Chapter 8 Class 10 Introduction to Trignometry

Class 10
Important Questions for Exam - Class 10

### Transcript

Example 2 If ∠ B and ∠ Q are acute angles such that sin B = sin Q, then prove that ∠ B = ∠ Q. Given: sin B = sin Q To prove: ∠ B = ∠ Q Proof: Let’s take two right angle triangles ABC & PQR Since, sin B = sin Q sin B = (𝑠𝑖𝑑𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑡𝑜 𝐵)/𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 sin B = 𝑨𝑪/𝑨𝑩 sin Q = (𝑠𝑖𝑑𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑡𝑜 𝑄)/𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 sin Q = 𝑷𝑹/𝑷𝑸 𝐴𝐶/𝑃𝑅=𝐴𝐵/𝑃𝑄 Let 𝑨𝑪/𝑷𝑹=𝑨𝑩/𝑷𝑸= k So, AC = k PR & AB = k PQ Now, Using Pythagoras theorem (Hypotenuse)2 = (Height)2 + (Base)2 In right Δ ABC, (AB)2 = AC2 + BC2 BC2 = AB2 – AC2 BC = √(𝐴𝐵2 −𝐴𝐶2) In right Δ PQR, (PQ)2 = (PR)2 + (QR)2 QR2 = PQ2 – PR2 QR = √(𝑃𝑄2−𝑃𝑅2) Now, 𝑩𝑪/𝑸𝑹 = √(𝑨𝑩^𝟐 − 𝑨𝑪^𝟐 )/√(𝑷𝑸^𝟐 − 𝑷𝑹^𝟐 ) From (1) AB = k(PQ) and AC = k(PR) From (1) and (2) 𝐴𝐶/𝑃𝑅 = 𝐴𝐵/𝑃𝑄 = 𝐵𝐶/𝑄𝑅 = k 𝑨𝑪/𝑷𝑹 = 𝑨𝑩/𝑷𝑸 = 𝑩𝑪/𝑸𝑹 Since corresponding sides of Δ ABC & Δ PQR are in the same ratio Thus, ∆ ABC ~ ∆ PQR Since corresponding angles of similar triangles are equal ∴ ∠ B = ∠ Q Hence proved 𝐵𝐶/𝑄𝑅 = √((𝑘𝑃𝑄)^2 − (𝑘𝑃𝑅)^2 )/√(𝑃𝑄^2 − 𝑃𝑅^2 ) 𝑩𝑪/𝑸𝑹= √(𝒌^𝟐 𝑷𝑸^𝟐− 𝒌^𝟐 𝑷𝑹^𝟐 )/√(𝑷𝑸^𝟐 −𝑷𝑹^𝟐 ) 𝐵𝐶/𝑄𝑅 = (𝑘√(𝑃𝑄^2 −𝑃𝑅^2 ))/√(𝑃𝑄^2 −𝑃𝑅^2 ) 𝑩𝑪/𝑸𝑹 = k

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#### Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.