Pertanyaan

Persamaan garis normal dari y = 2 sin x + 3 cos x pada titik ( 6 π , 1 + 2 3 3 ) adalah ....

Persamaan garis normal dari y = 2 sin x + 3 cos x pada titik $(\frac{π}{6}, 1 + \frac{3 3}{2})$ adalah ....

$begin mathsize 14px style y space equals space open parentheses negative fraction numerator 2 open parentheses 2 square root of 3 plus 3 close parentheses over denominator 3 end fraction close parentheses open parentheses x minus straight pi over 6 close parentheses space plus space open parentheses 1 plus fraction numerator 3 square root of 3 over denominator 2 end fraction close parentheses end style$
$begin mathsize 14px style y space equals space open parentheses negative fraction numerator 2 open parentheses 2 square root of 3 minus 3 close parentheses over denominator 3 end fraction close parentheses open parentheses x minus straight pi over 6 close parentheses space plus space open parentheses 1 plus fraction numerator 3 square root of 3 over denominator 2 end fraction close parentheses end style$
$begin mathsize 14px style y equals space open parentheses fraction numerator 2 open parentheses 2 square root of 3 plus 3 close parentheses over denominator 3 end fraction close parentheses open parentheses x minus straight pi over 6 close parentheses space plus space open parentheses 1 plus fraction numerator 3 square root of 3 over denominator 2 end fraction close parentheses end style$
$begin mathsize 14px style y space equals space open parentheses fraction numerator 2 open parentheses 2 square root of 3 minus 3 close parentheses over denominator 3 end fraction close parentheses open parentheses x minus straight pi over 6 close parentheses space plus space open parentheses 1 plus fraction numerator 3 square root of 3 over denominator 2 end fraction close parentheses end style$
$begin mathsize 14px style y space equals space open parentheses fraction numerator open parentheses 2 square root of 3 minus 3 close parentheses over denominator 3 end fraction close parentheses open parentheses x minus straight pi over 6 close parentheses space plus space open parentheses 1 plus fraction numerator 3 square root of 3 over denominator 2 end fraction close parentheses end style$

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Klaim

H. Nufus

Master Teacher

Mahasiswa/Alumni Universitas Negeri Surabaya

Jawaban terverifikasi

Pembahasan

Kita cari dulu gradien garis singgungnya dengan cara mencari turunan pertama dari kurva tersebut kemudian subtitusikan Ingat bahwa garis normal tegak lurus dengan garis singgung sehingga hubungan kedua gradiennya adalah Maka persamaan garis normal yang dimaksud adalah

Kita cari dulu gradien garis singgungnya dengan cara mencari turunan pertama dari kurva tersebut kemudian subtitusikan

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell y to the power of apostrophe end cell equals cell 2 c o s invisible function application x minus 3 s i n invisible function application x end cell row cell m subscript 1 end cell equals cell 2 c o s invisible function application pi over 6 minus 3 s i n invisible function application pi over 6 equals 2 fraction numerator square root of 3 over denominator 2 end fraction minus 3 open parentheses 1 half close parentheses equals fraction numerator 2 square root of 3 minus 3 over denominator 2 end fraction end cell end table end style$

Ingat bahwa garis normal tegak lurus dengan garis singgung sehingga hubungan kedua gradiennya adalah

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell m subscript 1 times m subscript 2 end cell equals cell negative 1 end cell row cell open parentheses fraction numerator 2 square root of 3 minus 3 over denominator 2 end fraction close parentheses m subscript 2 end cell equals cell negative 1 end cell row cell m subscript 2 end cell equals cell negative fraction numerator 2 over denominator 2 square root of 3 minus 3 end fraction end cell row cell m subscript 2 end cell equals cell negative fraction numerator 2 over denominator 2 square root of 3 minus 3 end fraction cross times fraction numerator 2 square root of 3 plus 3 over denominator 2 square root of 3 plus 3 end fraction end cell row cell m subscript 2 end cell equals cell negative fraction numerator 2 left parenthesis 2 square root of 3 plus 3 right parenthesis over denominator 3 end fraction end cell end table end style$

Maka persamaan garis normal yang dimaksud adalah

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell y minus y subscript 1 end cell equals cell m subscript 2 left parenthesis x minus x subscript 1 right parenthesis end cell row cell y minus open parentheses 1 plus fraction numerator 3 square root of 3 over denominator 2 end fraction close parentheses end cell equals cell open parentheses negative fraction numerator 2 left parenthesis 2 square root of 3 plus 3 right parenthesis over denominator 3 end fraction close parentheses open parentheses x minus pi over 6 close parentheses end cell row y equals cell open parentheses negative fraction numerator 2 left parenthesis 2 square root of 3 plus 3 over denominator 3 end fraction close parentheses open parentheses x minus pi over 6 close parentheses plus open parentheses 1 plus fraction numerator 3 square root of 3 over denominator 2 end fraction close parentheses end cell end table end style$