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.