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Asked by swayamagarwal2114 | 18 Jul, 2022, 03:06: PM
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Given colon space straight f left parenthesis straight x right parenthesis equals straight a subscript 1 space cos left parenthesis straight alpha subscript 1 plus straight x right parenthesis plus straight a subscript 2 space cos left parenthesis straight alpha subscript 2 plus straight x right parenthesis plus straight a subscript 3 space cos left parenthesis straight alpha subscript 3 plus straight x right parenthesis space plus space... space plus space straight a subscript straight n space cos left parenthesis straight alpha subscript straight n plus straight x right parenthesis straight f left parenthesis straight x right parenthesis space vanishes space for space straight x equals 0 space and space straight x equals straight x subscript 1 space where space straight x subscript 1 not equal to kπ Now comma space straight f left parenthesis 0 right parenthesis equals straight a subscript 1 space cos space straight alpha subscript 1 space plus thin space straight a subscript 2 space cos space straight alpha subscript 2 space plus thin space straight a subscript 3 space cos space straight alpha subscript 3 space plus space... space plus straight a subscript straight n space cos space straight alpha subscript straight n rightwards double arrow straight a subscript 1 space cos space straight alpha subscript 1 space plus thin space straight a subscript 2 space cos space straight alpha subscript 2 space plus thin space straight a subscript 3 space cos space straight alpha subscript 3 space plus space... space plus straight a subscript straight n space cos space straight alpha subscript straight n space equals space 0 space space... space left parenthesis 1 right parenthesis Also comma space straight f left parenthesis straight x subscript 1 right parenthesis space equals space straight a subscript 1 space cos left parenthesis straight alpha subscript 1 plus straight x subscript 1 right parenthesis plus straight a subscript 2 space cos left parenthesis straight alpha subscript 2 plus straight x subscript 1 right parenthesis plus straight a subscript 3 space cos left parenthesis straight alpha subscript 3 plus straight x subscript 1 right parenthesis space plus space... space plus space straight a subscript straight n space cos left parenthesis straight alpha subscript straight n plus straight x subscript 1 right parenthesis space space space space space space space space equals straight a subscript 1 space left parenthesis cosα subscript 1 space cosx subscript 1 space minus space sinα subscript 1 space sinx subscript 1 right parenthesis plus straight a subscript 2 space left parenthesis cosα subscript 2 space cosx subscript 1 space minus space sinα subscript 2 space sinx subscript 1 right parenthesis plus space... space plus space straight a subscript straight n space left parenthesis cosα subscript straight n space cosx subscript 1 space minus space sinα subscript straight n space sinx subscript 1 right parenthesis space space space space space space space space equals straight a subscript 1 space cosα subscript 1 space cosx subscript 1 space plus thin space straight a subscript 2 space cosα subscript 2 space cosx subscript 1 space plus thin space straight a subscript 3 space cosα subscript 3 space cosx subscript 1 space plus space... space plus straight a subscript straight n space cosα subscript straight n space cosx subscript 1 space minus space left parenthesis straight a subscript 1 space sinα subscript 1 space sinx subscript 1 space plus space straight a subscript 2 space sinα subscript 2 space sinx subscript 1 plus... plus space straight a subscript straight n space sinα subscript straight n space sinx subscript 1 right parenthesis space space space space space space space space equals cosx subscript 1 left parenthesis straight a subscript 1 space cos space straight alpha subscript 1 space plus thin space straight a subscript 2 space cos space straight alpha subscript 2 space plus thin space straight a subscript 3 space cos space straight alpha subscript 3 space plus space... space plus straight a subscript straight n space cos space straight alpha subscript straight n right parenthesis space minus space sinx subscript 1 space left parenthesis straight a subscript 1 space sinα subscript 1 space plus space straight a subscript 2 space sinα subscript 2 space plus... plus space straight a subscript straight n space sinα subscript straight n space right parenthesis space space space space space space space space equals space cosx subscript 1 space left parenthesis 0 right parenthesis space minus space sinx subscript 1 space left parenthesis straight a subscript 1 space sinα subscript 1 space plus space straight a subscript 2 space sinα subscript 2 space plus... plus space straight a subscript straight n space sinα subscript straight n space right parenthesis space space space space space space space space space space space space space space space space space space space... space From space left parenthesis 1 right parenthesis space space space space space space space space equals negative sinx subscript 1 space left parenthesis straight a subscript 1 space sinα subscript 1 space plus space straight a subscript 2 space sinα subscript 2 space plus... plus space straight a subscript straight n space sinα subscript straight n space right parenthesis rightwards double arrow negative sinx subscript 1 space left parenthesis straight a subscript 1 space sinα subscript 1 space plus space straight a subscript 2 space sinα subscript 2 space plus... plus space straight a subscript straight n space sinα subscript straight n space right parenthesis equals 0 Since comma space straight x subscript 1 not equal to kπ space rightwards double arrow sinx subscript 1 not equal to 0 Thus comma space straight a subscript 1 space sinα subscript 1 space plus space straight a subscript 2 space sinα subscript 2 space plus... plus space straight a subscript straight n space sinα subscript straight n space equals space 0
Answered by Renu Varma | 19 Jul, 2022, 12:35: PM
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