Pertanyaan

Jika 3 tan θ = x + 1 dan 0 < θ < 2 π , nyatakan ekspresi 2 θ + 1 − tan 2 θ 2 tan θ dalam fungsi dari x

Jika $3 tan θ = x + 1$ dan $0 < θ < \frac{π}{2}$ , nyatakan ekspresi $2 θ + \frac{2 tan θ}{1 - tan ^{2} θ}$ dalam fungsi dari $x$

W. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sriwijaya

Jawaban terverifikasi

Jawaban

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$table attributes columnalign right center left columnspacing 0px end attributes row cell 2 theta plus fraction numerator 2 space tan space theta over denominator 1 minus tan squared space theta end fraction end cell equals cell 2 open square brackets tan to the power of negative 1 end exponent space open parentheses fraction numerator x plus 1 over denominator 3 end fraction close parentheses close square brackets plus fraction numerator 6 x plus 6 over denominator 8 minus x squared minus 2 x end fraction end cell end table$ .

Pembahasan

Jika dan , maka: Interval berada pada kuadran I, berarti seluruh nilai perbandingan trigonometri bernilai positif. Sehingga ekspresi dalam fungsi dari dapat ditentukan dengan cara: Jadi, .

Jika 3 space tan space theta equals x plus 1 dan 0 less than theta less than straight pi over 2 , maka:

$table attributes columnalign right center left columnspacing 0px end attributes row cell 3 space tan space theta end cell equals cell x plus 1 end cell row cell tan space theta end cell equals cell fraction numerator x plus 1 over denominator 3 end fraction end cell row theta equals cell tan to the power of negative 1 end exponent space open parentheses fraction numerator x plus 1 over denominator 3 end fraction close parentheses end cell end table$

Interval 0 less than theta less than straight pi over 2 berada pada kuadran I, berarti seluruh nilai perbandingan trigonometri bernilai positif. Sehingga ekspresi $2 theta plus fraction numerator 2 space tan space theta over denominator 1 minus tan squared space theta end fraction$ dalam fungsi dari dapat ditentukan dengan cara:

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell 2 theta plus fraction numerator 2 space tan space theta over denominator 1 minus tan squared space theta end fraction end cell equals cell 2 open square brackets tan to the power of negative 1 end exponent space open parentheses fraction numerator x plus 1 over denominator 3 end fraction close parentheses close square brackets plus fraction numerator 2 space open parentheses begin display style fraction numerator x plus 1 over denominator 3 end fraction end style close parentheses over denominator 1 minus open parentheses begin display style fraction numerator x plus 1 over denominator 3 end fraction end style close parentheses squared end fraction end cell row blank equals cell 2 open square brackets tan to the power of negative 1 end exponent space open parentheses fraction numerator x plus 1 over denominator 3 end fraction close parentheses close square brackets plus fraction numerator begin display style fraction numerator 2 open parentheses x plus 1 close parentheses over denominator 3 end fraction end style over denominator 1 minus begin display style fraction numerator x squared plus 2 x plus 1 over denominator 9 end fraction end style end fraction end cell row blank equals cell 2 open square brackets tan to the power of negative 1 end exponent space open parentheses fraction numerator x plus 1 over denominator 3 end fraction close parentheses close square brackets plus fraction numerator begin display style fraction numerator 2 x plus 2 over denominator 3 end fraction end style over denominator begin display style fraction numerator 9 minus open parentheses x squared plus 2 x plus 1 close parentheses over denominator 9 end fraction end style end fraction end cell row blank equals cell 2 open square brackets tan to the power of negative 1 end exponent space open parentheses fraction numerator x plus 1 over denominator 3 end fraction close parentheses close square brackets plus open parentheses fraction numerator 2 x plus 2 over denominator 3 end fraction close parentheses times open parentheses fraction numerator 9 over denominator 9 minus open parentheses x squared plus 2 x plus 1 close parentheses end fraction close parentheses end cell row blank equals cell 2 open square brackets tan to the power of negative 1 end exponent space open parentheses fraction numerator x plus 1 over denominator 3 end fraction close parentheses close square brackets plus open parentheses fraction numerator 2 x plus 2 over denominator 3 end fraction close parentheses times open parentheses fraction numerator 9 over denominator 9 minus x squared minus 2 x minus 1 end fraction close parentheses end cell row blank equals cell 2 open square brackets tan to the power of negative 1 end exponent space open parentheses fraction numerator x plus 1 over denominator 3 end fraction close parentheses close square brackets plus open parentheses fraction numerator 2 x plus 2 over denominator up diagonal strike 3 end fraction close parentheses times open parentheses fraction numerator up diagonal strike 9 space 3 over denominator 8 minus x squared minus 2 x end fraction close parentheses end cell row blank equals cell 2 open square brackets tan to the power of negative 1 end exponent space open parentheses fraction numerator x plus 1 over denominator 3 end fraction close parentheses close square brackets plus fraction numerator 6 x plus 6 over denominator 8 minus x squared minus 2 x end fraction end cell end table end style$

Jadi, $table attributes columnalign right center left columnspacing 0px end attributes row cell 2 theta plus fraction numerator 2 space tan space theta over denominator 1 minus tan squared space theta end fraction end cell equals cell 2 open square brackets tan to the power of negative 1 end exponent space open parentheses fraction numerator x plus 1 over denominator 3 end fraction close parentheses close square brackets plus fraction numerator 6 x plus 6 over denominator 8 minus x squared minus 2 x end fraction end cell end table$ .