Learn All Concepts of Chapter 2 Class 11 Relations and Function - FREE. Check - Trigonometry Class 11 - All Concepts




Last updated at Feb. 13, 2020 by Teachoo
Learn All Concepts of Chapter 2 Class 11 Relations and Function - FREE. Check - Trigonometry Class 11 - All Concepts
Transcript
Example 29 Prove that cos2 ๐ฅ+cos2 (๐ฅ+๐/3) + cos2 (๐ฅโ๐/3) = 3/2 Lets first calculate all 3 terms separately We know that cos 2x = 2 cos2 x โ 1 cos 2x + 1 = 2cos2 x ๐๐๐ โกใ2๐ฅ + 1ใ/2 = cos2 x So, cos2 x = ๐๐จ๐ฌโกใ๐๐ + ๐ใ/๐ Replacing x with ("x + " ๐/3) is about cos2 ("x" +๐/3) = cosโกใ2(๐ฅ + ๐/3)+1ใ/2 = cosโกใ(2๐ฅ + 2๐/3) + 1ใ/2 Similarly, Replacing x with ("x โ" ๐/3) in cos2 x = cosโกใ2๐ฅ + 1ใ/2 cos2 ("x" โ๐/3) = cosโกใ2(๐ฅ โ ๐/3)+ 1ใ/2 = cosโกใ(2๐ฅ โ 2๐/3)+ 1ใ/2 Solving L.H.S cos2 x + cos2 (๐ฅ+ ๐/3) + cos2 (๐ฅโ๐/3) = (1 + cosโก2๐ฅ)/2 + (1 + cosโก(2๐ฅ + 2๐/3))/2 + (1 + cosโก(2๐ฅ โ 2๐/3))/2 = 1/2 [1+cosโกใ2๐ฅ+1+๐๐๐ (2๐ฅ+2๐/3)+1+๐๐๐ (2๐ฅโ2๐/3)ใ ] = 1/2 [3+cosโกใ2๐ฅ+๐๐๐ (2๐ฅ+2๐/3)+๐๐๐ (2๐ฅโ2๐/3)ใ ] = 1/2 [3+cosโกใ2๐ฅ+2๐๐๐ ((2๐ฅ + 2๐/3 + 2๐ฅ โ 2๐/3)/2).๐๐๐ ((2๐ฅ + 2๐/3 โ(2๐ฅ โ 2๐/3))/2)ใ ] Using cos x + cos y = 2 cos ((๐ฅ + ๐ฆ)/2). cos ((๐ฅ โ ๐ฆ)/2) Replace x by ("2" ๐ฅ" + " 2๐/3) & y by ("2x โ " 2๐/3) = 1/2 [3+cosโกใ2๐ฅ+2๐๐๐ ((4๐ฅ + 0)/2).๐๐๐ ((0 + 4๐/3)/2)ใ ] = 1/2 [3+cosโกใ2๐ฅ+2๐๐๐ (4๐ฅ/2).๐๐๐ ((4๐/3)/2)ใ ] = 1/2 [3+cosโกใ2๐ฅ+2 cosโก2๐ฅ cosโกใ2๐/3ใ ใ ] = 1/2 [3+cosโกใ2๐ฅ+2 cosโก2๐ฅ cosโก(๐โ๐/3) ใ ] = 1/2 [3+cosโกใ2๐ฅ+2 cosโก2๐ฅ ใ (ใโcosใโก(๐/3) ) ] = 1/2 [3+cosโกใ2๐ฅ+2 cosโก2๐ฅ ใ (โ1/2) ] (As cos (ฯ โ ๐) = โcos ๐) = 1/2 [3+cosโกใ2๐ฅโ2 ร1/2รcosโก2๐ฅ ใ ] = 1/2 [3+cosโกใ2๐ฅโcosโก2๐ฅ ใ ] = 1/2 [3+0] = 3/2 = R.H.S. Hence, L.H.S. = R.H.S. Hence Proved
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