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Himpunan nilai x yang memenuhi sin(x+15∘)=21​2​ dengan −180∘<x<180∘ adalah ....

Pertanyaan

Himpunan nilai begin mathsize 14px style x end style yang memenuhi begin mathsize 14px style sin left parenthesis x plus 15 degree right parenthesis equals 1 half square root of 2 end style dengan begin mathsize 14px style negative 180 degree less than x less than 180 degree end style adalah ....

  1. begin mathsize 14px style open curly brackets 45 degree comma 120 degree close curly brackets end style

  2. begin mathsize 14px style open curly brackets 30 degree comma 330 degree close curly brackets end style

  3. begin mathsize 14px style open curly brackets 45 degree comma 315 degree close curly brackets end style

  4. begin mathsize 14px style open curly brackets 30 degree comma 120 degree close curly brackets end style

  5. begin mathsize 14px style open curly brackets 45 degree comma 135 degree close curly brackets end style

M. Robo

Master Teacher

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah D.

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 360 degree end style

dan

begin mathsize 14px style x equals left parenthesis 180 degree minus straight a right parenthesis plus k times 360 degree end style

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

Ingat bahwa begin mathsize 14px style sin 45 degree equals 1 half square root of 2 end style.

Maka, dengan  begin mathsize 14px style sin left parenthesis x plus 15 degree right parenthesis equals sin 45 degree end style  kita punya nilai undefined yang memenuhi yaitu

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

atau

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell x plus 15 degree end cell equals cell left parenthesis 180 degree minus 45 degree right parenthesis plus k times 360 degree end cell row cell x plus 15 degree end cell equals cell 135 degree plus k times 360 degree end cell row x equals cell 120 degree plus k times 360 degree end cell end table end style

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

 

Kemungkinan 1: table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px x end table table attributes columnalign right center left columnspacing 0px end attributes row blank size 14px equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 30 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px degree end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px k end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px times end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 360 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px degree end tablebegin mathsize 14px style k element of straight natural numbers end style.

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

begin mathsize 14px style x equals 30 degree plus left parenthesis negative 1 right parenthesis times 360 degree x equals 30 degree minus 360 degree x equals negative 330 degree 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 x equals 30 degree plus 0 times 360 degree x equals 30 degree 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 x equals 30 degree plus 1 times 360 degree x equals 30 degree plus 360 degree space x equals 390 degree 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 besar, pasti tidak memenuhi syarat begin mathsize 14px style x end style juga.

 

Kemungkinan 2: table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px x end table table attributes columnalign right center left columnspacing 0px end attributes row blank size 14px equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 120 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px degree end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px k end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px times end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px 360 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank size 14px degree end tablebegin mathsize 14px style k element of straight natural numbers end style.

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

begin mathsize 14px style x equals 120 degree plus left parenthesis negative 1 right parenthesis times 360 degree x equals 120 degree minus 360 degree x equals negative 240 degree 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 undefined, maka

begin mathsize 14px style x equals 120 degree plus 0 times 360 degree x equals 120 degree 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 x equals 120 degree plus 1 times 360 degree x equals 120 degree plus 360 degree x equals 480 degree 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 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 30 degree comma 120 degree close curly brackets end style.

Dengan demikian, jawaban yang tepat adalah D.

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