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Selesaikan persamaan trigonometri di bawah ini, untuk 0∘≤x≤360∘. c. cotan x.cos x−sin x=1

Pertanyaan

Selesaikan persamaan trigonometri di bawah ini, untuk 0 degree less or equal than x less or equal than 360 degree.

c. cotan space x. cos space x minus sin space x equals 1  

E. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sebelas Maret

Jawaban terverifikasi

Jawaban

himpunan penyelesaian dari persamaan trigonometri tersebut adalah open curly brackets 30 degree comma space 150 degree comma space 270 degree close curly brackets.

Pembahasan

Ingat!

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space x end cell equals cell sin space alpha end cell row x equals cell alpha plus k.360 degree end cell row blank blank atau row x equals cell left parenthesis 180 minus alpha right parenthesis plus k.360 degree end cell end table 

Maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell cotan space x. cos space x minus sin space x end cell equals 1 row cell fraction numerator cos space x over denominator sin space x end fraction. cos space x minus sin space x end cell equals 1 row cell fraction numerator cos squared x over denominator sin space x end fraction minus sin space x end cell equals 1 row cell fraction numerator cos squared x minus sin squared x over denominator sin space x end fraction end cell equals 1 row cell cos squared x minus sin squared x end cell equals cell sin space x end cell row cell left parenthesis 1 minus sin squared x right parenthesis minus sin squared x minus sin space x end cell equals 0 row cell 1 minus 2 space sin squared x minus sin space x end cell equals 0 row cell negative 2 space sin squared x minus sin space x plus 1 end cell equals 0 row cell 2 space sin squared x plus sin space x minus 1 end cell equals 0 row cell left parenthesis 2 space sin space x minus 1 right parenthesis left parenthesis sin space x plus 1 right parenthesis end cell equals 0 end table   

sin space x equals 1 half space atau space sin space x equals negative 1 

  • sin space x equals 1 half 

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space x end cell equals cell 1 half end cell row cell sin space x end cell equals cell sin space 30 degree end cell row x equals cell 30 degree plus k.360 degree end cell row blank blank blank row k equals 0 row blank rightwards double arrow cell x equals 30 degree plus 0.360 degree end cell row blank equals cell 30 degree end cell row blank blank blank row x equals cell left parenthesis 180 degree minus 30 degree right parenthesis plus k.360 degree end cell row blank equals cell 150 degree plus k.360 degree end cell row blank blank blank row k equals 0 row blank rightwards double arrow cell x equals 150 degree plus 0.360 degree end cell row blank equals cell 150 degree end cell end table 

  • 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 degree plus k.360 degree end cell row blank blank blank row k equals 0 row x equals cell 270 degree plus 0.360 degree end cell row blank equals cell 270 degree end cell end table 

Jadi, himpunan penyelesaian dari persamaan trigonometri tersebut adalah open curly brackets 30 degree comma space 150 degree comma space 270 degree close curly brackets.

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