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Himpunan penyelesaian persamaan  untuk  adalah ...

Himpunan penyelesaian persamaan begin mathsize 14px style square root of 3 cos space x degree plus sin space x degree equals square root of 2 end style untuk begin mathsize 14px style 0 less or equal than x less or equal than 2 straight pi end style adalah ...  

  1. begin mathsize 14px style open curly brackets 5 over 12 straight pi comma space 19 over 12 straight pi close curly brackets end style

  2. begin mathsize 14px style open curly brackets straight pi over 12 comma space 19 over 12 straight pi close curly brackets end style

  3. begin mathsize 14px style open curly brackets straight pi over 12 comma space 23 over 12 straight pi close curly brackets end style

  4. begin mathsize 14px style open curly brackets straight pi over 12 comma space 5 over 12 straight pi close curly brackets end style

  5. begin mathsize 14px style open curly brackets 5 over 12 straight pi comma space 23 over 12 straight pi close curly brackets end style

Jawaban:

begin mathsize 14px style square root of 3 cos space x degree plus sin space x degree equals square root of 2 end style 

begin mathsize 14px style A space cos space x degree plus B space sin x degree equals k space cos space left parenthesis x minus theta right parenthesis end style dengan begin mathsize 14px style k equals square root of A squared plus B squared end root end style dan begin mathsize 14px style tan space theta equals B over A end style, maka:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row k equals cell square root of A squared plus B squared end root end cell row blank equals cell square root of open parentheses square root of 3 close parentheses squared plus 1 squared end root end cell row blank equals cell square root of 3 plus 1 end root end cell row blank equals cell square root of 4 end cell row blank equals 2 end table end style 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell tan space theta end cell equals cell B over A end cell row blank equals cell fraction numerator 1 over denominator square root of 3 end fraction end cell row blank equals cell 1 third square root of 3 end cell row blank blank blank row theta equals cell 30 degree space a t a u space theta equals 210 degree end cell end table end style 

Karena Koefisien begin mathsize 14px style sin space x end style dan begin mathsize 14px style cos space x end style positif, begin mathsize 14px style theta end style berada di kuadran I, maka nilai begin mathsize 14px style theta end style yang memenuhi adalah begin mathsize 14px style 30 degree end style.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell square root of 3 cos space x degree plus sin space x degree end cell equals cell 2 space cos space open parentheses x degree minus 30 degree close parentheses end cell row blank blank blank row cell square root of 3 cos space x degree plus sin space x degree end cell equals cell square root of 2 end cell row cell 2 space cos space open parentheses x degree minus 30 degree close parentheses end cell equals cell square root of 2 end cell row cell cos space open parentheses x degree minus 30 degree close parentheses end cell equals cell 1 half square root of 2 end cell row cell cos space open parentheses x degree minus 30 degree close parentheses end cell equals cell cos space 45 degree end cell row cell x degree minus 30 degree end cell equals cell 45 degree plus k.360 degree space end cell row cell atau space x degree minus 30 degree end cell equals cell negative 45 degree plus k.360 degree space end cell row blank blank blank row blank blank blank end table end style  

Kemungkinan pertama:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell x degree minus 30 degree end cell equals cell 45 degree plus k.360 degree end cell row x equals cell 75 degree plus k.360 degree end cell end table end style 

Karena nilai dari x memiliki batas begin mathsize 14px style 0 less or equal than x less or equal than 2 straight pi end style, nilai k yang memenuhi adalah ketika begin mathsize 14px style k equals 0 end style menghasilkan begin mathsize 14px style x equals 75 degree end style atau begin mathsize 14px style x equals 5 over 12 straight pi end style.

Kemungkinan kedua:

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell x degree minus 30 degree end cell equals cell negative 45 degree plus k.360 degree end cell row x equals cell negative 15 degree plus k.360 degree end cell end table end style    

Karena nilai dari x memiliki batas begin mathsize 14px style 0 less or equal than x less or equal than 2 straight pi end style, nilai k yang memenuhi adalah ketika begin mathsize 14px style k equals 1 end style menghasilkan begin mathsize 14px style x equals 345 degree end style atau begin mathsize 14px style x equals 23 over 12 straight pi end style.

Maka himpunan penyelesaiannya adalah begin mathsize 14px style open curly brackets 5 over 12 straight pi comma space 23 over 12 straight pi close curly brackets end style.

Jadi, jawaban yang tepat adalah E

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