Tunjukan bahwa: c. (cos 2x−sin 2x)=1−sin x

Pertanyaan

Tunjukan bahwa:

c. $\left(\cos\;\frac x2-\sin\;\frac x2\right)=1-\sin\;x$

E. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sebelas Maret

Jawaban terverifikasi

Jawaban

terbukti bahwa, $\left(\cos\;\frac x2-\sin\;\frac x2\right)=1-\sin\;x$ .

Pembahasan

Ingat kembali rumus:

$table attributes columnalign right center left columnspacing 0px end attributes row cell sin space straight A over 2 end cell equals cell plus-or-minus square root of fraction numerator 1 minus cos space straight A over denominator 2 end fraction end root end cell row cell cos straight A over 2 end cell equals cell plus-or-minus space square root of fraction numerator 1 plus cos space straight A over denominator 2 end fraction end root end cell end table$

$1-\cos^2\mathrm A=\sin^2\;\mathrm A$

$\left(a-b\right)^2=a^2+b^2-2ab\\\left(a-b\right)\left(a+b\right)=a^2-b^2$

Akan ditunjukkan bahwa, $\left(\cos\;\frac x2-\sin\;\frac x2\right)=1-\sin\;x$ , kita asumsikan sudut $\frac x2$ berada di kuadran pertama, maka $\cos\;\frac x2$ dan $\sin\;\frac x2$ bernilai positif, sehingga:

$table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses cos space x over 2 minus sin space x over 2 close parentheses end cell row blank equals cell open parentheses square root of fraction numerator 1 plus cos space straight A over denominator 2 end fraction end root minus square root of fraction numerator 1 minus cos space straight A over denominator 2 end fraction end root close parentheses squared end cell row blank equals cell fraction numerator 1 plus cos space straight A over denominator 2 end fraction plus fraction numerator 1 minus cos space straight A over denominator 2 end fraction minus 2 open parentheses square root of fraction numerator 1 plus cos space straight A over denominator 2 end fraction end root times square root of fraction numerator 1 minus cos space straight A over denominator 2 end fraction end root close parentheses end cell row blank equals cell 2 over 2 minus 2 square root of fraction numerator open parentheses 1 plus cos space straight A close parentheses open parentheses 1 minus cos space straight A close parentheses over denominator 4 end fraction end root end cell row blank equals cell 1 minus 2 square root of fraction numerator 1 squared minus cos squared straight A over denominator 4 end fraction end root end cell row blank equals cell 1 minus 2 square root of fraction numerator sin squared straight A over denominator 4 end fraction end root end cell row blank equals cell 1 minus fraction numerator up diagonal strike 2 space sinA over denominator up diagonal strike 2 end fraction end cell row blank equals cell 1 minus sin space straight A space bold left parenthesis bold terbukti bold right parenthesis end cell end table$

Jadi, terbukti bahwa, $\left(\cos\;\frac x2-\sin\;\frac x2\right)=1-\sin\;x$ .