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Misc 4 - Prove sin-1 8/17 + sin-1 3/5 = tan-1 77/36 - Miscellaneous

Misc. 4 - Chapter 2 Class 12 Inverse Trigonometric Functions - Part 2
Misc. 4 - Chapter 2 Class 12 Inverse Trigonometric Functions - Part 3
Misc. 4 - Chapter 2 Class 12 Inverse Trigonometric Functions - Part 4

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Misc 4 Prove sin–1 8/17 + sin–1 3/5 = tan–1 77/36 Let a = sin–1 8/17 & b = sin–1 3/5 We convert sin–1 to tan–1 & then use tan (a – b) formula Let a = sin–1 𝟖/𝟏𝟕 sin a = 8/17 Now, cos a = √(1 −𝑠𝑖𝑛2 𝑎) = √(1 −(8/17)^2 ) = √(225/289) = 15/17 Thus, tan a = (sin 𝑎)/(cos a) = (8/17)/(15/17) = 8/17 × 17/15 = 𝟖/𝟏𝟓 Let b = sin–1 𝟑/𝟓 sin b = 3/5 Now, cos b = √(1 −𝑠𝑖𝑛2 𝑏) = √(1 −(3/5)^2 )= √(16/25) = 4/5 Thus, tan b = sin⁡𝑏/cos⁡𝑏 = (3/5)/(4/5) = 3/5 × 5/4 = 𝟑/𝟒 Now tan (a + b) = tan⁡〖𝑎 +〖 tan〗⁡〖𝑏 〗 〗/(1 − tan⁡〖𝑎 tan⁡𝑏 〗 ) = (8/15 + 3/4)/(1 − 8/15 × 3/4) = ((8(4) + 3(15) )/(15 × 4) )/( (15 × 4 − 8 × 3)/(15 × 4) ) = ((32 + 45 )/(15 × 4) )/( (60 − 24)/(15 × 4) ) = (32 + 45)/(15 × 4) × (15 × 4)/(60 −24) = 77/36 Hence, tan (a + b) = 𝟕𝟕/𝟑𝟔 a + b = tan–1 77/36 Putting value of a & b sin–1 8/17 + sin–1 3/5 = tan–1 77/36 Hence Proved

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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.