Trigonometric Identities Part 2

Duration: 11 min

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Inside: a video lesson and guided study material.

Module outline

  1. General Intelligence & Reasoning: Series (Number and Letter Series) (Numerical Relations and Reasoning), Coding Decoding, Non Verbal Reasoning (Spatial Aptitude) (Spatial Reasoning) (Visual Reasoning), Analogy, Analytical Reasoning (Counting Figures Reasoning), Deductive and Inductive Reasoning (Logical Deduction and Induction) (Prepositional Reasoning), Syllogisms, Puzzle Solving (Puzzles), Venn Diagram, Clerical Aptitude, Verbal Classification
  2. General Awareness: Polity, Physics, Biology, Chemistry, History, Geography, Indian Economy, Environment, GK- Current Affairs, Static General Knowledge
  3. Quantitative Aptitude: Ratio and Proportion (Ratios), Data Interpretation, Mixture and Alligation, Number System, Speed Time and Distance, Percentage, Average, Arithmetic, Simple Interest and Compound Interest, Profit and Loss, Partnership, Time and Work, Algebra, Mensuration and Geometry, Trigonometry, Linear Algebra
  4. English Comprehension: Vocabulary, Noun, Subject Verb Agreement (Verb Noun Agreement), Adjectives, Comprehension / Reading Comprehension / Unseen Passages (Critical Reasoning) (Paragraph Questions), Pronoun, Preposition, Articles, Adverb, Interjection, Conjunction, Modals, Determiner, Active Passive Voice (Voice), Direct and Indirect Speech, Verbal Analogy (Word Based Analogy), Cloze Test (Cloze Passage), Sentence Re-arrangements (Para Jumbles) (Narrative Sequencing), Sentence Correction (Error Correction), Sentence Completion (Fill in the blanks)
  5. MOCK 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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