ट्रिग्नोमेट्री इन डिटेल्स

Jul 12, 2024

ट्रिग्नोमेट्री इन डिटेल्स - Lecture Summary (By Master DJ Harsh Priyam)

Overview

  • Focus on analyzing previous years' questions to understand trends in trigonometry questions.
  • Emphasis on practice; theory session aims to cover everything needed theoretically.
  • Aim to provide enough knowledge so external references aren't needed—though practice is essential.
  • Topics include trigonometric formulas, their applications, and problem-solving for different exam levels.

Key Concepts and Formulas

Basics of Trigonometry Functions

  • Sine (sin): $ rac{P}{H}$
  • Cosine (cos): $ rac{B}{H}$
  • Tangent (tan): $ rac{P}{B}$

Trigonometric Identities

  1. $ ext{sin}^2 heta + ext{cos}^2 heta = 1$
  2. $1 + ext{tan}^2 heta = ext{sec}^2 heta$

Trigonometric Ratios and Quadrants

  • First: $0^ ext{°} ext{to } 90^ ext{°}$ - All functions are positive
  • Second: $90^ ext{°} ext{to } 180^ ext{°}$ - Sin and cosec are positive
  • Third: $180^ ext{°} ext{to } 270^ ext{°}$ - Tan and cot are positive
  • Fourth: $270^ ext{°} ext{to } 360^ ext{°}$ - Cos and sec are positive

Conversion Between Degrees and Radians

  • $180^ ext{°} = ext{π radians}$
  • To convert degrees to radians: multiply by $ rac{π}{180}$
  • To convert radians to degrees: multiply by $ rac{180}{π}$

Compound Angle Formulas

  • $ ext{sin}(A ext{±} B) = ext{sin}A ext{cos}B ext{±} ext{cos}A ext{sin}B$
  • $ ext{cos}(A ext{±} B) = ext{cos}A ext{cos}B ext{∓} ext{sin}A ext{sin}B$
  • $ ext{tan}(A ext{±} B) = rac{ ext{tan}A ext{±} ext{tan}B}{1 ext{∓} ext{tan}A ext{tan}B}$

Multiple Angle Formulas

  • $ ext{sin}(2A) = 2 ext{sin}A ext{cos}A$
  • $ ext{cos}(2A) = ext{cos}^2A - ext{sin}^2A = 2 ext{cos}^2A - 1 = 1 - 2 ext{sin}^2A$
  • $ ext{tan}(2A) = rac{2 ext{tan}A}{1 - ext{tan}^2A}$

Triple Angle Formulas

  • $ ext{sin}(3A) = 3 ext{sin}A - 4 ext{sin}^3A$
  • $ ext{cos}(3A) = 4 ext{cos}^3A - 3 ext{cos}A$
  • $ ext{tan}(3A) = rac{3 ext{tan}A - ext{tan}^3A}{1 - 3 ext{tan}^2A}$

Transformations

  • $ ext{sin}C + ext{sin}D = 2 ext{sin} ( rac{C+D}{2}) ext{cos} ( rac{C-D}{2})$
  • $ ext{sin}C - ext{sin}D = 2 ext{cos} ( rac{C+D}{2}) ext{sin} ( rac{C-D}{2})$
  • $ ext{cos}C + ext{cos}D = 2 ext{cos} ( rac{C+D}{2}) ext{cos} ( rac{C-D}{2})$
  • $ ext{cos}C - ext{cos}D = -2 ext{sin} ( rac{C+D}{2}) ext{sin} ( rac{C-D}{2})$

Sin and Cos Values at Specific Angles

  • Important values:
    • $ ext{sin} rac{π}{6} = rac{1}{2} $
    • $ ext{cos} rac{π}{3} = rac{1}{2} $
    • $ ext{tan} 45^ ext{°} = 1 $

Maximum and Minimum Values of Trigonometric Expressions

  • For expressions of the type $a ext{sin}θ + b ext{cos}θ$, maximum value is $ ext{√}(a^2 + b^2)$, minimum value is $- ext{√}(a^2 + b^2)$.
  • Similar for $a ext{cos}θ + b ext{sin}θ$.

Examples and Applications

Question 1: Convert 30° to radians.

  • $30° = 30 imes rac{π}{180} = rac{π}{6}$ radians.

Question 2: Find $ ext{sin}(RA)$ if $ ext{cos}(RA) = 0.5$ and angle lies in 2nd quadrant.

  • $ ext{sin}(RA) = ext{√}(1 - ext{cos}^2(RA)) = ext{√}(0.75) = rac{ ext{√}3}{2}$

Question 3: If $ ext{tanP} = 3$, find $ ext{sec}^2P$.

  • $1 + ext{tan}^2P= ext{sec}^2P = 1 + 9 = 10$

Trigonometric Equations and Quadratic Forms

  • Convert trigonometric equations into quadratic forms for easy solving.
  • Use identity transformations like ${ 1 - ext{cos}x}$ in solving equations.

Advanced Examples

  • Solving equations involving multiple angles (e.g., $2x, 3x$)
  • Trig values in different quadrants.

Graphs of Trigonometric Functions

  • Sinx Graph: Oscillates between -1 and 1.
  • Cosx Graph: Starts at 1, similar oscillation to sinx.
  • Tanx Graph: Vertical asymptotes at $ rac{π}{2}, - rac{π}{2}$.

True/False Statements - K-value questions

  • Max/Min values: $a ext{sin}θ ext{+/-} b ext{cos}θ$
    • $ ext{Max value} = ext{√}(a^2 + b^2)$
    • $ ext{Min value} = ext{-√}(a^2 + b^2)$

Tools And Recommended Practice

  • Regular use of values and identities in solving problems.
  • Use graphing tools for better visualization of trigonometric functions.
  • Practice with elements from texts like Vedantu's Tattva module.

Recommendation: Keep revising these fundamentals and approach problems with a fresh mindset. Focus on understanding the transformation of trigonometric expressions and solving them using standard identities.

Next Steps: Explore combinatorial usage of these formulas in calculus and algebraic identities.

Key Takeaway: Mastering formulas, understanding their application in different quadrants, and visualizing functions graphically are pivotal in excelling in trigonometry.

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