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Pertanyaan

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

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

I. Roy

Master Teacher

Mahasiswa/Alumni Universitas Negeri Surabaya

Jawaban terverifikasi

Jawaban

benar bahwa S i n space A plus sin space B plus sin space C equals 4 space cos space open parentheses A over 2 close parentheses cos space open parentheses B over 2 close parentheses space cos space open parentheses C over 2 close parentheses

Pembahasan

Ingat bahwa :

Rumus jumlah dan selisihTrigonometri yaitu

c o s space A minus cos space B equals negative 2 space sin 1 half open parentheses A plus B close parentheses sin 1 half open parentheses A minus B close parentheses sin space A plus sin space B equals 2 space sin space 1 half open parentheses A plus B close parentheses cos 1 half open parentheses A minus B close parentheses

Sudut berelasi 

sin space open parentheses 90 degree minus alpha close parentheses equals cos space alpha

sin space open parentheses 180 degree minus alpha close parentheses equals sin space alpha

sin open parentheses negative alpha close parentheses equals negative sin space alpha cos open parentheses negative alpha close parentheses equals cos space alpha

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 A plus B close parentheses end cell row cell B plus D end cell equals cell 2 straight pi minus open parentheses straight A plus straight C close parentheses end cell row cell straight B plus straight C end cell equals cell 2 straight pi minus open parentheses straight A plus straight D close parentheses end cell row cell sin space open parentheses fraction numerator C plus D over denominator 2 end fraction close parentheses end cell equals cell sin 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 blank equals cell sin open parentheses 180 degree minus open parentheses fraction numerator A plus B over denominator 2 end fraction close parentheses close parentheses end cell row blank equals cell sin space open parentheses fraction numerator A plus B over denominator 2 end fraction close parentheses end cell end table

Sehingga 

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

Dengan demikian benar bahwa S i n space A plus sin space B plus sin space C equals 4 space cos space open parentheses A over 2 close parentheses cos space open parentheses B over 2 close parentheses space cos space open parentheses C over 2 close parentheses

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

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