Misc 20 - Using the fact that sin (A + B) = sin A cos B - Miscellaneou

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Misc 20 - Chapter 5 Class 12 Continuity and Differentiability - Part 2

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Misc 20 - Chapter 5 Class 12 Continuity and Differentiability - Part 3

  1. Chapter 5 Class 12 Continuity and Differentiability (Term 1)
  2. Serial order wise

Transcript

Misc 20 Using the fact that sin⁑(𝐴 + 𝐡)=sin⁑𝐴 cos⁑𝐡+cos⁑𝐴 sin⁑𝐡 and the differentiation, obtain the sum formula for cosines.Given sin⁑(𝐴 + 𝐡)=sin⁑𝐴 cos⁑𝐡+cos⁑𝐴 sin⁑𝐡 Consider A & B are function of π‘₯ Differentiating both side 𝑀.π‘Ÿ.𝑑.π‘₯. 𝑑(sin⁑(𝐴 + 𝐡) )/𝑑π‘₯ = 𝑑(sin⁑𝐴 cos⁑𝐡 + cos⁑𝐴 sin⁑𝐡)/𝑑π‘₯ 𝑑(sin⁑(𝐴 + 𝐡) )/𝑑π‘₯ = 𝑑(sin⁑𝐴 . cos⁑𝐡)/𝑑π‘₯ + 𝑑(cos⁑〖𝐴 γ€—. sin⁑𝐡)/𝑑π‘₯ cos (𝐴+𝐡) . 𝑑(𝐴 + 𝐡)/𝑑π‘₯ = 𝑑(sin⁑𝐴 . cos⁑𝐡)/𝑑π‘₯ + 𝑑(cos⁑〖𝐴 γ€—. sin⁑𝐡)/𝑑π‘₯ Using Product rule As (𝑒𝑣)’ = 𝑒’𝑣 + 𝑣’𝑒 𝒄𝒐𝒔 (𝑨+𝑩) . (𝒅𝑨/𝒅𝒙 + 𝒅𝑩/𝒅𝒙) = (𝑑(sin⁑𝐴 )/𝑑π‘₯. cos⁑𝐡" +" 𝑑(cos⁑𝐡 )/𝑑π‘₯ " " 𝑠𝑖𝑛⁑"A" ) + (𝑑(cos⁑𝐴 )/𝑑π‘₯. 𝑠𝑖𝑛⁑𝐡" +" 𝑑(sin⁑𝐡 )/𝑑π‘₯ ". " 𝑐"os A" ) = cos⁑𝐴.𝑑𝐴/𝑑π‘₯ ". cos B "βˆ’sin⁑𝐡.𝑑𝐡/𝑑π‘₯ " " sin⁑𝐴 βˆ’ sin⁑𝐴. 𝑑𝐴/𝑑π‘₯.sin⁑𝐡+cos⁑𝐡. 𝑑𝐡/𝑑π‘₯ ". " 𝑐"os A" = cos⁑𝐴.𝑑𝐴/𝑑π‘₯ ". cos B "βˆ’sin⁑𝐴 .𝑑𝐴/𝑑π‘₯ " " 𝑠𝑖𝑛⁑"B" βˆ’ sin⁑𝐡. 𝑑𝐡/𝑑π‘₯. 𝑠𝑖𝑛 𝐴⁑"+ cos B" . 𝑑𝐡/𝑑π‘₯ ". " 𝑐"os A" = 𝑑𝐴/𝑑π‘₯ (cos⁑𝐴 cosβ‘π΅βˆ’sin⁑𝐴 sin⁑𝐡 ) + 𝑑𝐡/𝑑π‘₯ (βˆ’sin⁑𝐡 sin⁑𝐴+cos⁑𝐡 cos⁑𝐴 ) (cos⁑𝐴 cosβ‘π΅βˆ’sin⁑𝐴 sin⁑𝐡 ) (𝑑𝐴/𝑑π‘₯ + 𝑑𝐡/𝑑π‘₯) Thus, cos (𝐴+𝐡) . (𝑑𝐴/𝑑π‘₯ + 𝑑𝐡/𝑑π‘₯) = (cos⁑𝐴 cosβ‘π΅βˆ’sin⁑𝐴 sin⁑𝐡 ) (𝑑𝐴/𝑑π‘₯ + 𝑑𝐡/𝑑π‘₯) 𝒄𝒐𝒔" " (𝑨+𝑩) = 𝒄𝒐𝒔⁑𝑨 π’„π’π’”β‘π‘©βˆ’π’”π’Šπ’β‘π‘¨ π’”π’Šπ’β‘π‘© Hence proved

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.