Lecture Notes: Trigonometry in 20 Minutes

Introduction

• Aim: Explain key concepts of the chapter within 20 minutes
• Initial Checks: Confirming audio and video clarity
• Request: Like, share, and subscribe to the channel
• Focus: Trigonometry concepts overview

Key Concepts of Trigonometry

Basics of Trigonometry

• Usage: Applied in triangles, specifically right-angled triangles
• Right-Angled Triangle Components:
  • Hypotenuse
  • Opposite side (Perpendicular)
  • Adjacent side (Base)
• Identification: Based on the reference angle in the triangle

Trigonometric Ratios

• Primary Ratios:
  • sin(θ) = Opposite / Hypotenuse
  • cos(θ) = Adjacent / Hypotenuse
  • tan(θ) = Opposite / Adjacent
• Reciprocal Ratios:
  • cosec(θ) = 1/sin(θ)
  • sec(θ) = 1/cos(θ)
  • cot(θ) = 1/tan(θ)
• Mnemonics for Ratios:
  • Pandit Badri Prasad Har Har Bhole (Sine, Cosine, Tangent)
  • Alternates: Pakistan Bhooka Pyasa, etc.

Simplifying Trigonometric Expressions

• Example Problem: Given cosec(θ) = 13/12, find all trigonometric ratios
  • Build right triangle assuming Hypotenuse = 13k and Base = 12k
  • Calculate using Pythagoras theorem for the perpendicular (5k)
  • Find ratios:
    • sin(θ) = 5/13
    • cos(θ) = 12/13
    • tan(θ) = 5/12
    • Reciprocals follow

Values of Trigonometric Ratios at Specific Angles

• Angles: 0°, 30°, 45°, 60°, 90°
• Values:
  • Memorization trick: Write and divide 0, 1, 2, 3, 4 by 4, take the square root
  • sin and cos values are complementary
  • Tangent values: ratio of sine and cosine
• Example Calculation: Substituting values in identities

Trigonometric Identities

• Key Identity: sin²(θ) + cos²(θ) = 1
• Derivations:
  • sin²(θ) = 1 - cos²(θ)
  • cos²(θ) = 1 - sin²(θ)
• Ratios of others derived similarly

Proof and Application

• Example Proof:
  • Simplifying a given trigonometric equation to show equivalence
  • Transforming and substituting values to prove identities
• Practical Problem Solving:
  • Use given ratios to calculate values and apply identities

Conclusion

• Summary: Quick overview and solving example problems
• Interaction: Engagement through solving and commenting
• Reminder: Join Vedantu group for further queries
• Engagement: Like, share, and subscribe

Acknowledgements

• Thanks to viewers and students for interaction and support