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Ex 2.2, 13 - Inverse Trigonometry - tan 1/2 [sin-1 2x/1+x2

Ex 2.2, 13 - Chapter 2 Class 12 Inverse Trigonometric Functions - Part 2
Ex 2.2, 13 - Chapter 2 Class 12 Inverse Trigonometric Functions - Part 3 Ex 2.2, 13 - Chapter 2 Class 12 Inverse Trigonometric Functions - Part 4 Ex 2.2, 13 - Chapter 2 Class 12 Inverse Trigonometric Functions - Part 5


Transcript

Ex 2.2, 13 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 (Using 1 + tan2 x = sec2 x) = 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)) (Using 2 sin x cos x = sin 2x) 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 (Using cos 2x = cos2 x – sin2 x) = 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) Putting x = tan ΞΈ & y = tan Ξ±

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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 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.