Slide21.JPG

Slide22.JPG
Slide23.JPG
Slide24.JPG
Slide25.JPG

Go Ad-free

Transcript

Ex 2.2, 9 tan 1/2 [sinβˆ’1 2π‘₯/(1 + π‘₯2) + cosβˆ’1 (1 βˆ’ 𝑦2)/(1 + 𝑦2) ] We solve sinβˆ’1 2π‘₯/(1 + π‘₯2) & cosβˆ’1 ((1 βˆ’ 𝑦2)/(1 + 𝑦2)) separately Solving sinβˆ’1 πŸπ’™/(𝟏 + π’™πŸ) sinβˆ’1 2π‘₯/(1 + π‘₯2) Putting x = tan ΞΈ = sinβˆ’1 ((2 tanβ‘πœƒ)/(𝟏 + π’•π’‚π’πŸ 𝜽)) = sinβˆ’1 ((2 tanβ‘πœƒ)/(π’”π’†π’„πŸ 𝜽)) We write sin-1 & cosβˆ’1 in terms of tan-1 When there is 1 + x2 , we put x = tan ΞΈ = sinβˆ’1 ((2 sin⁑θ/cos⁑θ )/(1/cos2⁑θ )) = sinβˆ’1 (2 sin⁑θ/cos⁑θ Γ—cos2⁑θ/1) = sinβˆ’1 (𝟐 π’”π’Šπ’β‘πœ½ π’„π’π’”β‘πœ½ ) = sinβˆ’1 (sin 2ΞΈ) = 2ΞΈ ∴ sinβˆ’1 2π‘₯/(1 + π‘₯2) = 2ΞΈ Solving cosβˆ’1 ((𝟏 βˆ’ 𝐲𝟐)/(𝟏 + 𝐲𝟐)) cosβˆ’1 ((𝟏 βˆ’ 𝐲𝟐)/(𝟏 + 𝐲𝟐)) Putting y = tan 𝜢 = cosβˆ’1 ((1 βˆ’γ€– tan2〗⁑𝛼)/(𝟏 +γ€– π­πšπ§πŸγ€—β‘πœΆ )) = cosβˆ’1 ((1 βˆ’γ€– tan2〗⁑𝛼)/𝐬𝐞𝐜𝟐⁑𝜢 ) = cosβˆ’1 ((1 βˆ’ (𝑠𝑖𝑛2 𝛼 )/(π‘π‘œπ‘ 2 𝛼))/((1 )/(π‘π‘œπ‘ 2 𝛼))) = cosβˆ’1 (( (π‘π‘œπ‘ 2 𝛼 βˆ’ 𝑠𝑖𝑛2 𝛼 )/(π‘π‘œπ‘ 2 𝛼))/((1 )/(π‘π‘œπ‘ 2 𝛼))) = cosβˆ’1 ((π‘π‘œπ‘ 2 𝛼 βˆ’ 𝑠𝑖𝑛2 𝛼 )/(π‘π‘œπ‘ 2 𝛼)Γ—(π‘π‘œπ‘ 2 𝛼)/1) = cosβˆ’1 (π’„π’π’”πŸ 𝜢 βˆ’ π’”π’Šπ’πŸ 𝜢) = cosβˆ’1 (π’„π’π’”β‘πŸπœΆ ) = 2Ξ± Hence, cosβˆ’1 ((𝟏 βˆ’ π’šπŸ)/(𝟏 + π’šπŸ)) = 2Ξ± Now, we have sinβˆ’1 2π‘₯/(1 + π‘₯2) = 2ΞΈ & cosβˆ’1 ((1 βˆ’ 𝑦2)/(1 + 𝑦2)) = 2Ξ± Solving tan 𝟏/𝟐 ["sinβˆ’1 " πŸπ’™/(𝟏 + π’™πŸ) " + cosβˆ’1 " (𝟏 βˆ’ π’šπŸ)/(𝟏 + π’šπŸ)] Putting values = cosβˆ’1 (π’„π’π’”πŸ 𝜢 βˆ’ π’”π’Šπ’πŸ 𝜢) = cosβˆ’1 (π’„π’π’”β‘πŸπœΆ ) = 2Ξ± Hence, cosβˆ’1 ((𝟏 βˆ’ π’šπŸ)/(𝟏 + π’šπŸ)) = 2Ξ± Now, we have sinβˆ’1 2π‘₯/(1 + π‘₯2) = 2ΞΈ & cosβˆ’1 ((1 βˆ’ 𝑦2)/(1 + 𝑦2)) = 2Ξ± Solving tan 𝟏/𝟐 ["sinβˆ’1 " πŸπ’™/(𝟏 + π’™πŸ) " + cosβˆ’1 " (𝟏 βˆ’ π’šπŸ)/(𝟏 + π’šπŸ)] Putting values = tan 1/2 [2ΞΈ + 2Ξ±] = tan (ΞΈ + Ξ±) = (tan⁑θ + tan⁑𝛼)/(γ€–1 βˆ’ tan〗⁑θ tan⁑𝛼 ) = (𝒙 + π’š)/(𝟏 βˆ’ π’™π’š)

Davneet Singh's photo - Co-founder, Teachoo

Made by

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 and Computer Science at Teachoo