Nilai x→21πlim1−sin2 x(sin 21x−cos 21x)2 adalah ....

Pertanyaan

Nilai $limit as x rightwards arrow 1 half straight pi of fraction numerator open parentheses sin space begin display style 1 half end style x minus cos space 1 half x close parentheses squared over denominator 1 minus sin squared space x end fraction$ adalah ....

$-1$
$-0,5$
$0$
$0,5$
$1$

P. Tessalonika

Master Teacher

Mahasiswa/Alumni Universitas Negeri Medan

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah D.

Pembahasan

Ingat identitas trigonometri yaitu : $\sin^2\;x+\cos^2\;x=1$ .

Ingat bahwa $\sin\;2x=2\;\sin\;x\;\cos\;x$ maka berlaku $\sin\;x=2\;\sin\;\frac12x\;\cos\;\frac12x$ , akibatnya diperoleh :

$limit as x rightwards arrow 1 half straight pi of fraction numerator open parentheses sin space begin display style 1 half end style x minus cos space begin display style 1 half end style x close parentheses squared over denominator 1 minus sin squared space x end fraction equals limit as x rightwards arrow 1 half straight pi of fraction numerator sin squared open parentheses begin display style 1 half end style x close parentheses plus cos squared space open parentheses begin display style 1 half end style x close parentheses minus 2 space sin space begin display style 1 half end style x space cos space begin display style 1 half end style x over denominator 1 minus sin squared space x end fraction equals limit as x rightwards arrow 1 half straight pi of fraction numerator 1 minus 2 space sin space begin display style 1 half end style x space cos space begin display style 1 half end style x over denominator 1 minus sin squared space x end fraction equals limit as x rightwards arrow 1 half straight pi of fraction numerator 1 minus sin space x over denominator 1 minus sin squared space x end fraction equals limit as x rightwards arrow 1 half straight pi of fraction numerator up diagonal strike 1 minus sin space x end strike over denominator up diagonal strike open parentheses 1 minus sin space x close parentheses end strike open parentheses 1 plus sin space x close parentheses end fraction equals limit as x rightwards arrow 1 half straight pi of fraction numerator 1 over denominator open parentheses 1 plus sin space x close parentheses end fraction equals fraction numerator 1 over denominator 1 plus sin space open parentheses begin display style 1 half end style straight pi close parentheses end fraction equals fraction numerator 1 over denominator 1 plus 1 end fraction equals 1 half equals 0 comma 5$

Dengan demikian, nilai dari $begin mathsize 12px style limit as x rightwards arrow 1 half straight pi of fraction numerator open parentheses sin space begin display style 1 half end style x minus cos space 1 half x close parentheses squared over denominator 1 minus sin squared space x end fraction equals 0 comma 5 end style$ .

Oleh karena itu, jawaban yang tepat adalah D.