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Himpunan nilai x yang memenuhi sin(2x+60∘)=cos50∘ dengan 0∘<x<360∘ adalah ....

Pertanyaan

Himpunan nilai begin mathsize 14px style x end style yang memenuhi begin mathsize 14px style sin left parenthesis 2 x plus 60 degree right parenthesis equals cos 50 degree end style dengan begin mathsize 14px style 0 degree less than x less than 360 degree end style adalah ....

  1. begin mathsize 14px style open curly brackets 40 degree comma 80 degree comma 260 degree comma 350 degree close curly brackets end style

  2. begin mathsize 14px style open curly brackets 220 degree comma 350 degree close curly brackets end style

  3. begin mathsize 14px style open curly brackets 40 degree comma 170 degree close curly brackets end style

  4. begin mathsize 14px style open curly brackets 40 degree comma 170 degree comma 220 degree comma 350 degree close curly brackets end style

  5. begin mathsize 14px style open curly brackets 80 degree comma 160 degree comma 260 degree comma 340 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

atau

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 cos x equals sin left parenthesis 90 degree minus x right parenthesis end style.

Maka,

begin mathsize 14px style cos 50 degree equals sin left parenthesis 90 degree minus 50 degree right parenthesis equals sin 40 degree end style

Sehingga kita punya persamaan

begin mathsize 14px style sin left parenthesis 2 x plus 60 degree right parenthesis equals cos 50 degree sin left parenthesis 2 x plus 60 degree right parenthesis equals sin 40 degree end style

yang dipenuhi oleh

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x plus 60 degree end cell equals cell 40 degree plus k times 360 degree end cell row cell 2 x end cell equals cell negative 20 degree plus k times 360 degree end cell row x equals cell negative 10 degree plus k times 180 degree end cell end table end style

atau

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x plus 60 degree end cell equals cell left parenthesis 180 degree minus 40 degree right parenthesis plus k times 360 degree end cell row cell 2 x end cell equals cell 80 degree plus k times 360 degree end cell row x equals cell 40 degree plus k times 180 degree 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 minus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 10 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree 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 180 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree 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 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 0 end style, maka

begin mathsize 14px style x equals negative 10 degree plus 0 times 180 degree x equals negative 10 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 1 end style, maka

begin mathsize 14px style x equals negative 10 degree plus 1 times 180 degree x equals negative 10 degree plus 180 degree x equals 170 degree 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 x equals negative 10 degree plus 2 times 180 degree x equals negative 10 degree plus 360 degree x equals 350 degree end style

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

Jika begin mathsize 14px style k equals 3 end style, maka

begin mathsize 14px style x equals negative 10 degree plus 3 times 180 degree x equals negative 10 degree plus 540 degree x equals 530 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: 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 40 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree 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 180 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree 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 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 x equals 40 degree plus left parenthesis negative 1 right parenthesis times 180 degree x equals 40 degree minus 180 degree x equals negative 140 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 40 degree plus 0 times 180 degree x equals 40 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 40 degree plus 1 times 180 degree x equals 40 degree plus 180 degree x equals 220 degree 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 x equals 40 degree plus 2 times 180 degree x equals 40 degree plus 360 degree x equals 400 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.

Dengan demikian, kita punya himpunan nilai begin mathsize 14px style x end style yang memenuhi yaitu begin mathsize 14px style open curly brackets 40 degree comma 170 degree comma 220 degree comma 350 degree close curly brackets end style.

Jadi, jawaban yang tepat adalah D.

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