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Pertanyaan

Jika A, B, dan C merupakan sudut-sudut dalam sebuah segitiga ABC dan cos space theta open parentheses sin space B plus sin space C close parentheses equals sin space ABuktikan bahwa  

tan to the power of 2 space end exponent theta over 2 equals tan open parentheses B over 2 close parentheses tan open parentheses C over 2 close parentheses  

W. Wati

Master Teacher

Jawaban terverifikasi

Pembahasan

Ingat bahwa :

Sudut berelasi

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space open parentheses 90 degree minus alpha close parentheses end cell equals cell cos space alpha end cell row cell sin open parentheses 180 degree minus alpha close parentheses end cell equals cell sin space alpha end cell row cell sin open parentheses negative alpha close parentheses end cell equals cell negative sin space alpha end cell row cell cos open parentheses negative alpha close parentheses end cell equals cell cos space alpha end cell end table

Identitas trigonometri

tan space A equals fraction numerator sin space A over denominator cos space A end fraction

Rumus jumlah dan selisih Trigonometri yaitu

sin space A plus sin space B equals 2 space sin 1 half open parentheses A plus B close parentheses cos 1 half open parentheses A minus B close parentheses cos space A plus cos space B equals 2 space cos 1 half open parentheses A plus B close parentheses cos 1 half open parentheses A minus B close parentheses

Sudut rangkap pada sinus 

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

Rumus sudut setenga pada tangen

tan squared alpha over 2 equals fraction numerator 1 minus space cos space alpha over denominator 1 plus space sin space alpha end fraction

Pada segitiga ABC, maka berlaku

table attributes columnalign right center left columnspacing 0px end attributes row cell A plus B plus C end cell equals cell 180 degree end cell row cell A plus B end cell equals cell 180 degree minus C end cell row C equals cell 180 degree minus open parentheses A plus B close parentheses end cell row cell sin space C over 2 end cell equals cell sin open parentheses fraction numerator 180 degree minus open parentheses A plus B close parentheses over denominator 2 end fraction close parentheses end cell row blank equals cell sin space open parentheses 90 degree minus open parentheses fraction numerator A plus B over denominator 2 end fraction close parentheses close parentheses end cell row blank equals cell cos open parentheses fraction numerator A plus B over denominator 2 end fraction close parentheses end cell row cell sin open parentheses fraction numerator A plus B over denominator 2 end fraction close parentheses end cell equals cell sin space open parentheses fraction numerator 180 degree minus C over denominator 2 end fraction close parentheses end cell row blank equals cell sin open parentheses 90 degree minus C close parentheses end cell row blank equals cell cos C over 2 end cell end table

Dari soal diketahui

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space theta open parentheses sin space B plus sin space C close parentheses end cell equals cell sin space A end cell row cell cos space theta end cell equals cell fraction numerator sin space A over denominator sin space B plus sin space C end fraction end cell end table

Sebelumnya akan ditentukan terlebih dahulu hasil dari sin space A plus sin space B plus sin space C

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell sin space A plus sin space B plus sin space C end cell row blank equals cell 2 space sin open parentheses fraction numerator A plus B over denominator 2 end fraction close parentheses cos open parentheses fraction numerator A minus B over denominator 2 end fraction close parentheses plus sin space C end cell row blank equals cell 2 space cos C over 2 cos open parentheses fraction numerator A minus B over denominator 2 end fraction close parentheses plus 2 space sin C over 2 cos C over 2 end cell row blank equals cell 2 space cos space C over 2 open square brackets cos space open parentheses fraction numerator A minus B over denominator 2 end fraction close parentheses plus sin space C over 2 close square brackets end cell row blank equals cell 2 space cos space C over 2 open square brackets cos space open parentheses fraction numerator A minus B over denominator 2 end fraction close parentheses plus cos open parentheses fraction numerator A plus B over denominator 2 end fraction close parentheses close square brackets end cell row blank equals cell 2 space cos space C over 2 open square brackets 2 space cos 1 half open parentheses fraction numerator A minus B plus A plus B over denominator 2 end fraction close parentheses cos 1 half open parentheses fraction numerator A minus B minus A minus B over denominator 2 end fraction close parentheses close square brackets end cell row blank equals cell 2 space cos C over 2 open square brackets 2 space cos open parentheses fraction numerator 2 A over denominator 4 end fraction close parentheses cos open parentheses fraction numerator negative 2 B over denominator 4 end fraction close parentheses close square brackets end cell row blank equals cell 4 space cos A over 2 cos B over 2 cos C over 2 end cell end table

Dengan cara yang sama diperoleh

sin space B plus sin space C minus sin space A equals 4 space cos A over 2 sin B over 2 sin C over 2

Maka

table attributes columnalign right center left columnspacing 0px end attributes row cell tan to the power of 2 space end exponent theta over 2 end cell equals cell fraction numerator 1 minus cos theta over denominator 1 plus sin space theta end fraction end cell row blank equals cell fraction numerator 1 minus begin display style fraction numerator sin space A over denominator sin space B plus sin space C end fraction end style over denominator 1 plus begin display style fraction numerator sin space A over denominator sin space B plus end fraction end style end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator sin space B plus sin space C minus sin space A over denominator up diagonal strike sin space B plus sin space C end strike end fraction end style over denominator begin display style fraction numerator sin space B plus sin space C plus sin space A over denominator up diagonal strike sin space B plus sin space C end strike end fraction end style end fraction end cell row blank equals cell fraction numerator sin space B plus sin space C minus sin space A over denominator sin space B plus sin space C plus sin space A end fraction end cell row blank equals cell fraction numerator up diagonal strike up diagonal strike 4 space cos space begin display style A over 2 end style end strike space sin begin display style B over 2 end style space sin begin display style C over 2 end style over denominator up diagonal strike up diagonal strike up diagonal strike 4 space cos begin display style A over 2 end style end strike end strike space co s begin display style B over 2 end style cos space begin display style C over 2 end style end fraction end cell row blank equals cell tan open parentheses B over 2 close parentheses tan open parentheses C over 2 close parentheses open parentheses terbukti close parentheses end cell end table

Dengan demikian terbukti bahwa tan to the power of 2 space end exponent theta over 2 equals tan open parentheses B over 2 close parentheses tan open parentheses C over 2 close parentheses

 

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