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Find fraction numerator d y over denominator d x end fraction comma text  if  end text x equals 2 cos invisible function application theta minus cos invisible function application 2 theta text  and  end text y equals 2 sin invisible function application theta minus sin invisible function application 2 theta

  1. tan invisible function application fraction numerator 3 theta over denominator 2 end fraction
  2. negative tan invisible function application fraction numerator 3 theta over denominator 2 end fraction
  3. cot invisible function application fraction numerator 3 theta over denominator 2 end fraction
  4. negative cot invisible function application fraction numerator 3 theta over denominator 2 end fraction

The correct answer is: tan invisible function application fraction numerator 3 theta over denominator 2 end fraction


    Given x equals 2 cos invisible function application theta minus cos invisible function application 2 theta
    and y equals 2 sin invisible function application theta minus sin invisible function application 2 theta
    table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell fraction numerator dx over denominator straight d theta end fraction equals negative 2 sin invisible function application theta plus 2 sin invisible function application 2 theta end cell row cell text  and  end text fraction numerator dy over denominator straight d theta end fraction equals 2 cos invisible function application theta minus 2 cos invisible function application 2 theta end cell row cell therefore dy over dx equals fraction numerator 2 cos invisible function application theta minus 2 cos invisible function application 2 theta over denominator negative 2 sin invisible function application theta plus 2 sin invisible function application 2 theta end fraction end cell row cell equals fraction numerator cos invisible function application theta minus cos invisible function application 2 theta over denominator sin invisible function application 2 theta minus sin invisible function application theta end fraction end cell row cell equals fraction numerator 2 sin invisible function application open parentheses fraction numerator theta plus 2 theta over denominator 2 end fraction close parentheses sin invisible function application open parentheses fraction numerator 2 theta minus theta over denominator 2 end fraction close parentheses over denominator 2 cos invisible function application open parentheses fraction numerator theta plus 2 theta over denominator 2 end fraction close parentheses sin invisible function application open parentheses fraction numerator 2 theta minus theta over denominator 2 end fraction close parentheses end fraction equals tan invisible function application fraction numerator 3 theta over denominator 2 end fraction end cell end table

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