Introduction to Trigonometry and SOHCAHTOA

Overview

• Target Audience: Students about to take trigonometry in high school or college.
• Focus: Understanding the SOHCAHTOA mnemonic for remembering trig ratios.

SOHCAHTOA Mnemonic

• Soh: Sine ( θ) = Opposite / Hypotenuse
• Cah: Cosine ( θ) = Adjacent / Hypotenuse
• Toa: Tangent ( θ) = Opposite / Adjacent

Example 1: Right Triangle 3-4-5

• Triangle Properties: Right triangle with sides 3, 4, and 5.
• Angle: θ identified within the triangle.
• Sides:
  • Opposite to θ: 4
  • Adjacent to θ: 3
  • Hypotenuse: 5
• Calculations:
  • Sine ( θ) = Opposite / Hypotenuse = 4/5
  • Cosine ( θ) = Adjacent / Hypotenuse = 3/5
  • Tangent ( θ) = Opposite / Adjacent = 4/3

Example 2: Right Triangle 5-12-13

• Triangle Properties: Right triangle with sides 5, 12, and unknown hypotenuse.
• Using Pythagorean Theorem: a² + b² = c²
  • 5² + 12² = c² => 25 + 144 = 169 => c = 13
• Sides:
  • Opposite to θ: 5
  • Adjacent to θ: 12
  • Hypotenuse: 13
• Calculations:
  • Sine ( θ) = Opposite / Hypotenuse = 5/13
  • Cosine ( θ) = Adjacent / Hypotenuse = 12/13
  • Tangent ( θ) = Opposite / Adjacent = 5/12

Special Right Triangles to Memorize

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

Example 3: Right Triangle 14-48-50

• Identifying Special Right Triangle: 7-24-25 times 2.
• Calculations:
  • Sine ( θ) = 14/50 = 7/25
  • Cosine ( θ) = 48/50 = 24/25
  • Tangent ( θ) = 14/48 = 7/24

Reciprocal Trig Ratios

• Cosecant (csc θ) = 1 / Sine ( θ)
  • Cosecant example: csc θ = 25/7
• Secant (sec θ) = 1 / Cosine ( θ)
  • Secant example: sec θ = 25/24
• Cotangent (cot θ) = 1 / Tangent ( θ)
  • Cotangent example: cot θ = 24/7

Example: Sine (

θ) = 8/17

• Objective: Find Cosine ( θ) and Tangent ( θ).
• Right Triangle Properties: Right triangle with θ in the 1st quadrant.
• Sides:
  • Opposite: 8
  • Hypotenuse: 17
  • Missing side: 15 (8-15-17 triangle)
• Calculations:
  • Cosine ( θ) = 15/17
  • Tangent ( θ) = 8/15

Quadrants and Trigonometric Values

• Quadrants Explanation: Understanding angles and trig values in different quadrants.
• Example: Sine ( θ) = 2/5, θ between π/2 and π (2nd quadrant).
  • Use Pythagorean theorem to find missing side.
  • Results:
    • Cosine ( θ) = -√21/5
    • Tangent ( θ) = 2/-√21

Special and Reference Angles

• Convert radians to degrees: Example – Cos(π/4).
• Triangles to Memorize: 30-60-90 and 45-45-90 triangles.
  • Example: Cos(30°) = √3/2

Coterminal Angles

• Example: Evaluating Cosecant (-13π/6).
  • Convert angle, use reference angles, and triangle properties.
  • Result: Cosecant = -2

ASTC Mnemonic

• All Students Take Calculus (ASTC)
  • Quadrant I: All positive
  • Quadrant II: Sine positive
  • Quadrant III: Tangent positive
  • Quadrant IV: Cosine positive

Example Problems

• Tangent (-120°)
  • Find reference angle, special triangle, evaluate tangent.
  • Result: Tan(-120°) = √3
• Secant (225°)
  • Convert, find reference angle, evaluate secant.
  • Result: Sec(225°) = -√2

Conclusion

• Tools: Utilize the concepts in the video for future study.
• Additional Resources: Check out the provided trigonometry playlist for detailed examples and additional practice problems.
• Call to Action: Subscribe to the channel for updates and new content.