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Pertanyaan

Jika fraction numerator sec space 2 x over denominator 1 plus cos space 2 x end fraction equals 1 half dan 0 less or equal than x less or equal than 2 straight pi, maka nilai x  yang memenuhi adalah...

  1. left curly bracket 0 degree comma space 150 degree comma space 360 degree right curly bracket  

  2. left curly bracket 30 degree comma space 120 degree comma space 330 degree right curly bracket  

  3. left curly bracket 0 degree comma space 180 degree comma space 360 degree right curly bracket 

  4. left curly bracket 60 degree comma space 180 degree comma space 330 degree right curly bracket  

  5. left curly bracket 30 degree comma space 120 degree comma space 360 degree right curly bracket  

E. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sebelas Maret

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah C.

Pembahasan

Ingat kembali rumus berikut.

  • sec space x equals fraction numerator 1 over denominator cos space x end fraction  
  • Jika diketahui cos space x equals cos space alpha maka x subscript 1 equals alpha plus k times 360 degree space dan space x subscript 2 equals negative alpha 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 fraction numerator sec space 2 x over denominator 1 plus cos space 2 x end fraction end cell equals cell 1 half end cell row cell 2 space sec space 2 x end cell equals cell 1 plus cos space 2 x end cell row cell fraction numerator 2 over denominator cos space 2 x end fraction end cell equals cell 1 plus cos space 2 x end cell end table  

Misalkan p equals cos space 2 x.

Persamaan di atas dapat diubah menjadi

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

Jadi diperoleh cos space space 2 x equals 1 space atau space cos space 2 x equals negative 2.

Pada rentang 0 less or equal than x less or equal than 2 straight pi, nilai cosinus adalah minimal negative 1 dan maksimal 1, jadi untuk cos space 2 x equals negative 2 tidak memenuhi.

Nilai yang digunakan adalah cos space 2 x equals 1.

Akan ditentukan x untuk cos space 2 x equals 1.

cos space 2 x equals 1 cos space 2 x equals cos space 0 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 0 degree plus 0 times 360 degree end cell row cell 2 x end cell equals cell 0 degree end cell row x equals cell 0 degree end cell end table 

 

  • table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x end cell equals cell 0 degree plus 1 times 360 degree end cell row cell 2 x end cell equals cell 360 degree end cell row x equals cell 180 degree end cell end table  

 

  • table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x end cell equals cell 0 degree plus 2 times 360 degree end cell row cell 2 x end cell equals cell 720 degree end cell row x equals cell 360 degree end cell end table  

 

  • table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x end cell equals cell 0 degree plus 3 times 360 degree end cell row cell 2 x end cell equals cell 1080 degree end cell row x equals cell 540 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 negative 0 degree plus 0 times 360 degree end cell row cell 2 x end cell equals cell 0 degree end cell row x equals cell 0 degree end cell end table 

 

  • table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x end cell equals cell negative 0 degree plus 1 times 360 degree end cell row cell 2 x end cell equals cell 360 degree end cell row x equals cell 180 degree end cell end table   

 

  • table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x end cell equals cell negative 0 degree plus 2 times 360 degree end cell row cell 2 x end cell equals cell 720 degree end cell row x equals cell 360 degree end cell end table

 

  • table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x end cell equals cell negative 0 degree plus 3 times 360 degree end cell row cell 2 x end cell equals cell 1080 degree end cell row x equals cell 540 degree space left parenthesis tidak space memenuhi right parenthesis end cell end table 

Jadi nilai x  yang memenuhi adalah  left curly bracket 0 degree comma space 180 degree comma space 360 degree right curly bracket.

Oleh karena itu, jawaban yang benar adalah C.

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