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Himpunan nilai begin mathsize 14px style x end style yang memenuhi begin mathsize 14px style 2 sin open parentheses 3 x plus straight pi over 6 close parentheses equals square root of 2 end style dengan begin mathsize 14px style 0 less than x less than straight pi end style adalah ....

  1. begin mathsize 14px style open curly brackets pi over 36 comma fraction numerator 25 pi over denominator 36 end fraction close curly brackets end style 

  2. begin mathsize 14px style open curly brackets fraction numerator 7 pi over denominator 36 end fraction comma fraction numerator 31 pi over denominator 36 end fraction close curly brackets end style 

  3. begin mathsize 14px style open curly brackets pi over 36 comma fraction numerator 19 pi over denominator 36 end fraction comma fraction numerator 25 pi over denominator 36 end fraction close curly brackets end style 

  4. begin mathsize 14px style open curly brackets fraction numerator 7 pi over denominator 36 end fraction comma fraction numerator 19 pi over denominator 36 end fraction comma fraction numerator 31 pi over denominator 36 end fraction close curly brackets end style 

  5. begin mathsize 14px style open curly brackets pi over 36 comma fraction numerator 7 pi over denominator 36 end fraction comma fraction numerator 25 pi over denominator 36 end fraction comma fraction numerator 31 pi over denominator 36 end fraction close curly brackets end style 

A. Acfreelance

Master Teacher

Jawaban terverifikasi

Pembahasan

Persamaan begin mathsize 14px style sin x equals sina end style terpenuhi oleh

begin mathsize 14px style x equals straight a plus k times 2 straight pi end style

dan

begin mathsize 14px style x equals open parentheses straight pi minus straight a close parentheses plus k times 2 straight pi end style

dengan begin mathsize 14px style k element of straight natural numbers end style.

Perhatikan bahwa

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 2 space sin open parentheses 3 x plus straight pi over 6 close parentheses end cell equals cell square root of 2 end cell row cell sin left parenthesis 3 x plus straight pi over 6 right parenthesis end cell equals cell 1 half square root of 2 end cell end table end style

Ingat bahwa begin mathsize 14px style sin straight pi over 4 equals 1 half square root of 2 end style.

Maka, dengan begin mathsize 14px style sin open parentheses 3 x plus straight pi over 6 close parentheses equals sin straight pi over 4 end style kita punya nilai begin mathsize 14px style x end style yang memenuhi yaitu

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 3 x plus straight pi over 6 end cell equals cell straight pi over 4 plus k times 2 straight pi end cell row cell 3 x end cell equals cell straight pi over 12 plus k times 2 straight pi end cell row x equals cell straight pi over 36 plus k times fraction numerator 2 straight pi over denominator 3 end fraction end cell end table end style 

atau

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 3 x plus straight pi over 6 end cell equals cell open parentheses straight pi minus straight pi over 4 close parentheses plus k times 2 straight pi end cell row cell 3 x end cell equals cell fraction numerator 7 straight pi over denominator 12 end fraction plus k times 2 straight pi end cell row x equals cell fraction numerator 7 straight pi over denominator 36 end fraction plus k times fraction numerator 2 straight pi over denominator 3 end fraction end cell end table end style

dengan begin mathsize 14px style k element of straight natural numbers end style.

 

Kemungkinan 1: begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell straight pi over 36 end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank k end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank times end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 2 straight pi over denominator 3 end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank comma end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank k end table table attributes columnalign right center left columnspacing 0px end attributes row blank element of blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight natural numbers end table end style.

Jika begin mathsize 14px style k equals negative 1 end style, maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row x equals cell straight pi over 36 plus left parenthesis negative 1 right parenthesis times fraction numerator 2 straight pi over denominator 3 end fraction end cell row blank equals cell straight pi over 36 minus fraction numerator 2 straight pi over denominator 3 end fraction end cell row blank equals cell negative fraction numerator 23 straight pi over denominator 36 end fraction end cell end table end style

Nilai ini tidak memenuhi syarat begin mathsize 14px style x end style, sehingga jika nilai begin mathsize 14px style k end style makin kecil, pasti tidak memenuhi syarat begin mathsize 14px style x end style juga.

