Trigonometric Identities Part 2

Duration: 11 min

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Module outline

  1. Quantitative Aptitude: Number System, Divisibility Rules, HCF LCM, Percentage, Ratio and Proportion (Ratios), Average, Profit and Loss, Simple Interest and Compound Interest, Mixture and Alligation, Partnership, Age Problems, Time and Work, Pipe and Cistern, Speed Time and Distance, Algebra, Arithmetic, Data Interpretation, Sequence and Series (Series), Logarithm, Powers and Exponents (Surds and Indices), Permutation and Combination, Probability, Speed Mathematics and Basic Mathematics, Trigonometry
  2. Logical Reasoning: Coding Decoding, Blood Relations, Directions (Direction Test), Ranking, Order and Sequence, Seating Arrangements, Puzzle Solving (Puzzles), Data Arrangement, Syllogisms, Inequality, Data Sufficiency, Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Series (Number and Letter Series) (Numerical Relations and Reasoning), Venn Diagram, Logical Sequence Of Words, Calendar, Clock, Cubes & Dices, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning)
  3. Verbal Ability: Noun, Pronoun, Adjectives, Verb, Adverb, Tenses, Subject Verb Agreement (Verb Noun Agreement), Articles, Preposition, Conjunction, Modals, Determiner, Vocabulary, Comprehension / Reading Comprehension / Unseen Passages (Critical Reasoning) (Paragraph Questions), Cloze Test (Cloze Passage), Sentence Correction (Error Correction), Sentence Completion (Fill in the blanks), Sentence Re-arrangements (Para Jumbles) (Narrative Sequencing), Active Passive Voice (Voice), Direct and Indirect Speech, Sentence Construction
  4. Data Structures (Domain Test): Introduction to DS, Array, Stack, Queue, Linked List, Tree, Graphs, Hashing
  5. Algorithms (Domain Test): Algorithm Analysis, Sorting Algorithms, Time Complexity Analysis, Greedy Algorithms
  6. OOPs (Domain Test): Object Oriented Programming (OOPS)
  7. DBMS (Domain Test): Basics of DBMS, ER Diagram, Relational Model & Functional Dependencies, Keys & Integrity Constraints, Normalization (1NF - BCNF), Decomposition Properties & 4NF, File Organization & Indexing, Relational Algebra, Relational Calculus, Transaction Management, Concurrency Control, Database Recovery, ORDBMS, Database Security & Authorization, Query Processing & Optimization, Enhanced Data Models, Data Warehousing & Mining, Big Data Systems, NoSQL
  8. SQL (Domain Test): SQL
  9. Operating Systems (Domain Test): Introduction to OS, Process Management, CPU Scheduling, Process Synchronization, Deadlock, Threads & Process Creation, Memory Management, Virtual Memory, Disc Scheduling, File Management, Windows OS, Linux OS, Security, Distributed Systems, Virtual Machines
  10. Computer Networks (Domain Test): Introduction to CN, Data Communication, DLL: Access Control, DLL: Flow Control, DLL: Error Control, DLL: Framing, Data Link Layer - Ethernet, Net Layer: IPv4 & Proto, Net Layer: IP Addressing, Net Layer:Routing Protocol, Transport Layer Services, TL: Congestion & UDP, Application Layer, Hardware basics, Network Security, Mobile Technology, Cloud Computing and IoT, Cyber and Info Security
  11. Coding: Coding Questions by Yash Sir, C Programming by Yash Sir, Programming in C++, Java, Python for Placements by Yash Sir
  12. Written English Test: English Writing Ability
  13. Psychometric Test: Behavioral Assessment (Behavioral Competency Profiling) (Psychometric Test)
AI summary & chapters

AI Summary

An AI-generated summary of this video lecture.

