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Pertanyaan

Pada triangle A B C selalu berlaku hubungan: a space sin squared open parentheses C over 2 close parentheses plus c space sin squared space open parentheses A over 2 close parentheses equals b over 2. Buktikan bahwa a minus b plus c equals b

I. Roy

Master Teacher

Mahasiswa/Alumni Universitas Negeri Surabaya

Jawaban terverifikasi

Jawaban

terbukti bahwa pada hubungan a space sin squared open parentheses C over 2 close parentheses plus c space sin squared space open parentheses A over 2 close parentheses equals b over 2 terbukti a minus b plus c equals b

Pembahasan

Ingat bahwa:

Identitas pada trigonometri

sin squared A plus cos squared A equals 1

Besar sudut segitiga dengan aturan cosinus

cos space A equals fraction numerator b squared plus c squared minus a squared over denominator 2 b c end fraction cos space C equals fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction

Rumus sudut rangkap pada cosinus

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 2 A end cell equals cell 2 space cos squared A minus 1 end cell row cell 1 plus cos space 2 A end cell equals cell 2 space cos squared A end cell row cell 1 plus cos space A end cell equals cell 2 space cos squared space 1 half A end cell end table

Dari soal diketahui

a space sin squared open parentheses C over 2 close parentheses plus c space sin squared space open parentheses A over 2 close parentheses equals b over 2

Akan dibuktikan a minus b plus c equals b

Jawab:

table attributes columnalign right center left columnspacing 0px end attributes row cell a space sin squared open parentheses C over 2 close parentheses plus c space sin squared space open parentheses A over 2 close parentheses end cell equals cell b over 2 end cell row cell a open parentheses 1 minus cos squared C over 2 close parentheses plus c space open parentheses 1 minus cos squared A over 2 close parentheses end cell equals cell b over 2 open parentheses Kalikan space kedua space ruas space dengan space 2 close parentheses end cell row cell 2 a open parentheses 1 minus cos squared C over 2 close parentheses plus 2 c open parentheses 1 minus cos squared A over 2 close parentheses end cell equals b row cell 2 a minus 2 a space cos squared C over 2 plus 2 c minus 2 c space cos squared space A over 2 end cell equals b row cell 2 a minus a open parentheses 2 space cos squared C over 2 close parentheses plus 2 c minus c open parentheses space cos squared space A over 2 close parentheses end cell equals b row cell 2 a minus a open parentheses 1 plus cos space C close parentheses plus 2 c minus c open parentheses 1 plus cos space A space close parentheses end cell equals b row cell 2 a minus a minus a space cos space C plus 2 c minus c minus c space cos space A end cell equals b row cell a minus a space cos space C plus c minus c space cos space A end cell equals b row cell a minus up diagonal strike a open parentheses fraction numerator a squared plus b squared minus c squared over denominator 2 up diagonal strike a b end fraction close parentheses plus c minus up diagonal strike c open parentheses fraction numerator b squared plus c squared minus a squared over denominator 2 b up diagonal strike c end fraction close parentheses end cell equals cell b space open parentheses kalikan space kedua space ruas space dengan space 2 straight b close parentheses end cell row cell 2 a b minus up diagonal strike a squared end strike minus b squared plus up diagonal strike c squared end strike plus 2 b c minus b squared minus up diagonal strike c squared end strike plus up diagonal strike a squared end strike end cell equals cell 2 b squared end cell row cell 2 a b minus 2 b squared plus 2 b c end cell equals cell 2 b squared open parentheses bagi space kedua space ruas space dengan space 2 straight b close parentheses end cell row cell a minus b plus c end cell equals cell b open parentheses terbukti close parentheses end cell end table

Dengan demikian terbukti bahwa pada hubungan a space sin squared open parentheses C over 2 close parentheses plus c space sin squared space open parentheses A over 2 close parentheses equals b over 2 terbukti a minus b plus c equals b

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