Pertanyaan

Buktikan: c. sin α + sin ( α + 3 2 π ) + sin ( α + 3 4 π ) = 0 .

Buktikan:
c. $sin α + sin (α + \frac{2}{3} π) + sin (α + \frac{4}{3} π) = 0$ .

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Klaim

R. Febrianti

Master Teacher

Mahasiswa/Alumni Universitas Negeri Malang

Jawaban terverifikasi

Jawaban

terbukti bahwa .

terbukti bahwa sin space alpha plus sin open parentheses alpha plus 2 over 3 straight pi close parentheses plus sin open parentheses alpha plus 4 over 3 straight pi close parentheses equals 0

sin space alpha plus sin open parentheses alpha plus 2 over 3 straight pi close parentheses plus sin open parentheses alpha plus 4 over 3 straight pi close parentheses equals 0

Pembahasan

Ingat rumus jumlah dua sudut pada sinus yaitu Karena terletak pada kuadran II maka nilai sehingga sebagai berikut. Karena dan terletak pada kuadran IIImaka nilai dan sehingga yaitu Maka sebagai berikut. Jadi terbukti bahwa .

Ingat rumus jumlah dua sudut pada sinus yaitu

sin open parentheses text A end text plus text B end text close parentheses equals sin space text A end text times cos space text B end text plus cos space text A end text space sin space text B end text

Karena cos space 2 over 3 straight pi terletak pada kuadran II maka nilai sehingga sin open parentheses alpha plus 2 over 3 straight pi close parentheses sebagai berikut.

$table attributes columnalign right center left columnspacing 0px end attributes row cell sin open parentheses alpha plus 2 over 3 straight pi close parentheses end cell equals cell sin space alpha cos space 2 over 3 straight pi plus cos space αsin space 2 over 3 straight pi end cell row blank equals cell sin space αcos space fraction numerator 2 times begin display style stack up diagonal strike 180 with 60 on top end style over denominator up diagonal strike 3 end fraction plus cos space αsin space fraction numerator 2 times begin display style stack up diagonal strike 180 with 60 on top end style over denominator up diagonal strike 3 end fraction end cell row blank equals cell sin space αcos space 120 degree plus cos space αsin space 120 degree end cell row blank equals cell sin space straight alpha open parentheses negative 1 half close parentheses plus cos space straight alpha 1 half square root of 3 end cell row blank equals cell negative 1 half sin space straight alpha plus 1 half square root of 3 cos space straight alpha end cell end table$

Karena cos space 4 over 3 straight pi dan terletak pada kuadran III maka nilai cos space 2 over 3 straight pi equals negative cos space 1 third straight pi dan sin space 4 over 3 straight pi equals negative sin space 1 third straight pi sehingga sin open parentheses alpha plus 4 over 3 straight pi close parentheses yaitu

$table attributes columnalign right center left columnspacing 0px end attributes row cell sin open parentheses alpha plus 2 over 3 straight pi close parentheses end cell equals cell sin space alpha cos space 4 over 3 straight pi plus cos space αsin space 4 over 3 straight pi end cell row blank equals cell sin space αcos space fraction numerator 4 times begin display style stack up diagonal strike 180 with 60 on top end style over denominator up diagonal strike 3 end fraction plus cos space αsin space fraction numerator 4 times begin display style stack up diagonal strike 180 with 60 on top end style over denominator up diagonal strike 3 end fraction end cell row blank equals cell sin space αcos space 240 degree plus cos space αsin space 240 degree end cell row blank equals cell sin space straight alpha open parentheses negative 1 half close parentheses plus cos space straight alpha open parentheses negative 1 half square root of 3 close parentheses end cell row blank equals cell negative 1 half sin space straight alpha minus 1 half square root of 3 cos space straight alpha end cell end table$

Maka sin space alpha plus sin open parentheses alpha plus 2 over 3 straight pi close parentheses plus sin open parentheses alpha plus 4 over 3 straight pi close parentheses sebagai berikut.

Jadi terbukti bahwa sin space alpha plus sin open parentheses alpha plus 2 over 3 straight pi close parentheses plus sin open parentheses alpha plus 4 over 3 straight pi close parentheses equals 0 .