Pertanyaan

Jika tan 2 α = 4 sin α cos α untuk 2 π < α < π , maka cos α = ...

Jika $tan 2 α = 4 sin α cos α$ untuk $\frac{π}{2} < α < π$ , maka $cos α = ...$

D. Nuryani

Master Teacher

Mahasiswa/Alumni Universitas Padjadjaran

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah D.

Pembahasan

Sebelum masuk ke pembahasan, ingatlah kembali rumus-rumus terkait trigonometri berikut: Rumus rangkap pada tangen yaitu: Identitas trigonometri: Dengan menggunakan kedua rumus, tersebut diperoleh perhitungan sebagai berikut. Karena atau terdapat dikuadran 2, maka nilai kosinus selalu negatif, sehingga pilih nilai . Oleh karena itu, jawaban yang benar adalah D.

Sebelum masuk ke pembahasan, ingatlah kembali rumus-rumus terkait trigonometri berikut:

Rumus rangkap pada tangen yaitu: $tan space 2 alpha equals fraction numerator 2 space tan space alpha over denominator 1 minus tan squared alpha end fraction$
Identitas trigonometri:

Dengan menggunakan kedua rumus, tersebut diperoleh perhitungan sebagai berikut.

$table attributes columnalign right center left columnspacing 0px end attributes row cell tan space 2 alpha end cell equals cell 4 space sin space alpha space cos space alpha end cell row cell cos space alpha end cell equals cell fraction numerator tan space 2 alpha over denominator 4 space sin space alpha space end fraction end cell row cell cos space alpha end cell equals cell fraction numerator begin display style fraction numerator 2 space tan space alpha over denominator 1 minus tan squared space alpha end fraction end style over denominator 4 space sin space alpha end fraction end cell row cell cos space alpha end cell equals cell fraction numerator 2 space tan space alpha over denominator 4 times open parentheses 1 minus tan squared space alpha close parentheses times sin space alpha end fraction end cell row cell cos space alpha end cell equals cell fraction numerator begin display style fraction numerator sin space alpha space over denominator cos space alpha end fraction end style space over denominator 2 times open parentheses 1 minus open parentheses begin display style fraction numerator sin space alpha over denominator cos space alpha end fraction end style close parentheses squared close parentheses times sin space alpha end fraction end cell row cell cos space alpha end cell equals cell 1 half times fraction numerator up diagonal strike sin space alpha end strike over denominator cos space alpha end fraction times fraction numerator 1 over denominator open parentheses 1 minus begin display style fraction numerator sin squared alpha over denominator cos squared alpha end fraction end style close parentheses times up diagonal strike sin space alpha end strike end fraction end cell row cell cos space alpha end cell equals cell 1 half times fraction numerator 1 over denominator cos space alpha end fraction times fraction numerator 1 over denominator open parentheses begin display style fraction numerator cos squared alpha minus sin squared alpha over denominator cos squared alpha end fraction end style close parentheses end fraction end cell row cell cos space alpha end cell equals cell 1 half times fraction numerator 1 over denominator up diagonal strike cos space alpha end strike end fraction times fraction numerator up diagonal strike cos squared alpha end strike over denominator cos squared alpha minus sin squared alpha end fraction end cell row cell cos space alpha end cell equals cell 1 half times fraction numerator cos space alpha over denominator cos squared alpha minus open parentheses 1 minus cos squared alpha close parentheses end fraction end cell row cell cos space alpha end cell equals cell 1 half times fraction numerator cos space alpha over denominator 2 cos squared alpha minus 1 end fraction end cell row cell fraction numerator cos space alpha over denominator fraction numerator cos space alpha over denominator 2 cos squared alpha minus 1 end fraction end fraction end cell equals cell 1 half end cell row cell fraction numerator up diagonal strike cos space alpha end strike times open parentheses 2 cos squared alpha minus 1 close parentheses over denominator up diagonal strike cos space alpha end strike end fraction end cell equals cell 1 half end cell row cell 2 cos squared alpha minus 1 end cell equals cell 1 half end cell row cell 2 cos squared alpha end cell equals cell 1 half plus 1 end cell row cell 2 cos squared alpha end cell equals cell 3 over 2 end cell row cell cos squared alpha end cell equals cell 3 over 4 end cell row cell cos space alpha end cell equals cell plus-or-minus square root of 3 over 4 end root end cell row cell cos space alpha end cell equals cell 1 half square root of 3 space atau space minus 1 half square root of 3 end cell end table$

Karena straight pi over 2 less than alpha less than straight pi atau terdapat dikuadran 2, maka nilai kosinus selalu negatif, sehingga pilih nilai cos space alpha equals negative 1 half square root of 3 .

Oleh karena itu, jawaban yang benar adalah D.