Jika begin mathsize 14px style k equals 0 end style, maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row x equals cell straight pi over 36 plus 0 times fraction numerator 2 straight pi over denominator 3 end fraction end cell row blank equals cell straight pi over 36 end cell end table end style

 Nilai ini memenuhi syarat begin mathsize 14px style x end style.

Jika begin mathsize 14px style k equals 1 end style, maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row x equals cell straight pi over 36 plus 1 times fraction numerator 2 straight pi over denominator 3 end fraction end cell row blank equals cell straight pi over 36 plus fraction numerator 2 straight pi over denominator 3 end fraction end cell row blank equals cell fraction numerator 25 straight pi over denominator 36 end fraction end cell end table end style

Nilai ini memenuhi syarat begin mathsize 14px style x end style.

Jika begin mathsize 14px style k equals 2 end style, maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row x equals cell straight pi over 36 plus 2 times fraction numerator 2 straight pi over denominator 3 end fraction end cell row blank equals cell straight pi over 36 plus fraction numerator 4 straight pi over denominator 3 end fraction end cell row blank equals cell fraction numerator 49 straight pi over denominator 36 end fraction end cell end table end style

Nilai ini tidak memenuhi syarat begin mathsize 14px style x end style, sehingga jika nilai undefined makin besar, pasti tidak memenuhi syarat begin mathsize 14px style x end style juga.

 

Kemungkinan 2: begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 7 straight pi over denominator 36 end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank k end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank times end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator 2 straight pi over denominator 3 end fraction end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank comma end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank k end table table attributes columnalign right center left columnspacing 0px end attributes row blank element of blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight natural numbers end table end style

Jika begin mathsize 14px style k equals negative 1 end style, maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row x equals cell fraction numerator 7 straight pi over denominator 36 end fraction plus left parenthesis negative 1 right parenthesis times fraction numerator 2 straight pi over denominator 3 end fraction end cell row blank equals cell fraction numerator 7 straight pi over denominator 36 end fraction minus fraction numerator 2 straight pi over denominator 3 end fraction end cell row blank equals cell negative fraction numerator 17 straight pi over denominator 36 end fraction end cell end table end style

Nilai ini tidak memenuhi syarat begin mathsize 14px style x end style, sehingga jika nilai undefined makin kecil, pasti tidak memenuhi syarat begin mathsize 14px style x end style juga.

Jika undefined, maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row x equals cell fraction numerator 7 straight pi over denominator 36 end fraction plus 0 times fraction numerator 2 straight pi over denominator 3 end fraction end cell row blank equals cell fraction numerator 7 straight pi over denominator 36 end fraction end cell end table end style

 Nilai ini memenuhi syarat begin mathsize 14px style x end style.

Jika undefined, maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row x equals cell fraction numerator 7 straight pi over denominator 36 end fraction plus 1 times fraction numerator 2 straight pi over denominator 3 end fraction end cell row blank equals cell fraction numerator 7 straight pi over denominator 36 end fraction plus fraction numerator 2 straight pi over denominator 3 end fraction end cell row blank equals cell fraction numerator 31 straight pi over denominator 36 end fraction end cell end table end style

Nilai ini memenuhi syarat begin mathsize 14px style x end style.

Jika undefined, maka

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row x equals cell fraction numerator 7 straight pi over denominator 36 end fraction plus 2 times fraction numerator 2 straight pi over denominator 3 end fraction end cell row blank equals cell fraction numerator 7 straight pi over denominator 36 end fraction plus fraction numerator 4 straight pi over denominator 3 end fraction end cell row blank equals cell fraction numerator 55 straight pi over denominator 36 end fraction end cell end table end style

Nilai ini tidak memenuhi syarat begin mathsize 14px style x end style, sehingga jika nilai undefined makin besar, pasti tidak memenuhi syarat begin mathsize 14px style x end style juga.

Jadi, kita punya himpunan nilai begin mathsize 14px style x end style yang memenuhi yaitu begin mathsize 14px style open curly brackets pi over 36 comma fraction numerator 7 pi over denominator 36 end fraction comma fraction numerator 25 pi over denominator 36 end fraction comma fraction numerator 31 pi over denominator 36 end fraction close curly brackets end style.

Dengan demikian, jawaban yang tepat adalah E.

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