# If 0 < θ < π, then minimum value of 3 sin θ + cosec3 θ is :-

1
by Atreyi

2014-08-03T22:41:39+05:30

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Minimum value of y is when dy / dФ        here Ф is theta.

y = 3 sin Ф + cosec 3Ф
dy / dФ  =  3 cosФ - 3 cot 3Ф cosec 3Ф  = 0
cos Ф =  cot 3Ф  cosec 3Ф
=>  cos Ф =  cos 3Ф / sin² 3Ф
=> cos Ф  sin² 3Ф  = cos 3Ф
=> cos Ф sin² 3Ф =  cos 2Ф cos Ф - sin 2Ф sin Ф          as cos ( 2Ф + Ф)

=>  cos Ф [ sin 2Ф cosФ + cos 2Ф sin Ф ]² = cos 2Ф cos Ф - sin 2Ф sin Ф
cosФ [2sinФ cos²Ф + (2cos²Ф - 1) sinФ]² = (cos²Ф - sin²Ф)cosФ - 2 sin²ФcosФ
cancel cosФ on both sides
=>  (4 sinФ cos²Ф - sin Ф)² =  cos³Ф - 3 sin²Ф cosФ
=>  sin² Ф (4 cos² Ф - 1)² =  cos³ Ф - 3 (1-cos²Ф) cosФ
=>  (1-cos²Ф] [16 cos^4 Ф  +1 - 8 cos² Ф ]  = cosФ [ cos² Ф - 3 + 3 cos² Ф]
- 16 cos^6 Ф + 24 cos^4 Ф - 9 cos² Ф + 1 = cosФ [ 4 cos² Ф -3 ]

The given expression is minimum between Ф = 21.61 deg and 30 degrees. cosec 3Ф meets 3 sinФ at Ф=21.61 deg. During this domain, cosec 3Ф is decreasing & sin Ф is increasing.
minimum value comes at theta = 18 degrees
minimum value is possibly 2.18