Question

$Find space dy over dx when space straight x equals straight a left parenthesis straight theta plus sinθ right parenthesis comma space straight y equals straight a left parenthesis 1 minus cosθ right parenthesis$

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Solution

space space space space space space straight x equals straight a left parenthesis straight theta plus sinθ right parenthesis comma space straight y equals straight a left parenthesis 1 minus cosθ right parenthesis dy over dθ equals straight a left parenthesis 1 plus cosθ right parenthesis equals straight a open parentheses 2 space cos squared straight theta over 2 close parentheses equals 2 straight a space cos squared straight theta over 2 space space space space space space straight y equals straight a left parenthesis 1 minus cosθ right parenthesis therefore space space space space space dy over dθ equals straight a left parenthesis sinθ right parenthesis equals straight a open parentheses 2 space sin straight theta over 2 cos straight theta over 2 close parentheses equals 2 straight a space sin straight theta over 2 cos straight theta over 2 Now space dy over dx equals fraction numerator begin display style dy over dθ end style over denominator begin display style dx over dθ end style end fraction equals fraction numerator 2 space sin straight theta over 2 cos straight theta over 2 over denominator 2 straight a space cos squared straight theta over 2 end fraction equals fraction numerator sin begin display style straight theta over 2 end style over denominator cos begin display style straight theta over 2 end style end fraction equals tan straight theta over 2

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