Subscribe to our Youtube Channel - https://www.youtube.com/channel/UCZBx269Tl5Os5NHlSbVX4Kg

Slide77.JPG

Slide78.JPG
Slide79.JPG Slide80.JPG Slide81.JPG

  1. Chapter 3 Class 11 Trigonometric Functions
  2. Serial order wise

Transcript

Example 29 Prove that cos2 ๐‘ฅ+cos2 (๐‘ฅ+๐œ‹/3) + cos2 (๐‘ฅโˆ’๐œ‹/3) = 3/2 Lets first calculate all 3 terms separately We know that cos 2x = 2 cos2 x โˆ’ 1 cos 2x + 1 = 2cos2 x ๐‘๐‘œ๐‘ โกใ€–2๐‘ฅ + 1ใ€—/2 = cos2 x So, cos2 x = ๐œ๐จ๐ฌโกใ€–๐Ÿ๐’™ + ๐Ÿใ€—/๐Ÿ Replacing x with ("x + " ๐œ‹/3) is about cos2 ("x" +๐œ‹/3) = cosโกใ€–2(๐‘ฅ + ๐œ‹/3)+1ใ€—/2 = cosโกใ€–(2๐‘ฅ + 2๐œ‹/3) + 1ใ€—/2 Similarly, Replacing x with ("x โˆ’" ๐œ‹/3) in cos2 x = cosโกใ€–2๐‘ฅ + 1ใ€—/2 cos2 ("x" โˆ’๐œ‹/3) = cosโกใ€–2(๐‘ฅ โˆ’ ๐œ‹/3)+ 1ใ€—/2 = cosโกใ€–(2๐‘ฅ โˆ’ 2๐œ‹/3)+ 1ใ€—/2 Solving L.H.S cos2 x + cos2 (๐‘ฅ+ ๐œ‹/3) + cos2 (๐‘ฅโˆ’๐œ‹/3) = (1 + cosโก2๐‘ฅ)/2 + (1 + cosโก(2๐‘ฅ + 2๐œ‹/3))/2 + (1 + cosโก(2๐‘ฅ โˆ’ 2๐œ‹/3))/2 = 1/2 [1+cosโกใ€–2๐‘ฅ+1+๐‘๐‘œ๐‘ (2๐‘ฅ+2๐œ‹/3)+1+๐‘๐‘œ๐‘ (2๐‘ฅโˆ’2๐œ‹/3)ใ€— ] = 1/2 [3+cosโกใ€–2๐‘ฅ+๐‘๐‘œ๐‘ (2๐‘ฅ+2๐œ‹/3)+๐‘๐‘œ๐‘ (2๐‘ฅโˆ’2๐œ‹/3)ใ€— ] = 1/2 [3+cosโกใ€–2๐‘ฅ+2๐‘๐‘œ๐‘ ((2๐‘ฅ + 2๐œ‹/3 + 2๐‘ฅ โˆ’ 2๐œ‹/3)/2).๐‘๐‘œ๐‘ ((2๐‘ฅ + 2๐œ‹/3 โˆ’(2๐‘ฅ โˆ’ 2๐œ‹/3))/2)ใ€— ] Using cos x + cos y = 2 cos ((๐‘ฅ + ๐‘ฆ)/2). cos ((๐‘ฅ โˆ’ ๐‘ฆ)/2) Replace x by ("2" ๐‘ฅ" + " 2๐œ‹/3) & y by ("2x โˆ’ " 2๐œ‹/3) = 1/2 [3+cosโกใ€–2๐‘ฅ+2๐‘๐‘œ๐‘ ((4๐‘ฅ + 0)/2).๐‘๐‘œ๐‘ ((0 + 4๐œ‹/3)/2)ใ€— ] = 1/2 [3+cosโกใ€–2๐‘ฅ+2๐‘๐‘œ๐‘ (4๐‘ฅ/2).๐‘๐‘œ๐‘ ((4๐œ‹/3)/2)ใ€— ] = 1/2 [3+cosโกใ€–2๐‘ฅ+2 cosโก2๐‘ฅ cosโกใ€–2๐œ‹/3ใ€— ใ€— ] = 1/2 [3+cosโกใ€–2๐‘ฅ+2 cosโก2๐‘ฅ cosโก(๐œ‹โˆ’๐œ‹/3) ใ€— ] = 1/2 [3+cosโกใ€–2๐‘ฅ+2 cosโก2๐‘ฅ ใ€— (ใ€–โˆ’cosใ€—โก(๐œ‹/3) ) ] = 1/2 [3+cosโกใ€–2๐‘ฅ+2 cosโก2๐‘ฅ ใ€— (โˆ’1/2) ] (As cos (ฯ€ โˆ’ ๐œƒ) = โˆ’cos ๐œƒ) = 1/2 [3+cosโกใ€–2๐‘ฅโˆ’2 ร—1/2ร—cosโก2๐‘ฅ ใ€— ] = 1/2 [3+cosโกใ€–2๐‘ฅโˆ’cosโก2๐‘ฅ ใ€— ] = 1/2 [3+0] = 3/2 = R.H.S. Hence, L.H.S. = R.H.S. Hence Proved

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.