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Buktikan bahwa cos x1+sin x​+1+sin xcos x​=cos x2​ dan cari semua besar sudut antara −180∘ dan 180∘ yang merupakan solusi dari persamaan (1+sinx)2+cos2x=4cos x (1+sin x).

Pertanyaan

Buktikan bahwa fraction numerator 1 plus sin space x over denominator cos space x end fraction plus fraction numerator cos space x over denominator 1 plus sin space x end fraction equals fraction numerator 2 over denominator cos space x end fraction dan cari semua besar sudut antara negative 180 degree dan 180 degree yang merupakan solusi dari persamaan open parentheses 1 plus sin x close parentheses squared plus cos squared x equals 4 cos space x space left parenthesis 1 plus sin space x right parenthesis.

I. Sutiawan

Master Teacher

Mahasiswa/Alumni Universitas Pasundan

Jawaban terverifikasi

Jawaban

himpunan penyelesaian dari persamaan trigonometri tersebut adalah open curly brackets negative 60 degree comma space minus 90 degree comma space 60 degree close curly brackets.

Pembahasan

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator 1 plus sin space x over denominator cos space x end fraction plus fraction numerator cos space x over denominator 1 plus sin space x end fraction end cell equals cell fraction numerator 1 plus sin space x over denominator cos space x end fraction plus fraction numerator cos space x over denominator 1 plus sin space x end fraction cross times fraction numerator 1 minus sin space x over denominator 1 minus sin space x end fraction end cell row blank equals cell fraction numerator 1 plus sin space x over denominator cos space x end fraction plus fraction numerator cos space x space left parenthesis 1 minus sin space x right parenthesis over denominator 1 minus sin squared x end fraction end cell row blank equals cell fraction numerator 1 plus sin space x over denominator cos space x end fraction plus fraction numerator cos space x space left parenthesis 1 minus sin space x right parenthesis over denominator cos squared x end fraction end cell row blank equals cell fraction numerator 1 plus sin space x over denominator cos space x end fraction plus fraction numerator 1 minus sin space x over denominator cos space x end fraction end cell row blank equals cell fraction numerator 1 plus sin space x plus 1 minus sin space x over denominator cos space x end fraction end cell row blank equals cell fraction numerator 2 over denominator cos space x end fraction space left parenthesis terbukti right parenthesis end cell end table end style 

begin mathsize 12px style open parentheses 1 plus sin x close parentheses squared plus cos squared x equals 4 cos space x space left parenthesis 1 plus sin space x right parenthesis end style 
begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses 1 plus sin x close parentheses squared plus cos squared x minus 4 space cos space x space left parenthesis 1 plus sin space x right parenthesis end cell equals 0 row cell open parentheses 1 plus sin x close parentheses squared plus cos squared x minus 4 space cos space x space left parenthesis 1 plus sin space x right parenthesis end cell equals 0 row cell 1 plus 2 space sin space x plus sin squared x plus cos squared x minus 4 space cos space x minus 4 space cos space x space sin space x end cell equals 0 row cell 1 plus 2 space sin space x plus 1 minus 4 space cos space x minus 4 space cos space x space sin space x end cell equals 0 row cell 2 space sin space x minus 4 space cos space x minus 4 space cos space x space sin space x plus 2 end cell equals 0 row cell sin space x minus 2 space cos space x minus 2 space cos space x space sin space x plus 1 end cell equals 0 row cell left parenthesis sin space x plus 1 right parenthesis left parenthesis negative 2 space cos space x plus 1 right parenthesis end cell equals 0 end table end style   

begin mathsize 12px style sin space x equals negative 1 space atau space cos space x equals 1 half end style  

  • sin space x equals negative 1

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space x end cell equals cell negative 1 end cell row cell sin space x end cell equals cell sin space 270 degree end cell row x equals cell 270 plus k.360 degree end cell row blank blank blank row k equals 0 row blank rightwards double arrow cell x equals 270 degree plus 0.360 end cell row blank equals cell 270 degree space left parenthesis tidak space memenuhi right parenthesis end cell row blank blank blank row x equals cell left parenthesis 180 minus 270 right parenthesis plus k.360 end cell row blank equals cell negative 90 degree plus k.360 end cell row blank blank blank row k equals 0 row x equals cell negative 90 degree plus 0.360 degree end cell row blank equals cell negative 90 degree end cell row blank blank blank row k equals 1 row x equals cell negative 90 degree plus 1.360 degree end cell row blank equals cell 270 degree space left parenthesis tidak space memenuhi right parenthesis end cell end table  

  • cos space x equals 1 half 

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space x end cell equals cell 1 half end cell row cell cos space x end cell equals cell cos space 60 degree end cell row x equals cell 60 degree plus k.360 degree end cell row blank blank blank row k equals 0 row blank rightwards double arrow cell x equals 60 degree plus 0.360 degree end cell row blank equals cell 60 degree end cell row blank blank blank row x equals cell negative 60 degree plus k.360 degree end cell row blank blank blank row k equals 0 row blank rightwards double arrow cell x equals negative 60 degree plus 0.360 degree end cell row blank equals cell negative 60 degree end cell end table 

Jadi, himpunan penyelesaian dari persamaan trigonometri tersebut adalah open curly brackets negative 60 degree comma space minus 90 degree comma space 60 degree close curly brackets.

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