
  • Co-terminal Angles:
    • Same terminal side; example: 30°, 390°, -330° (found by adding/subtracting 360°).
Graphs of Trigonometric Functions
  • Sine and Cosine:
    • Period of sine and cosine functions is $360°$ (or $2π$ radians).
  • Tangent Function:
    • Period of tangent function is $180°$ (or $π$ radians).
Applications of Trigonometry
  • Used in various real-world applications, e.g., finding heights, distances, angles of elevation, and depression.
  • Practical problems involving trigonometric ratios to calculate unknown sides or angles in triangles.
Summary of Important Concepts

Key Definitions:

  • Complementary angles: Add to 90°.
  • Supplementary angles: Add to 180°.
  • Co-terminal angles: Same terminal side differing by full rotations (360°).
  • Trigonometric identities: Fundamental relationships between sin, cos, and tan.

Conclusions:

  • Review the formulas, ensure understanding of angles in different quadrants, utilize reference angles effectively, and master the conversions between degrees and radians for exam preparation.