Pertanyaan

Jika 2 cos ( x + 45 ) ∘ = cos ( x − 45 ) ∘ , maka tan 2 x = ...

Jika $2 cos (x + 45)^{\circ} = cos (x - 45)^{\circ}$ , maka $tan 2 x = ...$

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Klaim

E. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sebelas Maret

Jawaban terverifikasi

Pembahasan

Ingat kembali rumus berikut. Dari rumus di atas, diperoleh perhitungan sebagai berikut. Akan ditentukan Jadi, nilai adalah . Oleh karena itu, jawaban yang benar adalah D.

Ingat kembali rumus berikut.

$fraction numerator sin space A over denominator cos space A end fraction equals tan space A$
$tan space 2 A equals fraction numerator 2 space tan space A over denominator 1 minus tan squared space A end fraction$

Dari rumus di atas, diperoleh perhitungan sebagai berikut.

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell 2 space cos space left parenthesis x plus 45 right parenthesis degree end cell equals cell cos space left parenthesis x minus 45 right parenthesis degree end cell row cell 2 left parenthesis cos space x space cos space 45 degree minus sin space x space sin space 45 degree right parenthesis end cell equals cell cos space x space cos space 45 degree plus sin space x space sin space 45 degree end cell row cell 2 left parenthesis cos space x times 1 half square root of 2 minus sin space x times 1 half square root of 2 right parenthesis end cell equals cell cos space x times 1 half square root of 2 plus sin space x times 1 half square root of 2 end cell row cell square root of 2 space cos space x minus square root of 2 space sin space x end cell equals cell 1 half square root of 2 space cos space x plus 1 half square root of 2 space sin space x end cell row cell square root of 2 space cos space x minus 1 half square root of 2 space cos space x end cell equals cell 1 half square root of 2 space sin space x plus square root of 2 space sin space x end cell row cell cos space x space left parenthesis square root of 2 minus 1 half square root of 2 right parenthesis end cell equals cell sin space x left parenthesis 1 half square root of 2 plus square root of 2 right parenthesis end cell row cell cos space x times 1 half square root of 2 end cell equals cell sin space x times 3 over 2 square root of 2 end cell row cell fraction numerator 1 half square root of 2 over denominator 3 over 2 square root of 2 end fraction end cell equals cell fraction numerator sin space x over denominator cos space x end fraction end cell row cell fraction numerator square root of 2 over denominator 2 end fraction times fraction numerator 2 over denominator 3 square root of 2 end fraction end cell equals cell tan space x end cell row cell 1 third end cell equals cell tan space x end cell end table end style$

Akan ditentukan tan space 2 x

$table attributes columnalign right center left columnspacing 0px end attributes row cell tan space 2 x end cell equals cell fraction numerator 2 space tan space x over denominator 1 minus tan squared space x end fraction end cell row blank equals cell fraction numerator 2 times begin display style 1 third end style over denominator 1 minus left parenthesis begin display style 1 third end style right parenthesis squared end fraction end cell row blank equals cell fraction numerator begin display style 2 over 3 end style over denominator 1 minus begin display style 1 over 9 end style end fraction end cell row blank equals cell fraction numerator begin display style 2 over 3 end style over denominator begin display style 8 over 9 end style end fraction end cell row blank equals cell 2 over 3 times 9 over 8 end cell row blank equals cell 3 over 4 end cell end table$

Jadi, nilai tan space 2 x adalah 3 over 4 .

Oleh karena itu, jawaban yang benar adalah D.

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