This educational video is a comprehensive lecture on trigonometric functions, focusing on the sum and difference of two angles. The instructor systematically presents a series of fundamental identities, starting with the core formulas for sin(A+B), sin(A-B), cos(A+B), and cos(A-B), and provides a mnemonic trick to remember the sign patterns. The lesson progresses to double-angle formulas for sine, cosine, and tangent, and then to product-to-sum identities, which are derived by manipulating the sum and difference formulas. The video concludes with a worked example of a trigonometric proof, demonstrating the application of these identities to simplify an expression involving cotangents. The content is delivered on a digital blackboard with handwritten-style text, and the instructor is visible in a small window, guiding the viewer through the concepts.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video opens with a title slide, "Trigonometric Functions of Sum and Difference of Two Angles :". The instructor presents the fundamental identities for the sum and difference of angles. The first four formulas are listed: 1) sin(A+B) = sinAcosB + cosAsinB, 2) sin(A-B) = sinAcosB - cosAsinB, 3) cos(A+B) = cosAcosB - sinAsinB, and 4) cos(A-B) = cosAcosB + sinAsinB. A mnemonic trick is provided in a diagram: for sine, the signs are opposite (opp pair) and the sign of the result is the same as the function (same sign); for cosine, the signs are the same (same pair) and the sign of the result is opposite (opp sign). The instructor then introduces the double-angle formulas: 5) sin2x = 2sinxcosx and 6) cos2x = cos²x - sin²x.

  2. 2:00 – 5:00 02:00-05:00

    The instructor continues to build on the double-angle formulas. The identity for cos2x is shown to have three equivalent forms: cos2x = cos²x - sin²x, cos2x = 2cos²x - 1, and cos2x = 1 - 2sin²x. The corresponding tangent double-angle formula, tan2x = 2tanx / (1 - tan²x), is also presented. The video then transitions to the product-to-sum identities. The instructor begins to derive the first one, 2sinAcosB = sin(A+B) + sin(A-B), by writing the sum and difference formulas for sine and adding them together. The derivation is shown step-by-step on the board.

  3. 5:00 – 10:00 05:00-10:00

    The derivation of the product-to-sum identities is completed. The instructor shows that 2sinAcosB = sin(A+B) + sin(A-B). The other three identities are then presented: 2cosAsinB = sin(A+B) - sin(A-B), 2cosAcosB = cos(A+B) + cos(A-B), and 2sinAsinB = cos(A-B) - cos(A+B). The video then moves to a proof problem. The task is to prove that cotx cot2x - cot2x cot3x - cot3x cotx = 1. The instructor begins by using the identity cot(A+B) = (cotA cotB - 1) / (cotA + cotB) to express cot3x as cot(x+2x), which leads to the equation cot3x = (cotx cot2x - 1) / (cotx + cot2x). This is rearranged to cotx cot2x - 1 = cot3x (cotx + cot2x).

  4. 10:00 – 10:37 10:00-10:37

    The proof is completed. The instructor substitutes the expression for cotx cot2x - 1 into the original equation. The left-hand side (LHS) becomes (cot3x (cotx + cot2x)) - cot2x cot3x - cot3x cotx. This simplifies to cot3x cotx + cot3x cot2x - cot2x cot3x - cot3x cotx, which further simplifies to 0 + 0 = 0. The instructor then states that the right-hand side (RHS) is 1, and the proof is complete. The video also briefly shows the triple-angle formulas: sin3x = 3sinx - 4sin³x, cos3x = 4cos³x - 3cosx, and tan3x = (3tanx - tan³x) / (1 - 3tan²x).

The video provides a structured and logical progression of trigonometric identities. It begins with the foundational sum and difference formulas, which are essential for understanding more complex relationships. The instructor uses a mnemonic to aid memorization and then derives the double-angle formulas from the core identities. The lesson then expands to the product-to-sum identities, which are powerful tools for simplifying expressions. The video culminates in a practical application, demonstrating how to use these identities to prove a trigonometric equation, thereby reinforcing the concepts through problem-solving. The overall flow is from basic definitions to advanced applications, making it a comprehensive tutorial.

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