Functions 11 - Chapter 5: Trigonometric Ratios

Introduction to Trigonometry Beyond Triangles

• Understanding reciprocal trigonometric ratios
• Relating six trigonometric ratios to the unit circle
• Solving problems using trig ratios
• Proving trigonometric identities

Chapter Content Overview

• Practice problems from the Nelson Textbook
• Sections to focus on: 5.1, 5.6, 5.7, 5.8, 5.2, 5.3/5.4, 5.5
• Emphasis on asking questions and seeking help

Section 5.1: Trigonometric Ratios of Acute Angles

• Learning Goal: Evaluate reciprocal trigonometric ratios
• Mnemonic: SOH CAH TOA for right angle triangles
• Primary and Reciprocal Trig Ratios
• Examples include determining csc, sec, cot, angles, and using reciprocal ratios
• Success Criteria:
  • Use SOH CAH TOA for primary and reciprocal ratios
  • Solve reciprocal trigonometric ratios without a calculator

Section 5.6: The Sine Law

• Learning Goal: Use the Sine law for non-right angle triangles
• Requirements: Must have a corresponding angle and side
• Forms of the Sine Law:
  • Angle form and side form
• Ambiguous Case: Occurs when the given angle is acute
• Success Criteria:
  • Recognize and apply the Sine law
  • Identify ambiguous cases

Section 5.7: The Cosine Law

• Learning Goal: Use the Cosine law for non-right angle triangles
• When to use:
  • SOH CAH TOA for right angles
  • Sine Law when a corresponding pair is present
  • Cosine Law otherwise
• Forms of Cosine Law:
  • Side and angle forms
• Success Criteria:
  • Apply cosine law for S-A-S or S-S-S triangles
  • Rearrange the cosine law for unknown angles

Section 5.8: 3D Problems

• Learning Goal: Solve 3D problems using SOH CAH TOA, Sine, and Cosine Laws
• Key: Visualizing 3D problems can be challenging
• Examples: Problems from the textbook
• Success Criteria:
  • Sketch and solve 3D problems accurately
  • Choose correct methods for solving unknowns

Section 5.2: Trigonometric Ratios and Special Triangles

• Learning Goal: Find exact values for sin, cos, tan of specific angles
• Special Triangles: Memorize for 30°, 45°, 60°
• Success Criteria:
  • Draw special triangles
  • Identify exact values for angles
  • Evaluate without calculators

Sections 5.3 & 5.4: Trigonometric Ratios for Angles Larger than 90°

• Learning Goal: Use Cartesian plane for angles between 0° and 360°
• Concepts:
  • Principal angle and related acute angle
  • CAST Rule for determining signs
• Process: Draw standard position, determine related angle, use CAST and SOH CAH TOA
• Success Criteria:
  • Identify angle positions and signs using CAST
  • Find multiple principal angles

Section 5.5: Trigonometric Identities

• Learning Goal: Prove trigonometric identities
• Strategies:
  • Use reciprocal, quotient, and Pythagorean identities
  • Convert to sin and cos
  • Start from side with more information
• Success Criteria:
  • Prove identities using various strategies
  • Recognize forms for proving identities

This is a comprehensive overview of Chapter 5, which focuses on trigonometric concepts, laws, and applications beyond basic triangles, extending into non-right angles and 3D problems, as well as proving identities.