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Tentukan himpunan penyelesaian tiap persamaan berikut untuk 0∘≤x≤360∘. a. sin 2x+sin (2x−30∘)=0

Pertanyaan

Tentukan himpunan penyelesaian tiap persamaan berikut untuk 0 degree less or equal than x less or equal than 360 degree.

a. sin space 2 x plus sin space left parenthesis 2 x minus 30 degree right parenthesis equals 0  

S. Yoga

Master Teacher

Mahasiswa/Alumni Universitas Pendidikan Indonesia

Jawaban terverifikasi

Jawaban

 himpunan penyelesaiannya adalah HP equals left curly bracket 7 comma 5 degree semicolon space 97 comma 5 degree right curly bracket.

Pembahasan

Ingat bahwa:

  • sin space x plus sin space y equals 2 space sin space 1 half left parenthesis x plus y right parenthesis space cos space 1 half left parenthesis x minus y right parenthesis 
  • cos open parentheses x over 2 close parentheses equals square root of fraction numerator 1 plus cos left parenthesis x right parenthesis over denominator 2 end fraction end root 

Mencari himpunan penyelesaian:

table attributes columnalign right center left columnspacing 0px end attributes row cell sin open parentheses 2 x close parentheses plus sin open parentheses 2 x minus 30 degree close parentheses end cell equals 0 row cell 2 sin open parentheses fraction numerator 2 x plus 2 x minus 30 degree over denominator 2 end fraction close parentheses cos open parentheses fraction numerator 2 x minus open parentheses 2 x minus 30 degree close parentheses over denominator 2 end fraction close parentheses end cell equals 0 row cell 2 sin open parentheses fraction numerator 4 x minus 30 degree over denominator 2 end fraction close parentheses cos open parentheses fraction numerator 30 degree over denominator 2 end fraction close parentheses end cell equals 0 row cell 2 sin open parentheses fraction numerator 24 x minus 30 degree over denominator 12 end fraction close parentheses cos open parentheses fraction numerator 30 degree over denominator 2 end fraction close parentheses end cell equals 0 row cell 2 sin open parentheses fraction numerator 24 x minus 30 degree over denominator 12 end fraction close parentheses square root of fraction numerator 1 plus begin display style fraction numerator square root of 3 over denominator 2 end fraction end style over denominator 2 end fraction end root end cell equals 0 row cell sin open parentheses fraction numerator 24 x minus 180 degree over denominator 12 end fraction close parentheses square root of 2 plus square root of 3 end root end cell equals 0 row cell fraction numerator square root of 2 plus square root of 3 end root sin open parentheses fraction numerator 24 x minus 180 degree over denominator 12 end fraction close parentheses over denominator square root of 2 plus square root of 3 end root end fraction end cell equals cell fraction numerator 0 over denominator square root of 2 plus square root of 3 end root end fraction end cell row cell sin open parentheses fraction numerator 24 x minus 180 degree thin space over denominator 12 end fraction close parentheses end cell equals 0 end table 

table attributes columnalign right center left columnspacing 0px end attributes row blank rightwards arrow cell fraction numerator 24 x minus 180 degree thin space over denominator 12 end fraction equals 0 plus 360 degree n end cell row blank blank atau row cell fraction numerator 24 x minus 180 degree over denominator 12 end fraction end cell equals cell 180 degree plus 360 degree n end cell row blank blank blank row blank rightwards arrow cell x equals 180 degree n plus 7 comma 5 degree end cell row blank blank atau row straight x equals cell 97 comma 5 degree thin space plus 180 degree straight n end cell end table  

Untuk

table attributes columnalign center left center end attributes row blank cell x equals 180 degree n plus 7 comma 5 degree end cell blank row cell x equals 0 rightwards arrow end cell cell x equals 7 comma 5 degree end cell blank row cell x equals 1 rightwards arrow end cell cell x equals 367 comma 5 degree end cell blank row blank cell x equals 97 comma 5 degree plus 180 degree n end cell blank row cell x equals 1 rightwards arrow end cell cell x equals 97 comma 5 degree end cell blank row cell x equals 2 rightwards arrow end cell cell x equals 457 comma 5 degree end cell blank end table 

himpunan penyelesaian yang memenuhi 0 degree less or equal than x less or equal than 360 degree, adalah HP equals left curly bracket 7 comma 5 degree semicolon space 97 comma 5 degree right curly bracket.

Jadi, himpunan penyelesaiannya adalah HP equals left curly bracket 7 comma 5 degree semicolon space 97 comma 5 degree right curly bracket.

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