Understanding Right Triangle Trigonometry

Sep 11, 2024

Lecture on Right Triangle Trigonometry

Introduction

  • Expression "SOA": Refers to trigonometry functions in right triangles.
  • Focus: Right triangle trigonometry and trigonometric functions.

Triangle Basics

  • Angle Theta (θ):
    • Opposite Side: Opposite to angle θ.
    • Adjacent Side: Next to angle θ.
    • Hypotenuse: Across from the right angle.

Pythagorean Theorem

  • Equation: a² + b² = c²
  • Applies to Right Triangles: Relates the sides of the triangle.

Trigonometric Functions

  • Six Trigonometric Functions:
    • Sine (sin θ): Opposite / Hypotenuse
    • Cosine (cos θ): Adjacent / Hypotenuse
    • Tangent (tan θ): Opposite / Adjacent
    • Cosecant (csc θ): 1 / sin θ = Hypotenuse / Opposite
    • Secant (sec θ): 1 / cos θ = Hypotenuse / Adjacent
    • Cotangent (cot θ): 1 / tan θ = Adjacent / Opposite

Special Right Triangles

  • Common Ratios:
    • 3-4-5, 5-12-13, 8-15-17, 7-24-25
    • Multiples like 6-8-10, 9-12-15 etc.
  • Less Common Ratios:
    • 9-40-41, 11-60-61

Example Problems

Problem 1

  • Given Triangle: Sides 3 and 4
  • Find Missing Side Using Pythagorean Theorem:
    • Hypotenuse = 5
  • Trigonometric Values:
    • sin θ = 4/5
    • cos θ = 3/5
    • tan θ = 4/3
    • csc θ = 5/4
    • sec θ = 5/3
    • cot θ = 3/4

Problem 2

  • Given Triangle: Sides 8 and 17
  • Find Missing Side:
    • Opposite = 15 (Using 8-15-17 triangle)
  • Trigonometric Values:
    • sin θ = 15/17
    • cos θ = 8/17
    • tan θ = 15/8
    • csc θ = 17/15
    • sec θ = 17/8
    • cot θ = 8/15

Additional Problems

  • Using Trigonometric Ratios to Find:
    • Missing side lengths
    • Angle measures

Inverse Trigonometric Functions

  • Finding Angles:
    • Use inverse functions (e.g., arctan, arccos, arcsin)

Trigonometry Course Information

  • Course Available on Udemy:
    • Covers angles, unit circle, right triangles, graphing functions, inverse functions, applications, identities, and more.
  • Curriculum Includes:
    • Unit circle, graphing trig functions, verifying identities, solving equations.

Conclusion

  • Review Key Concepts: Practice finding side lengths and angles using trigonometric functions.
  • Further Learning: Consider exploring the course for more detailed explanations and exercises.