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19. Buktikan identitas berikut.

c. fraction numerator sin space alpha plus sin space 2 alpha plus sin space 3 alpha over denominator cos space alpha plus cos space 2 alpha plus cos space 3 alpha end fraction equals space tan space 2 alpha 

P. Tessalonika

Master Teacher

Mahasiswa/Alumni Universitas Negeri Medan

Jawaban terverifikasi

Pembahasan

Ingat rumus identitas penjumlahan sinus dan cosinus berikut:

  • sin space x plus sin space y equals 2 space sin space 1 half open parentheses x plus y close parentheses space cos space 1 half open parentheses x minus y close parentheses 
  • cos space x plus cos space y equals 2 space cos space 1 half open parentheses x plus y close parentheses space cos space 1 half open parentheses x minus y close parentheses 

Terlebih dahulu kita selesaikan bagian open parentheses sin space alpha plus sin space 3 alpha close parentheses dan open parentheses cos space alpha plus cos space 3 alpha close parentheses dengan rumus tersebut, diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 3 alpha plus sin space alpha end cell equals cell 2 space sin space 1 half open parentheses 3 alpha plus alpha close parentheses space cos space 1 half open parentheses 3 alpha minus alpha close parentheses end cell row blank equals cell 2 space sin space 1 half open parentheses 4 alpha close parentheses space cos space 1 half open parentheses 2 alpha close parentheses end cell row blank equals cell 2 space sin space 2 alpha space cos space alpha end cell end table 

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 3 alpha plus space cos space alpha end cell equals cell 2 space cos space 1 half open parentheses 3 alpha plus alpha close parentheses space cos space 1 half open parentheses 3 alpha minus alpha close parentheses end cell row blank equals cell 2 space cos space 1 half open parentheses 4 alpha close parentheses space cos space 1 half open parentheses 2 alpha close parentheses end cell row blank equals cell 2 space cos space 2 alpha space cos space alpha end cell end table 

Sehingga soal tersebut akan berubah menjadi:

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator sin space alpha plus sin space 2 alpha plus sin space 3 alpha over denominator cos space alpha plus cos space 2 alpha plus cos space 3 alpha end fraction end cell equals cell fraction numerator sin space 3 alpha plus sin space alpha plus sin space 2 alpha over denominator cos space 3 alpha plus cos space alpha plus cos space 2 alpha end fraction end cell row blank equals cell fraction numerator 2 space sin space 2 alpha space cos space alpha plus sin space 2 alpha over denominator 2 space cos space 2 alpha space cos space alpha plus cos space 2 alpha end fraction end cell row blank equals cell fraction numerator sin space 2 alpha up diagonal strike open parentheses 2 space cos space alpha plus 1 close parentheses end strike over denominator cos space 2 alpha up diagonal strike open parentheses 2 space cos space alpha plus 1 close parentheses end strike end fraction end cell row blank equals cell tan space 2 alpha end cell end table 

Dengan demikian,

terbukti bahwa fraction numerator sin space alpha plus sin space 2 alpha plus sin space 3 alpha over denominator cos space alpha plus cos space 2 alpha plus cos space 3 alpha end fraction equals space tan space 2 alpha.

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