Properties of Triangle - Half Angle Formula

Duration: 18 min

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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 video is a mathematics lecture that systematically presents the half-angle formulas for a triangle, derived from the semi-perimeter 's'. The instructor first writes out the general formulas for sin(A/2), cos(A/2), and tan(A/2) in terms of the sides a, b, c and the semi-perimeter s. The formulas are then extended to cover all three angles of the triangle. The lecture progresses to a worked example where the sides of a triangle are given as a=8, b=10, c=12, and the student is guided through the calculation of sin(A/2) and cos(A/2) using the derived formulas. The video concludes with two more problems: the first asks for the value of cot(B/2) * cot(C/2) given the condition 3a = b + c, and the second asks for the relationship between the sides a, b, and c given the values of tan(A/2) and tan(C/2). The entire lesson is delivered on a digital blackboard with the instructor visible in a small window, and the content is presented in a clear, step-by-step manner.

Chapters

  1. 0:00 – 2:00 00:00-02:00

    The video begins by introducing the 'Half Angle formula' for a triangle. The instructor writes the formulas for sin(A/2), cos(A/2), and tan(A/2) on a digital blackboard. The formulas are presented as: sin(A/2) = √[(s-b)(s-c)/bc], cos(A/2) = √[s(s-a)/bc], and tan(A/2) = √[(s-b)(s-c)/s(s-a)]. The instructor explains that 's' is the semi-perimeter of the triangle, defined as s = (a+b+c)/2.

  2. 2:00 – 5:00 02:00-05:00

    The instructor extends the half-angle formulas to cover all three angles of the triangle. The formulas for sin(B/2), sin(C/2), cos(B/2), cos(C/2), tan(B/2), and tan(C/2) are written out, showing the pattern of substituting the appropriate sides. For example, sin(B/2) = √[(s-a)(s-c)/ac] and tan(C/2) = √[(s-a)(s-b)/s(s-c)]. The instructor emphasizes that these formulas are derived from the standard half-angle identities and the Law of Cosines.

  3. 5:00 – 10:00 05:00-10:00

    The video transitions to a worked example. The problem states: 'In ΔABC, the sides are a=8, b=10, c=12. Find the values of sin(A/2) and cos(A/2)'. The instructor first calculates the semi-perimeter s = (8+10+12)/2 = 15. Then, using the formulas, sin(A/2) = √[(15-10)(15-12)/(10*12)] = √[5*3/120] = √[15/120] = √[1/8] = 1/√8. Similarly, cos(A/2) = √[15(15-8)/(10*12)] = √[15*7/120] = √[105/120] = √[7/8].

  4. 10:00 – 15:00 10:00-15:00

    The instructor presents the first problem: 'In ΔABC, if 3a = b + c, then the value of cot(B/2) * cot(C/2) is:'. The options are a) 1, b) 2, c) √3, d) √2. The instructor begins by writing the formulas for cot(B/2) and cot(C/2) as the reciprocals of tan(B/2) and tan(C/2). The product is then expressed as: cot(B/2) * cot(C/2) = √[(s-b)(s-c)/s(s-b)] * √[(s-a)(s-b)/s(s-c)] = √[(s-a)/s]. Using the given condition 3a = b + c, the instructor derives that 2s = a + b + c = a + 3a = 4a, so s = 2a. Substituting this into the expression gives √[(2a - a)/(2a)] = √[a/(2a)] = √[1/2] = 1/√2. The instructor notes that this is not among the options, indicating a possible error in the problem or options.

  5. 15:00 – 17:48 15:00-17:48

    The second problem is presented: 'If tan(A/2) = 5/6 and tan(C/2) = 2/5, then find the relation between a, b, and c'. The instructor writes the formulas: tan(A/2) = √[(s-b)(s-c)/s(s-a)] = 5/6 and tan(C/2) = √[(s-a)(s-b)/s(s-c)] = 2/5. By squaring both equations and multiplying them, the instructor derives: [(s-b)(s-c)/s(s-a)] * [(s-a)(s-b)/s(s-c)] = (5/6) * (2/5) = 1/3. This simplifies to (s-b)^2 / s^2 = 1/3, so (s-b)/s = 1/√3. This leads to s - b = s/√3, and after rearranging, the relation a + c = 2b is found. The instructor confirms this by substituting s = (a+b+c)/2 into the equation.

The video provides a comprehensive lesson on the half-angle formulas for a triangle, starting with the derivation of the general formulas for sin(A/2), cos(A/2), and tan(A/2) in terms of the sides and semi-perimeter. The instructor then demonstrates the application of these formulas through a detailed worked example, calculating the values of sin(A/2) and cos(A/2) for a triangle with given side lengths. The lesson concludes with two more complex problems that require the student to apply the formulas in a more abstract way, involving algebraic manipulation and the use of given conditions to find relationships between the sides of the triangle. The progression from basic formulas to complex problem-solving effectively illustrates the power and utility of these trigonometric identities.

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