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Untuk 0≤x≤2π, tentukan himpunan penyelesaian dari setiap sistem persamaan berikut.  c. {sin x−sin 3x=0sin 3x−cos x=0​

Pertanyaan

Untuk 0 less or equal than x less or equal than 2 straight pi, tentukan himpunan penyelesaian dari setiap sistem persamaan berikut. 

c. open curly brackets table attributes columnalign left end attributes row cell sin space x minus sin space 3 x equals 0 end cell row cell sin space 3 x minus cos space x equals 0 end cell end table close 

S. Yoga

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Pembahasan

Ingat bahwa:

  • sin space s minus sin space t equals 2 space sin space open parentheses fraction numerator s minus t over denominator 2 end fraction close parentheses space cos space open parentheses fraction numerator s minus t over denominator 2 end fraction close parentheses 
  • cos space x equals sin space open parentheses straight pi over 2 minus x close parentheses 
  • sin space left parenthesis 180 minus x right parenthesis degree equals sin space x degree space atau space sin left parenthesis pi minus x right parenthesis equals sin space x 
  • cos space left parenthesis negative x right parenthesis equals cos space x 

Sehingga, 

Untuk persamaan sin open parentheses x close parentheses minus sin open parentheses 3 x close parentheses equals 0 pada interval 0 less or equal than x less or equal than 2 straight pi.

table attributes columnalign right center left columnspacing 0px end attributes row cell sin open parentheses x close parentheses minus sin open parentheses 3 x close parentheses end cell equals 0 row cell negative sin open parentheses 3 x close parentheses plus sin open parentheses x close parentheses end cell equals 0 row cell 2 sin open parentheses fraction numerator x minus 3 x over denominator 2 end fraction close parentheses cos open parentheses fraction numerator x plus 3 x over denominator 2 end fraction close parentheses end cell equals 0 row cell negative 2 cos open parentheses 2 x close parentheses sin open parentheses x close parentheses end cell equals 0 end table 

cos open parentheses 2 x close parentheses equals 0 space atau space sin open parentheses x close parentheses equals 0 

 table attributes columnalign right center left columnspacing 0px end attributes row blank rightwards arrow cell cos open parentheses 2 x close parentheses equals 0 end cell row blank rightwards arrow cell 2 x equals pi over 2 plus 2 pi n end cell row x equals cell pi over 4 plus pi n end cell row cell Untuk space n end cell equals cell left curly bracket 0 comma space 1 right curly bracket end cell row x equals cell open curly brackets pi over 4 comma space fraction numerator 5 pi over denominator 4 end fraction close curly brackets end cell row blank blank blank row blank rightwards arrow cell 2 x equals fraction numerator 3 pi over denominator 2 end fraction plus 2 pi n end cell row x equals cell fraction numerator 3 pi over denominator 4 end fraction plus pi n end cell row cell Untuk space n end cell equals cell open curly brackets 0 comma space 1 close curly brackets end cell row x equals cell open curly brackets fraction numerator 3 pi over denominator 4 end fraction comma space fraction numerator 7 pi over denominator 4 end fraction close curly brackets end cell end table 


table attributes columnalign right center left columnspacing 0px end attributes row blank rightwards arrow cell sin open parentheses x close parentheses equals 0 end cell row blank rightwards arrow cell x equals 0 plus 2 pi n end cell row cell Untuk space n end cell equals 0 row x equals 0 row blank blank blank row blank rightwards arrow cell x equals pi plus 2 pi n end cell row cell Untuk space n end cell equals cell open curly brackets 0 comma space 1 close curly brackets end cell row x equals cell open curly brackets pi comma space 2 pi close curly brackets end cell end table 

Yang memenuhi persamaan sin open parentheses x close parentheses minus sin open parentheses 3 x close parentheses equals 0 adalah x equals open curly brackets pi over 4 comma thin space fraction numerator 3 pi over denominator 4 end fraction comma thin space fraction numerator 5 pi over denominator 4 end fraction comma thin space fraction numerator 7 pi over denominator 4 end fraction comma thin space 0 comma thin space pi comma thin space 2 pi close curly brackets.

Untuk persamaan sin open parentheses 3 x close parentheses minus cos open parentheses x close parentheses equals 0 pada interval 0 less or equal than x less or equal than 2 straight pi.

table attributes columnalign right center left columnspacing 0px end attributes row cell sin open parentheses 3 x close parentheses minus cos open parentheses x close parentheses end cell equals 0 row cell sin open parentheses 3 x close parentheses end cell equals cell cos open parentheses x close parentheses end cell row cell sin open parentheses 3 x close parentheses end cell equals cell sin open parentheses pi over 2 minus x close parentheses end cell row blank rightwards arrow cell 3 x equals pi over 2 minus x plus 2 pi n end cell row cell 4 x end cell equals cell pi over 2 plus 2 pi n end cell row x equals cell fraction numerator pi plus 4 pi n over denominator 8 end fraction end cell row cell Untuk space n end cell equals cell open curly brackets 0 comma space 1 comma space 2 comma space 3 close curly brackets end cell row x equals cell open curly brackets pi over 8 comma space fraction numerator 5 pi over denominator 8 end fraction comma space fraction numerator 9 pi over denominator 8 end fraction comma space fraction numerator 13 pi over denominator 8 end fraction close curly brackets end cell row blank blank blank row blank rightwards arrow cell 3 x equals pi minus open parentheses pi over 2 minus x close parentheses plus 2 pi n end cell row cell 2 x end cell equals cell pi minus pi over 2 plus 2 pi n end cell row x equals cell fraction numerator pi plus 4 pi n over denominator 4 end fraction end cell row cell Untuk space n end cell equals cell open curly brackets 0 comma space 1 close curly brackets end cell row x equals cell open curly brackets pi over 4 comma space fraction numerator 5 pi over denominator 4 end fraction close curly brackets end cell end table 

Yang memenuhi persamaan sin open parentheses 3 x close parentheses minus cos open parentheses x close parentheses equals 0 adalah x equals open curly brackets pi over 8 comma thin space pi over 4 comma thin space fraction numerator 5 pi over denominator 8 end fraction comma thin space fraction numerator 5 pi over denominator 4 end fraction comma thin space fraction numerator 9 pi over denominator 8 end fraction comma thin space fraction numerator 13 pi over denominator 8 end fraction close curly brackets.

Jadi, himpunan penyelesaian yang memenuhi persamaan tersebut adalah 

HP equals open curly brackets pi over 4 comma thin space fraction numerator 3 pi over denominator 4 end fraction comma thin space fraction numerator 5 pi over denominator 4 end fraction comma thin space fraction numerator 7 pi over denominator 4 end fraction comma thin space 0 comma thin space pi comma thin space 2 pi comma space pi over 8 comma space fraction numerator 5 pi over denominator 8 end fraction comma space fraction numerator 9 pi over denominator 8 end fraction comma space fraction numerator 13 pi over denominator 8 end fraction close curly brackets.

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