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ICSE Class 10 Answered

prove that:5 cos theta +3 cos (theta+60(degree))+3 lies between -4 and 10
Asked by kmondaldgms | 19 Aug, 2018, 12:23: AM
answered-by-expert Expert Answer
begin mathsize 16px style 5 cosθ space plus space 3 cos open parentheses straight theta space plus space straight pi over 3 close parentheses space plus space 3 space
equals space 5 cosθ space plus space 3 space open square brackets cos space open parentheses straight pi over 3 close parentheses cosθ space minus sin space open parentheses straight pi over 3 close parentheses sinθ space close square brackets plus space 3
open parentheses since comma space cos left parenthesis straight a space plus space straight b right parenthesis space equals space cosacosb space minus space sinasinb space close parentheses
equals space 5 cosθ space plus 3 open parentheses 1 half close parentheses cosθ space minus space 3 open parentheses fraction numerator square root of 3 over denominator 2 end fraction close parentheses sinθ space plus space 3
open parentheses since comma space cos open parentheses straight pi over 3 close parentheses space equals space 1 half comma space sin open parentheses straight pi over 3 close parentheses space equals space fraction numerator square root of 3 over denominator 2 end fraction close parentheses
equals space 13 over 2 cosθ space minus 3 open parentheses fraction numerator square root of 3 over denominator 2 end fraction close parentheses sinθ space plus space 3
minimum space value space is space 3 space minus space square root of 13 squared over 2 squared space space plus space space open parentheses 3 square root of 3 close parentheses squared over 2 squared end root
open parentheses acosa space plus space straight b space sina space plus space straight c...... minimum space value space is space straight c space minus space square root of straight a squared space space plus space straight b squared space space end root close parentheses
equals space 3 space minus space square root of 169 over 4 space space plus space space 27 over 24 end root
equals space 3 space minus space square root of 49
equals space 3 space minus space 7 space
equals space minus 4
minimum space value space is space 3 space plus space square root of 13 squared over 2 squared space space plus space space open parentheses 3 square root of 3 close parentheses squared over 2 squared end root
equals space space 3 space plus space square root of 169 over 4 space space plus space space 27 over 24 end root
equals space 3 space plus space square root of 49
equals space 3 space plus space 7 space
equals space 10
Hence space 5 cosθ space plus space 3 cos open parentheses straight theta space plus space straight pi over 3 close parentheses space plus space 3 space lies space between space minus 4 space to space 10. end style
Answered by Yasmeen Khan | 24 Aug, 2018, 12:17: PM
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