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Pertanyaan

Jika A plus B plus C plus D equals 2 straight pi, tunjukkan bahwa.

sin space A minus sin space B plus sin space C minus sin space D equals negative 4 space cos space open parentheses fraction numerator A plus B over denominator 2 end fraction close parentheses sin space open parentheses fraction numerator A plus C over denominator 2 end fraction close parentheses cos open parentheses fraction numerator A plus D over denominator 2 end fraction close parentheses

I. Roy

Master Teacher

Mahasiswa/Alumni Universitas Negeri Surabaya

Jawaban terverifikasi

Pembahasan

Ingat bahwa :

Rumus jumlah dan selisihTrigonometri 

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

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 space open parentheses negative alpha close parentheses end cell equals cell cos open parentheses alpha close parentheses end cell end table

Dari soal diketahui

table attributes columnalign right center left columnspacing 0px end attributes row cell A plus B plus C plus D end cell equals cell 2 straight pi end cell row cell C plus D end cell equals cell 2 straight pi minus open parentheses straight A plus straight B close parentheses end cell row cell cos space open parentheses fraction numerator C plus D over denominator 2 end fraction close parentheses end cell equals cell cos open parentheses fraction numerator 2 straight pi minus open parentheses A plus B close parentheses over denominator 2 end fraction close parentheses end cell row cell cos space open parentheses fraction numerator C plus D over denominator 2 end fraction close parentheses end cell equals cell cos space open parentheses straight pi minus open parentheses fraction numerator straight A plus straight B over denominator 2 end fraction close parentheses close parentheses end cell row blank equals cell negative cos open parentheses fraction numerator straight A plus straight B over denominator 2 end fraction close parentheses end cell end table

Sehingga 

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

Dengan demikian benar bahwa sin space A minus sin space B plus sin space C minus sin space D equals negative 4 space cos space open parentheses fraction numerator A plus B over denominator 2 end fraction close parentheses sin space open parentheses fraction numerator A plus C over denominator 2 end fraction close parentheses cos open parentheses fraction numerator A plus D over denominator 2 end fraction close parentheses

 

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Jika A+B+C+D=2π, tunjukkan bahwa. cosA−cosB+cosC−cosD=4sin(2A+B​)sin(2A+D​)cos(2A+C​)

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