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Prove that if 1/2 ≤ x ≤ 1, then cos −1 x + cos −1 [x/2+√(3 - 3x 2 )/2] = π/3

This is a question of CBSE Sample Paper - Class 12 - 2017/18.

You can download the question paper here  https://www.teachoo.com/cbse/sample-papers/


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Question 7 - CBSE Class 12 Sample Paper for 2018 Boards - Part 2
Question 7 - CBSE Class 12 Sample Paper for 2018 Boards - Part 3

Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Transcript

Question 7 Prove that if 1/2 ≀ x ≀ 1 then cosβˆ’1 x + cosβˆ’1 [π‘₯/2+√(3 βˆ’ 3π‘₯^2 )/2] = πœ‹/3 Solving LHS cosβˆ’1 x + cosβˆ’1 [π‘₯/2+√(3 βˆ’ 3π‘₯^2 )/2] = cosβˆ’1 x + cosβˆ’1 [π‘₯/2+√(3(1 βˆ’ π‘₯^2))/2] Let x = cos πœƒ i.e. cosβˆ’1 x = πœƒ Theory Since it is √(1βˆ’π‘₯^2 ) we can use x = sin πœƒ or x = cos πœƒ Since it is given cos-1 x, we use x = cos πœƒ = πœƒ + cosβˆ’1 [cosβ‘πœƒ/2+√(3(1βˆ’cos^2β‘γ€–πœƒ)γ€— )/2] Using sin ΞΈ = 1 – cos2 ΞΈ = πœƒ + cosβˆ’1 [cosβ‘πœƒ/2+√(3 sin^2β‘πœƒ )/2] = πœƒ + cosβˆ’1 [cosβ‘πœƒ/2+γ€–βˆš3 sinγ€—β‘πœƒ/2] = πœƒ + cosβˆ’1 [1/2 Γ—cosβ‘πœƒ+√3/2 Γ—sinβ‘πœƒ ] Theory We convert 1/2 Γ—cosβ‘πœƒ+√3/2 Γ—sinβ‘πœƒ into cos (x – y) Since cos (x – y) = cos x cos y + sin x sin y We use x = 60Β° and y = πœƒ = πœƒ + cosβˆ’1 [cos⁑〖60Β°γ€— Γ—cosβ‘πœƒ+sin⁑〖60Β°γ€—Γ—sinβ‘πœƒ ] = πœƒ + cosβˆ’1 [cosβ‘γ€–πœ‹/3γ€— Γ—cosβ‘πœƒ+sinβ‘γ€–πœ‹/3γ€—Γ—sinβ‘πœƒ ] Using cos (x – y) = cos x cos y + sin x sin y = πœƒ + cosβˆ’1 [cos⁑(πœ‹/3βˆ’πœƒ) ] = πœƒ + cosβˆ’1 [π‘π‘œπ‘ (πœ‹/3βˆ’πœƒ)] = πœƒ + πœ‹/3 βˆ’ πœƒ = πœ‹/3 = RHS 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 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.