Jika tan(2x+45∘)=a,tan(x+30∘)=b,dan a⋅b∈{1,−1,2,−2},maka tan(3x+75∘)⋅tan(x+15∘)=…

Pertanyaan

Jika $\tan(2x+45^\circ)=a,\tan(x+30^\circ)=b,$ dan $a\cdot b\not\in\left\{1,-1,\sqrt2,-\sqrt2\right\},$ maka $\tan\left(3x+75^\circ\right)\cdot\tan\left(x+15^\circ\right)=\dots$

$fraction numerator a squared minus b squared over denominator 1 plus a squared b squared end fraction$
$fraction numerator a squared plus b squared over denominator 1 minus a squared b squared end fraction$
$fraction numerator a squared minus b squared over denominator 1 minus a squared b squared end fraction$
$fraction numerator a squared minus b squared over denominator 1 minus 2 a squared b squared end fraction$
$fraction numerator a squared plus b squared over denominator 1 plus 2 a squared b squared end fraction$

I. Roy

Master Teacher

Mahasiswa/Alumni Universitas Negeri Surabaya

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah C

Pembahasan

Ingat bahwa :

rumus jumlah atau selisih dua sudut tan

$tan open parentheses A plus-or-minus B close parentheses equals fraction numerator tan space A plus-or-minus tan space B over denominator 1 minus-or-plus space tan space A space tan space B end fraction$

Sehingga

$table attributes columnalign right center left columnspacing 0px end attributes row cell tan space open parentheses 3 x minus 30 degree close parentheses end cell equals cell tan space open parentheses open parentheses 2 x minus 45 close parentheses plus open parentheses x plus 15 degree close parentheses close parentheses end cell row blank equals cell fraction numerator tan open parentheses 2 x minus 45 close parentheses plus tan open parentheses x plus 15 degree close parentheses over denominator 1 minus tan space open parentheses 2 x minus 45 close parentheses times tan space open parentheses x plus 15 degree close parentheses end fraction end cell row blank equals cell fraction numerator a plus b over denominator 1 minus a times b end fraction end cell end table$

$table attributes columnalign right center left columnspacing 0px end attributes row cell tan space open parentheses x minus 60 degree close parentheses end cell equals cell tan space open parentheses open parentheses 2 x minus 45 close parentheses minus open parentheses x plus 15 degree close parentheses close parentheses end cell row blank equals cell fraction numerator tan open parentheses 2 x minus 45 close parentheses minus tan open parentheses x plus 15 degree close parentheses over denominator 1 plus tan space open parentheses 2 x minus 45 close parentheses times tan space open parentheses x plus 15 degree close parentheses end fraction end cell row blank equals cell fraction numerator a minus b over denominator 1 plus a times b end fraction end cell end table$

Maka

$table attributes columnalign right center left columnspacing 0px end attributes row cell tan open parentheses 3 x plus 75 degree close parentheses times tan open parentheses x plus 15 degree close parentheses end cell equals cell open parentheses fraction numerator a plus b over denominator 1 minus a b end fraction close parentheses open parentheses fraction numerator a minus b over denominator 1 plus a b end fraction close parentheses end cell row blank equals cell fraction numerator a squared minus b squared over denominator 1 minus a squared b squared end fraction end cell end table$

Jadi, jawaban yang tepat adalah C