Himpunan nilai x yang memenuhi persamaan  sin 2x=cos 4x untuk 0∘≤x≤360∘ adalah...

Pertanyaan

Himpunan nilai x yang memenuhi persamaan space sin space 2 x equals cos space 4 x untuk 0 degree less or equal than x less or equal than 360 degree adalah...

  1. left curly bracket 75 degree comma space 135 degree comma space 225 degree comma space 305 degree right curly bracket 

  2. left curly bracket 45 degree comma space 75 degree comma space 135 degree comma space 315 degree right curly bracket 

  3. left curly bracket 15 degree comma space 45 degree comma space 195 degree comma space 255 degree comma space 315 degree right curly bracket

  4. left curly bracket 15 degree comma space 45 degree comma space 75 degree comma space 135 degree comma space 195 degree comma space 305 degree right curly bracket 

  5. left curly bracket 15 degree comma space 75 degree comma space 135 degree comma space 195 degree comma space 255 degree comma space 315 degree right curly bracket 

N. Puspita

Master Teacher

Jawaban terverifikasi

Pembahasan

Ingat kembali rumus berikut.

  • cos space a x equals 1 minus 2 space sin squared space left parenthesis a over 2 right parenthesis x 
  • Jika diketahui sin space x equals sin space alpha maka x subscript 1 equals alpha plus k times 360 degree space dan space x subscript 2 equals left parenthesis 180 degree minus alpha right parenthesis plus k times 360 degree

Dari rumus di atas, diperoleh perhitungan sebagai berikut.

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 2 x end cell equals cell cos space 4 x end cell row cell sin space 2 x end cell equals cell 1 minus 2 space sin squared space 2 x end cell row cell 2 space sin squared space 2 x plus sin space 2 x minus 1 end cell equals 0 end table 

Misalkan p equals sin space 2 x.

Persamaan di atas dapat diubah menjadi

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 p squared plus p minus 1 end cell equals 0 row cell left parenthesis 2 p minus 1 right parenthesis left parenthesis p plus 1 right parenthesis end cell equals 0 row cell 2 p minus 1 end cell equals cell 0 space atau space p plus 1 equals 0 end cell row p equals cell 1 half space space space space space space space space space space space space space p equals negative 1 end cell end table 

Jadi diperoleh sin space 2 x equals 1 half space atau space sin space 2 x equals negative 1.

Akan ditentukan x untuk sin space 2 x equals 1 half.

sin space 2 x equals 1 half sin space 2 x equals sin space 30 degree  

Untuk rentang 0 degree less or equal than x less or equal than 360 degree, diperoleh nilai x adalah

  • table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x end cell equals cell 30 degree plus 0 times 360 degree end cell row cell 2 x end cell equals cell 30 degree end cell row x equals cell 15 degree end cell end table 

 

  • table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x end cell equals cell 30 degree plus 1 times 360 degree end cell row cell 2 x end cell equals cell 30 degree plus 360 degree end cell row cell 2 x end cell equals cell 390 degree end cell row x equals cell 195 degree end cell end table 

 

  • table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x end cell equals cell 30 degree plus 2 times 360 degree end cell row cell 2 x end cell equals cell 30 degree plus 720 degree end cell row cell 2 x end cell equals cell 750 degree end cell row x equals cell 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 2 x end cell equals cell left parenthesis 180 degree minus 30 degree right parenthesis plus 0 times 360 degree end cell row cell 2 x end cell equals cell 150 degree end cell row x equals cell 75 degree end cell end table 

 

  • table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x end cell equals cell left parenthesis 180 degree minus 30 degree right parenthesis plus 1 times 360 degree end cell row cell 2 x end cell equals cell 150 degree plus 360 degree end cell row cell 2 x end cell equals cell 510 degree end cell row x equals cell 255 degree end cell end table 

 

  • table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x end cell equals cell left parenthesis 180 degree minus 30 degree right parenthesis plus 2 times 360 degree end cell row cell 2 x end cell equals cell 150 degree plus 720 degree end cell row cell 2 x end cell equals cell 870 degree end cell row x equals cell 435 degree space left parenthesis tidak space memenuhi right parenthesis end cell end table 

 

Akan ditentukan x untuk sin space 2 x equals negative 1.

sin space 2 x equals negative 1 sin space 2 x equals sin space 270 degree 

Untuk rentang 0 degree less or equal than x less or equal than 360 degree, diperoleh nilai x adalah

  • table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x end cell equals cell 270 degree plus 0 times 360 degree end cell row cell 2 x end cell equals cell 270 degree end cell row x equals cell 135 degree end cell end table 

 

  •  table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x end cell equals cell 270 degree plus 1 times 360 degree end cell row cell 2 x end cell equals cell 270 degree plus 360 degree end cell row cell 2 x end cell equals cell 630 degree end cell row x equals cell 315 degree end cell end table 

 

  • table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x end cell equals cell left parenthesis 180 degree minus 270 degree right parenthesis plus 0 times 360 degree end cell row cell 2 x end cell equals cell negative 90 degree end cell row x equals cell negative 45 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 2 x end cell equals cell left parenthesis 180 degree minus 270 degree right parenthesis plus 1 times 360 degree end cell row cell 2 x end cell equals cell negative 90 degree plus 360 degree end cell row cell 2 x end cell equals cell 270 degree end cell row x equals cell 135 degree space end cell end table

 

  • table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x end cell equals cell left parenthesis 180 degree minus 270 degree right parenthesis plus 2 times 360 degree end cell row cell 2 x end cell equals cell negative 90 degree plus 720 degree end cell row cell 2 x end cell equals cell 630 degree end cell row x equals cell 315 degree space end cell end table 

 

  • table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x end cell equals cell left parenthesis 180 degree minus 270 degree right parenthesis plus 3 times 360 degree end cell row cell 2 x end cell equals cell negative 90 degree plus 1080 degree end cell row cell 2 x end cell equals cell 990 degree end cell row x equals cell 495 degree space space left parenthesis tidak space memenuhi right parenthesis end cell end table

Jadi himpunan nilai x yang memenuhi persamaan space sin space 2 x equals cos space 4 x untuk 0 degree less or equal than x less or equal than 360 degree adalah left curly bracket 15 degree comma space 75 degree comma space 135 degree comma space 195 degree comma space 255 degree comma space 315 degree right curly bracket.

Oleh karena itu, jawaban yang benar adalah E.

 

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