Lecture on Right Triangle Trigonometry and SOHCAHTOA

Introduction

• Focus on right triangle trigonometry.
• Explanation of the expression "SOA", which refers to the mnemonic SOHCAHTOA for trigonometric functions.

Triangle Anatomy

• Angle Theta: Reference angle in a right triangle.
• Three Sides:
  • Opposite side
  • Adjacent side
  • Hypotenuse (longest side)
• Pythagorean Theorem: (a^2 + b^2 = c^2) for right triangles.

Trigonometric Functions

• SOHCAHTOA: Mnemonic for the primary trigonometric ratios:
  • Sine (Sin): (\frac{\text{Opposite}}{\text{Hypotenuse}})
  • Cosine (Cos): (\frac{\text{Adjacent}}{\text{Hypotenuse}})
  • Tangent (Tan): (\frac{\text{Opposite}}{\text{Adjacent}})
• Reciprocal Functions:
  • Cosecant (Csc): (\frac{1}{\text{Sin}} = \frac{\text{Hypotenuse}}{\text{Opposite}})
  • Secant (Sec): (\frac{1}{\text{Cos}} = \frac{\text{Hypotenuse}}{\text{Adjacent}})
  • Cotangent (Cot): (\frac{1}{\text{Tan}} = \frac{\text{Adjacent}}{\text{Opposite}})

Examples and Problem Solving

Example 1: Solving Right Triangle

• Given: Right triangle with sides 3, 4, and unknown hypotenuse.
• Solution:
  • Use Pythagorean theorem to find hypotenuse: 5.
  • Calculate trigonometric functions:
    • (\sin \theta = \frac{4}{5})
    • (\cos \theta = \frac{3}{5})
    • (\tan \theta = \frac{4}{3})
    • (\csc \theta = \frac{5}{4})
    • (\sec \theta = \frac{5}{3})
    • (\cot \theta = \frac{3}{4})

Example 2: Special Right Triangles

• Special Right Triangles:
  • 3-4-5
  • 5-12-13
  • 8-15-17
  • 7-24-25
• Other Special Triangles:
  • 9-40-41
  • 11-60-61

Example 3: Solving Using Trig Functions

• Given angle and one side, find missing side and trig functions.
• Use given data and known trigonometric relationships to solve.

Calculating Unknown Angles

• Using Inverse Trigonometric Functions:
  • If (\tan \theta = \frac{5}{4}), then (\theta = \tan^{-1}(\frac{5}{4})).
  • Use calculator in degree mode for calculations.

Course Information

• Overview of a trigonometry course available on Udemy.
• Covers topics like unit circle, right triangle trigonometry, graphing trig functions, identities, and more.

Course Curriculum Highlights

• Angles and Radians
• Unit Circle and Trig Functions
• Triangle Trigonometry: Includes SOHCAHTOA, special triangles.
• Applications and Graphing
• Advanced Topics: Verifying identities, sum and difference formulas, etc.

Conclusion

• Availability of further sections to be added to the course.
• Encouragement to ask questions if needed.