Right Triangle Trigonometry

Introduction to SOH-CAH-TOA

• SOH-CAH-TOA is an acronym used to remember the definitions of the trigonometric functions sine, cosine, and tangent.
  • Sine (SOH): Opposite side/Hypotenuse
  • Cosine (CAH): Adjacent side/Hypotenuse
  • Tangent (TOA): Opposite side/Adjacent side
• Reciprocal functions:
  • Cosecant: 1/Sine = Hypotenuse/Opposite
  • Secant: 1/Cosine = Hypotenuse/Adjacent
  • Cotangent: 1/Tangent = Adjacent/Opposite

Using the Pythagorean Theorem

• The theorem: a^2 + b^2 = c^2, where c is the hypotenuse.
• Special right triangles:
  • 3-4-5 triangle
  • 5-12-13 triangle
  • 8-15-17 triangle
  • 7-24-25 triangle
  • Multiples of these triangles (e.g., 6-8-10, 9-12-15) also work.

Example Problems

Problem 1: Given a Right Triangle

• Sides: 3, 4, and hypotenuse (to find)
• Use the Pythagorean theorem to find the hypotenuse: 5
• Trigonometric functions:
  • Sin(θ) = 4/5
  • Cos(θ) = 3/5
  • Tan(θ) = 4/3
  • Cosecant = 5/4
  • Secant = 5/3
  • Cotangent = 3/4

Problem 2: Another Triangle

• Sides: 8, hypotenuse 17, missing side
• Use known 8-15-17 triangle or Pythagorean theorem to find the missing side is 15.
• Trigonometric functions:
  • Sin(θ) = 15/17
  • Cos(θ) = 8/17
  • Tan(θ) = 15/8
  • Cosecant = 17/15
  • Secant = 17/8
  • Cotangent = 8/15

Problem 3: Triangle with Given Hypotenuse

• Hypotenuse: 25, one side: 15
• Recognize as a 3-4-5 triangle scaled by 5; missing side is 20.
• Trigonometric functions:
  • Sin(θ) = 4/5
  • Cos(θ) = 3/5
  • Tan(θ) = 4/3
  • Cosecant = 5/4
  • Secant = 5/3
  • Cotangent = 3/4

Solving for Angle using Trigonometric Functions

• Use inverse functions to find angles.
• Example:
  • Given opposite = 5, adjacent = 4, find θ using tan⁻¹(5/4).
  • Result: θ = 51.34°

Additional Examples

• Cosine and Sine used to find angles and missing sides.
• Calculators used for computations.

Trigonometry Course Overview

• Topics covered:
  • Angles, Radians, Degree Conversions
  • Unit Circle and Trig Functions
  • Special Right Triangles
  • Graphing Trig Functions
  • Applications, Trig Identities, and Formulas
• Available on Udemy with further sections in progress.