2. Jika tanθ=71, nilai dari (cosec2θ+secan2θcosec2θ−secan2θ) adalah ....

Pertanyaan

2. Jika $\tan\;\theta=\frac1{\sqrt7}$ , nilai dari $\left(\frac{\mathrm{cosec}^2\mathrm\theta-\mathrm{secan}^2\mathrm\theta}{\mathrm{cosec}^2\mathrm\theta+\mathrm{secan}^2\mathrm\theta}\right)$ adalah ....

$1 half$
$3 over 7$
$5 over 7$
$3 over 4$
$1$

Pembahasan Soal:

Terlebih dahulu kita definisikan $tan space theta equals fraction numerator 1 over denominator square root of 7 end fraction$ pada segitiga siku-siku. Perhatikan gambar berikut.

Ingat kembali teorema pythagoras untuk mencari panjang BC, sebagai berikut:

$table attributes columnalign right center left columnspacing 0px end attributes row BC equals cell square root of open parentheses square root of 7 close parentheses squared plus 1 squared end root end cell row blank equals cell square root of 7 plus 1 end root end cell row blank equals cell square root of 8 end cell end table$

Sehingga kita dapat mencari nilai cosec $theta$ dan secan $theta$ dengan perbandingan sisi pada siku-siku tersebut.

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos e c space theta end cell equals cell fraction numerator 1 over denominator sin space theta end fraction end cell row blank equals cell fraction numerator 1 over denominator begin display style fraction numerator 1 over denominator square root of 8 end fraction end style end fraction end cell row blank equals cell square root of 8 end cell end table$

$table attributes columnalign right center left columnspacing 0px end attributes row cell secan space theta end cell equals cell fraction numerator 1 over denominator cos space theta end fraction end cell row blank equals cell fraction numerator 1 over denominator begin display style fraction numerator square root of 7 over denominator square root of 8 end fraction end style end fraction end cell row blank equals cell fraction numerator square root of 8 over denominator square root of 7 end fraction end cell end table$

Sehingga diperoleh:

$table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses fraction numerator cosec squared straight theta minus secan squared straight theta over denominator cosec squared straight theta plus secan squared straight theta end fraction close parentheses end cell equals cell fraction numerator open parentheses square root of 8 close parentheses squared minus open parentheses begin display style fraction numerator square root of 8 over denominator square root of 7 end fraction end style close parentheses squared over denominator open parentheses square root of 8 close parentheses squared plus open parentheses begin display style fraction numerator square root of 8 over denominator square root of 7 end fraction end style close parentheses squared end fraction end cell row blank equals cell fraction numerator 8 minus begin display style 8 over 7 end style over denominator 8 plus begin display style 8 over 7 end style end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 56 minus 8 over denominator up diagonal strike 7 end fraction end style over denominator begin display style fraction numerator 56 plus 8 over denominator up diagonal strike 7 end fraction end style end fraction end cell row blank equals cell 48 over 64 end cell row blank equals cell 3 over 4 end cell end table$

Dengan demikian, nilai dari $open parentheses fraction numerator cosec squared straight theta minus secan squared straight theta over denominator cosec squared straight theta plus secan squared straight theta end fraction close parentheses equals 3 over 4$ .

Oleh karena itu, jawaban yang benar adalah D.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

S. Dwi

Terakhir diupdate 07 Oktober 2021