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Tentukan himpunan penyelesaian dari persamaan-persamaan trigonometri berikut! b. cos (3x−15∘)=21​3​, 0∘≤x≤360∘

Pertanyaan

Tentukan himpunan penyelesaian dari persamaan-persamaan trigonometri berikut!

b. cos space left parenthesis 3 x minus 15 degree right parenthesis equals 1 half square root of 3 comma space 0 degree less or equal than x less or equal than 360 degree space

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Jawaban terverifikasi

Jawaban

HP equals left curly bracket 15 degree comma space 115 degree comma space 135 degree comma space 235 degree comma space 255 degree comma space 355 degree right curly bracket.

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah bold HP bold equals bold left curly bracket bold 15 bold degree bold comma bold space bold 115 bold degree bold comma bold space bold 135 bold degree bold comma bold space bold 235 bold degree bold comma bold space bold 255 bold degree bold comma bold space bold 355 bold degree bold right curly bracket.

Ingat cos space x equals a diubah dahulu menjadi cos space x equals cos space alpha.

Jika cos space x equals cos space alpha, maka

  • x equals alpha plus k times 360 degree atau
  • x equals negative alpha plus k times 360 degree

Diketahui cos space left parenthesis 3 x minus 15 degree right parenthesis equals 1 half square root of 3 untuk 0 degree less or equal than x less or equal than 360 degree, maka

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space left parenthesis 3 x minus 15 degree right parenthesis end cell equals cell 1 half square root of 3 end cell row cell cos space left parenthesis 3 x minus 15 degree right parenthesis end cell equals cell cos space 30 degree end cell end table

Berdasarkan rumus di atas, dapat ditentukan:

  • table attributes columnalign right center left columnspacing 0px end attributes row cell bold 3 bold italic x bold minus bold 15 bold degree end cell bold equals cell bold 30 bold degree bold plus bold italic k bold times bold 360 bold degree end cell row cell 3 x end cell equals cell 45 degree plus k times 360 degree end cell row x equals cell 15 degree plus k times 120 degree end cell end table

           untuk k bilangan bulat, maka

table attributes columnalign right center left columnspacing 0px end attributes row k equals cell 0 comma space x equals 15 degree plus 0 times 120 degree equals 15 degree space left parenthesis memenuhi right parenthesis end cell row k equals cell 1 comma space x equals 15 degree plus 1 times 120 degree equals 135 degree space left parenthesis memenuhi right parenthesis end cell row k equals cell 2 comma space x equals 15 degree plus 2 times 120 degree equals 255 degree space left parenthesis memenuhi right parenthesis end cell row k equals cell 3 comma space x equals 15 degree plus 3 times 120 degree equals 375 degree space left parenthesis tidak space memenuhi right parenthesis end cell end table

  • table attributes columnalign right center left columnspacing 0px end attributes row cell bold 3 bold italic x bold minus bold 15 bold degree end cell bold equals cell bold minus bold 30 bold degree bold plus bold italic k bold times bold 360 bold degree end cell row cell 3 x end cell equals cell negative 15 degree plus k times 360 degree end cell row x equals cell negative 5 degree plus k times 120 degree end cell end table

           untuk k bilangan bulat, maka

table attributes columnalign right center left columnspacing 0px end attributes row k equals cell 0 comma space x equals negative 5 degree plus 0 times 120 degree equals negative 5 degree space left parenthesis tidak space memenuhi right parenthesis end cell row k equals cell 1 comma space x equals negative 5 degree plus 1 times 120 degree equals 115 degree space left parenthesis memenuhi right parenthesis end cell row k equals cell 2 comma space x equals negative 5 degree plus 2 times 120 degree equals 235 degree space left parenthesis memenuhi right parenthesis end cell row k equals cell 3 comma space x equals negative 5 degree plus 3 times 120 degree equals 355 degree space left parenthesis memenuhi right parenthesis end cell row k equals cell 4 comma space x equals negative 5 degree plus 4 times 120 degree equals 475 degree space left parenthesis tidak space memenuhi right parenthesis end cell end table    

Dengan demikian, HP equals left curly bracket 15 degree comma space 115 degree comma space 135 degree comma space 235 degree comma space 255 degree comma space 355 degree right curly bracket.

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