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Himpunan penyelesaian persamaan sin 2x=sin 3x+sin x , untuk 0<x<2π adalah ….

Pertanyaan

Himpunan penyelesaian persamaan begin mathsize 14px style sin space 2 straight x equals sin space 3 straight x plus sin space straight x end style , untuk begin mathsize 14px style 0 less than straight x less than 2 straight pi end style adalah ….

  1. begin mathsize 14px style open curly brackets straight pi over 3 comma straight pi over 2 comma straight pi comma fraction numerator 3 straight pi over denominator 2 end fraction comma fraction numerator 5 straight pi over denominator 3 end fraction comma straight 2 straight pi close curly brackets end style 

  2. begin mathsize 14px style open curly brackets straight pi over 3 comma fraction numerator 5 straight pi over denominator 3 end fraction comma straight 2 straight pi close curly brackets end style 

  3. begin mathsize 14px style open curly brackets 0 comma straight pi over 2 comma straight pi comma fraction numerator 5 straight pi over denominator 3 end fraction comma straight 2 straight pi close curly brackets end style 

  4. begin mathsize 14px style open curly brackets straight pi over 3 comma straight pi over 2 comma fraction numerator 3 straight pi over denominator 2 end fraction comma fraction numerator 5 straight pi over denominator 3 end fraction comma straight pi close curly brackets end style 

  5. begin mathsize 14px style open curly brackets 0 comma straight pi over 2 comma straight pi over 3 comma fraction numerator 5 straight pi over denominator 3 end fraction comma straight straight pi close curly brackets end style 

A. Acfreelance

Master Teacher

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah D.

Pembahasan

Gunakan rumus penjumlahan sin berikut ini :

begin mathsize 14px style sin space straight A plus sin space straight B equals 2 sin space open parentheses fraction numerator straight A plus straight B over denominator 2 end fraction close parentheses space cos space open parentheses fraction numerator straight A minus straight B over denominator 2 end fraction close parentheses end style 

Maka,

begin mathsize 14px style sin space 2 straight x equals sin space 3 straight x plus sin space straight x sin space 2 straight x equals 2 space sin space open parentheses fraction numerator 3 straight x plus straight x over denominator 2 end fraction close parentheses space cos space open parentheses fraction numerator 3 straight x minus straight x over denominator 2 end fraction close parentheses sin space 2 straight x equals 2 space sin space 2 straight x space cos space straight x 0 equals 2 space sin space 2 straight x space cos space straight x minus sin space 2 straight x 0 equals sin space 2 straight x space open parentheses 2 space cos space straight x minus 1 close parentheses 0 equals sin space 2 straight x space atau straight 2 space cos space straight x minus 1 equals 0 end style 

Karena interval yang diminta pada soal begin mathsize 14px style 0 less or equal than straight x less or equal than 2 straight pi end style maka, 

Untuk begin mathsize 14px style 2 space cos space straight x minus 1 equals 0 end style 

Error converting from MathML to accessible text. 

 

Untuk begin mathsize 14px style sin space 2 straight x equals 0 end style 

begin mathsize 14px style sin space 2 straight x equals sin space 0 2 straight x subscript 1 equals 0 plus 2 πk    dan straight 2 straight x subscript 2 equals open parentheses straight pi minus 0 close parentheses plus 2 πk end style 

Jadi himpunan penyelesaian persamaan tersebut adalah undefined.

Dengan demikian, jawaban yang tepat adalah D.

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