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Tentukan himpunan penyelesaian dari setiap persamaan trigonometri di bawah ini. a. sin ( x − 30 ) ∘ = sin 1 5 ∘ , 0 ∘ ≤ x ≤ 36 0 ∘

Tentukan himpunan penyelesaian dari setiap persamaan trigonometri di bawah ini.

a.  

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W. Lestari

Master Teacher

Mahasiswa/Alumni Universitas Sriwijaya

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solusi dan himpunan penyelesaian persamaan trigonometri adalah .

solusi dan himpunan penyelesaian persamaan trigonometri sin space open parentheses x minus 30 close parentheses degree equals sin space 15 degree comma space 0 degree less or equal than x less or equal than 360 degree adalah HP equals open curly brackets 45 degree comma space 195 degree close curly brackets.

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Pembahasan

Ingat kembali aturan penyelesaian persamaan trigonometri berikut: Diketahui persamaan trigonometri . Maka: Sehingga: - Persamaan - Persamaan Jadi, solusi dan himpunan penyelesaian persamaan trigonometri adalah .

Ingat kembali aturan penyelesaian persamaan trigonometri berikut:

sin space x equals sin space alpha rightwards double arrow open curly brackets table attributes columnalign left end attributes row cell x equals open square brackets alpha plus open parentheses 360 times k close parentheses close square brackets degree end cell row cell x equals open parentheses 180 minus straight alpha close parentheses degree plus open parentheses 360 times k close parentheses degree end cell end table close 

Diketahui persamaan trigonometri sin space open parentheses x minus 30 close parentheses degree equals sin space 15 degree comma space 0 degree less or equal than x less or equal than 360 degree. Maka:

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 cell sin space open parentheses x minus 30 close parentheses degree end cell equals cell sin space 15 degree end cell end table 

Sehingga:

- Persamaan 1 

table attributes columnalign right center left columnspacing 0px end attributes row x equals cell alpha plus open parentheses 360 degree times k close parentheses end cell row cell x minus 30 degree end cell equals cell 15 degree plus open parentheses 360 degree times k close parentheses end cell row x equals cell 15 degree plus open parentheses 360 degree times k close parentheses plus 30 degree end cell row x equals cell 45 degree plus open parentheses 360 degree times k close parentheses end cell end table 

begin mathsize 12px style k equals negative 1 rightwards arrow x equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank 45 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 cell open parentheses 360 degree times k close parentheses end cell end table equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank 45 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 cell open parentheses 360 degree times open parentheses negative 1 close parentheses close parentheses end cell end table equals negative 315 degree space open parentheses TM close parentheses k equals 0 rightwards arrow x equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank 45 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 cell open parentheses 360 degree times k close parentheses end cell end table equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank 45 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 cell open parentheses 360 degree times 0 close parentheses end cell end table equals 45 degree space space open parentheses straight M close parentheses k equals 1 rightwards arrow x equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank 45 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 cell left parenthesis 360 degree times k right parenthesis end cell end table equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank 45 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 cell left parenthesis 360 degree times 1 right parenthesis end cell end table equals 405 degree space open parentheses TM close parentheses end style 

- Persamaan 2 

table attributes columnalign right center left columnspacing 0px end attributes row x equals cell open parentheses 180 degree minus straight alpha close parentheses plus open parentheses 360 degree times k close parentheses end cell row cell x minus 30 degree end cell equals cell open parentheses 180 degree minus 15 degree close parentheses plus open parentheses 360 degree times k close parentheses end cell row cell x minus 30 degree end cell equals cell 165 degree plus open parentheses 360 degree times k close parentheses end cell row x equals cell 165 degree plus open parentheses 360 degree times k close parentheses plus 30 degree end cell row x equals cell 195 degree plus open parentheses 360 degree times k close parentheses end cell end table 

begin mathsize 12px style k equals negative 1 rightwards arrow x equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank 195 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 cell open parentheses 360 degree times k close parentheses end cell end table equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank 195 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 cell open parentheses 360 degree times open parentheses negative 1 close parentheses close parentheses end cell end table equals negative 165 degree space open parentheses TM close parentheses k equals 0 rightwards arrow x equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank 195 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 cell open parentheses 360 degree times k close parentheses end cell end table equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank 195 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 cell open parentheses 360 degree times 0 close parentheses end cell end table equals 195 degree space open parentheses straight M close parentheses k equals 1 rightwards arrow x equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank 195 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 cell open parentheses 360 degree times k close parentheses end cell end table equals table attributes columnalign right center left columnspacing 0px end attributes row blank blank 195 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 cell open parentheses 360 degree times 1 close parentheses end cell end table equals 555 degree space open parentheses TM close parentheses end style  

Jadi, solusi dan himpunan penyelesaian persamaan trigonometri sin space open parentheses x minus 30 close parentheses degree equals sin space 15 degree comma space 0 degree less or equal than x less or equal than 360 degree adalah HP equals open curly brackets 45 degree comma space 195 degree close curly brackets.

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