Basics of Trigonometry

Jul 18, 2024

Basics of Trigonometry

Introduction

  • Trigonometry: Study of the ratios of sides of triangles.
    • Trig: Triangle
    • Metry: Measure

Right Triangle

  • A triangle with one angle equal to 90 degrees.
  • Hypotenuse: Longest side, opposite the right angle.
  • Example:
    • Side lengths: 3, 4, 5
    • Pythagorean theorem: 3^2 + 4^2 = 5^2

Trigonometric Functions

  • Sine (sin): Opposite / Hypotenuse
  • Cosine (cos): Adjacent / Hypotenuse
  • Tangent (tan): Opposite / Adjacent

Mnemonic: SOH CAH TOA

  • SOH: Sin = Opposite / Hypotenuse
  • CAH: Cos = Adjacent / Hypotenuse
  • TOA: Tan = Opposite / Adjacent

Example Calculations

  • Right triangle with angle θ:
    • sin(θ) = Opposite / Hypotenuse
      • Opposite = 3
      • Hypotenuse = 5
      • sin(θ) = 3/5
    • cos(θ) = Adjacent / Hypotenuse
      • Adjacent = 4
      • Hypotenuse = 5
      • cos(θ) = 4/5
    • tan(θ) = Opposite / Adjacent
      • Opposite = 3
      • Adjacent = 4
      • tan(θ) = 3/4

Alternate Angle Example

  • Right triangle with angle x (instead of θ):
    • sin(x) = Opposite / Hypotenuse
      • Opposite = 4
      • Hypotenuse = 5
      • sin(x) = 4/5
    • cos(x) = Adjacent / Hypotenuse
      • Adjacent = 3
      • Hypotenuse = 5
      • cos(x) = 3/5
    • tan(x) = Opposite / Adjacent
      • Opposite = 4
      • Adjacent = 3
      • tan(x) = 4/3

Further Considerations

  • Future videos will cover more examples.
  • Examining behavior of angles approaching 90 degrees and beyond.
  • Potential introduction of new definitions for sine, cosine, and tangent for any angle.