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Tentukan himpunan penyelesaian dari persamaan-persamaan trigonometri berikut! sin 5x=21​2​, 0∘≤x≤360∘

Pertanyaan

Tentukan himpunan penyelesaian dari persamaan-persamaan trigonometri berikut!

  1. sin space 5 x equals 1 half square root of 2 comma space 0 degree less or equal than x less or equal than 360 degree space

A. Acfreelance

Master Teacher

Jawaban terverifikasi

Jawaban

begin mathsize 12px style HP equals left curly bracket 9 degree comma space 27 degree comma space 81 degree comma space 99 degree comma space 153 degree comma space 171 degree comma space 225 degree comma space 242 degree comma space 297 degree comma space 315 degree right curly bracket end style.

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah begin mathsize 12px style bold HP bold equals bold left curly bracket bold 9 bold degree bold comma bold space bold 27 bold degree bold comma bold space bold 81 bold degree bold comma bold space bold 99 bold degree bold comma bold space bold 153 bold degree bold comma bold space bold 171 bold degree bold comma bold space bold 225 bold degree bold comma bold space bold 242 bold degree bold comma bold space bold 297 bold degree bold comma bold space bold 315 bold degree bold right curly bracket end style.

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

Jika sin space x equals sin space alpha, maka

  • x equals alpha plus k times 360 degree atau
  • x equals left parenthesis 180 degree minus alpha right parenthesis plus k times 360 degree

Diketahui sin space 5 x equals 1 half square root of 2 untuk space 0 degree less or equal than x less or equal than 360 degree, maka

sin space 5 x equals 1 half square root of 2 sin space 5 x equals sin space 45 degree

Berdasarkan rumus di atas, dapat ditentukan:

  • table attributes columnalign right center left columnspacing 0px end attributes row cell bold 5 bold italic x end cell bold equals cell bold 45 bold degree bold plus bold italic k bold times bold 360 bold degree end cell row x equals cell 9 degree plus k times 72 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 9 degree plus 0 times 72 degree equals 9 degree space left parenthesis memenuhi right parenthesis end cell row k equals cell 1 comma space x equals 9 degree plus 1 times 72 degree equals 81 degree space left parenthesis memenuhi right parenthesis end cell row k equals cell 2 comma space x equals 9 degree plus 2 times 72 degree equals 153 degree space left parenthesis memenuhi right parenthesis end cell row k equals cell 3 comma space x equals 9 degree plus 3 times 72 degree equals 225 degree space left parenthesis memenuhi right parenthesis end cell row k equals cell 4 comma space x equals 9 degree plus 4 times 72 degree equals 297 degree space left parenthesis memenuhi right parenthesis end cell row k equals cell 5 comma space x equals 9 degree plus 5 times 72 degree equals 369 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 5 bold italic x end cell bold equals cell bold left parenthesis bold 180 bold degree bold minus bold 45 bold degree bold right parenthesis bold plus bold italic k bold times bold 360 bold degree end cell row cell 5 x end cell equals cell 135 degree plus k times 360 degree end cell row x equals cell 27 degree plus k times 72 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 27 degree plus 0 times 72 degree equals 27 degree space left parenthesis memenuhi right parenthesis end cell row k equals cell 1 comma space x equals 27 degree plus 1 times 72 degree equals 99 degree space left parenthesis memenuhi right parenthesis end cell row k equals cell 2 comma space x equals 27 degree plus 2 times 72 degree equals 171 degree space left parenthesis memenuhi right parenthesis end cell row k equals cell 3 comma space x equals 27 degree plus 3 times 72 degree equals 243 degree space left parenthesis memenuhi right parenthesis end cell row k equals cell 4 comma space x equals 27 degree plus 4 times 72 degree equals 315 degree space left parenthesis memenuhi right parenthesis end cell row k equals cell 5 comma space x equals 27 degree plus 5 times 72 degree equals 387 degree space left parenthesis tidak space memenuhi right parenthesis end cell end table

Dengan demikian, begin mathsize 12px style HP equals left curly bracket 9 degree comma space 27 degree comma space 81 degree comma space 99 degree comma space 153 degree comma space 171 degree comma space 225 degree comma space 242 degree comma space 297 degree comma space 315 degree right curly bracket end style.

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