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Ex 8.4, 2 - Write all trigonometric ratios in terms of sec A - Ex 8.4

  1. Class 10
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Ex 8.4, 2 Write all the other trigonometric ratios of โˆ  A in terms of sec A. cos A We know that cos A = ๐Ÿ/๐’”๐’†๐’„โก๐‘จ tan A We know that 1 + tan2 A = sec2 A tan2 A = sec2 A โˆ’ 1 tan A = ยฑ โˆš(๐‘ ๐‘’๐‘2 ๐ด โˆ’1) Here, A is acute (i.e. less than 90ยฐ) & tan A is positive when A is acute tan A = โˆš(๐‘ ๐‘’๐‘2 ๐ด โˆ’1) cot cot A = ๐Ÿ/(๐’•๐’‚๐’ ๐‘จ) Putting value of tan A found above cot A = 1/(โˆš(sec^2โกใ€– ๐ด โˆ’1ใ€— ) ) cosec A We know that 1 + cot2 A = cosec2 A cosec2 A = 1 + cot2 A cosec2 A = 1 + (1/(โˆš(sec^2โกใ€–๐ด โˆ’ 1ใ€— ) ))^2 cosec2 A = 1 + (1/(๐‘ ๐‘’๐‘2 ๐ด โˆ’ 1)) cosec2 A = ((1 ร— (๐‘ ๐‘’๐‘2 ๐ด โˆ’1) + 1)/(๐‘ ๐‘’๐‘2 ๐ด โˆ’1)) cosec2 A = (( ๐‘ ๐‘’๐‘2 ๐ด โˆ’ 1 + 1)/(๐‘ ๐‘’๐‘2 ๐ด โˆ’ 1)) cosec2 A = ( ๐‘ ๐‘’๐‘2 ๐ด )/(๐‘ ๐‘’๐‘2 ๐ด โˆ’ 1) cosec A = ยฑ โˆš(( ๐‘ ๐‘’๐‘2 ๐ด )/(๐‘ ๐‘’๐‘2 ๐ด โˆ’ 1)) cosec A = ยฑ secโก๐ด/โˆš(๐‘ ๐‘’๐‘2 ๐ด โˆ’ 1 ) Here, A is acute (i.e. less than 90ยฐ) & cosec A is positive when A is acute cosec A = secโก๐ด/โˆš(๐‘ ๐‘’๐‘2 ๐ด โˆ’ 1 ) sin A We know that sin A = ๐Ÿ/(๐’„๐’๐’”๐’†๐’„ ๐‘จ) sin A = 1/((secโก๐ด/โˆš(sec^2โกใ€–๐ด โˆ’1ใ€— )) ) sin A = โˆš(๐‘ ๐‘’๐‘2 ๐ด โˆ’ 1)/secโก๐ด

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