Lecture Notes: Right Triangle Trigonometry

Introduction

• Focus on right triangle trigonometry.
• Explanation of sides relative to angle θ:
  • Opposite: across from θ.
  • Adjacent: next to θ.
  • Hypotenuse: longest side, across from the right angle.

The Pythagorean Theorem

• Applies to right triangles: (a^2 + b^2 = c^2).
• Used to find missing sides.

Six Trigonometric Functions

Definition Using SOH-CAH-TOA

• 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)

Example Problems

Problem 1

• Triangle with sides: 3, 4, hypotenuse 5.
• Verify using Pythagorean theorem: 5 is the hypotenuse.
• Calculate trigonometric functions:
  • sin θ = 4/5
  • cos θ = 3/5
  • tan θ = 4/3
  • csc θ = 5/4
  • sec θ = 5/3
  • cot θ = 3/4

Special Right Triangles

• Examples: 3-4-5, 5-12-13, 8-15-17 triangles.
• Multiples of these (e.g., 6-8-10) are also valid.

Problem 2

• Triangle with sides: 8, hypotenuse 17.
• Missing side is 15 (8-15-17 triangle).
• 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

Problem 3

• Hypotenuse 25, one side 15 (similar to 3-4-5 triangle, scaled).
• Missing side is 20.
• Trigonometric functions calculated similarly.

Finding Angles Using Trigonometric Ratios

• Example with given angle and one side.
• Use appropriate function (e.g., tangent for opposite/adjacent) to solve for unknown.

Inverse Trigonometric Functions

• Finding angles:
  • Example: θ = inverse tangent (5/4).
  • Use calculator in degree mode.

Course Information

• Trigonometry course available on Udemy.
• Covers angles, unit circle, right triangle trigonometry, and more.
• Additional topics include graphing, applications, identities, and equations.

Additional Topics

• Sum and difference formulas, double angle, inverse functions.
• Graphing functions and identities.

Conclusion

• Key concepts include understanding and applying trigonometric ratios.
• Special attention should be given to verifying identities and solving equations.