Himpunan penyelesaian dari, untuk, adalah…

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PERTANYAAN :

Himpunan penyelesaian dari 4 space cos squared 3 x minus 3 equals 0 untuk 0 degree less or equal than x less or equal than 360 degree adalah…

  1. open curly brackets 10 degree comma space 50 degree comma space 70 degree comma space 110 degree comma space 130 degree comma space 170 degree comma space close curly brackets open curly brackets 190 degree comma space 230 degree comma space 250 degree comma space 290 degree comma space 310 degree comma space 350 degree close curly brackets  

  2. open curly brackets 10 degree comma space 50 degree comma space 80 degree comma space 110 degree comma space 140 degree comma space 170 degree comma space close curly brackets open curly brackets 190 degree comma space 210 degree comma space 250 degree comma space 290 degree comma space 310 degree comma space 350 degree close curly brackets  

  3. open curly brackets 10 degree comma space 40 degree comma space 70 degree comma space 100 degree comma space 130 degree comma space 170 degree comma space close curly brackets open curly brackets 180 degree comma space 230 degree comma space 260 degree comma space 290 degree comma space 310 degree comma space 350 degree close curly brackets   

  4. open curly brackets 10 degree comma space 40 degree comma space 80 degree comma space 110 degree comma space 130 degree comma space 160 degree comma space close curly brackets open curly brackets 190 degree comma space 230 degree comma space 250 degree comma space 290 degree comma space 310 degree comma space 350 degree close curly brackets   

  5. open curly brackets 10 degree comma space 50 degree comma space 70 degree comma space 100 degree comma space 120 degree comma space 160 degree comma space close curly brackets open curly brackets 190 degree comma space 230 degree comma space 250 degree comma space 270 degree comma space 300 degree comma space 350 degree close curly brackets   

Table of Contents

Jawaban

jawaban yang benar adalah A.

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah A.

Persamaan Trigonometri Dasar

Penyelesaian persamaan trigonometri dasar bentuk cos space x equals a dilakukan dengan cara: mengubah bentuk cos space x equals a menjadi cos space x equals cos space a.

Ingat persamaan trigonometri dasar berikut:

Jika cos space x equals cos space alpha, nilai x equals alpha plus k times 360 degree atau x equals negative alpha plus k times 360 degree.

Diketahui 4 space cos squared 3 x minus 3 equals 0 untuk 0 degree less or equal than x less or equal than 360 degree misal cos space 3 x equals p, maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell 4 space cos squared 3 x minus 3 end cell equals 0 row cell 4 p squared minus 3 end cell equals 0 row cell 4 p squared end cell equals 3 row cell p squared end cell equals cell 3 over 4 end cell row p equals cell plus-or-minus square root of 3 over 4 end root end cell row blank equals cell plus-or-minus fraction numerator square root of 3 over denominator square root of 4 end fraction end cell row blank equals cell plus-or-minus fraction numerator square root of 3 over denominator 2 end fraction end cell row p equals cell 1 half square root of 3 space atau space straight p equals negative 1 half square root of 3 end cell end table 

Untuk p equals 1 half square root of 3, karena cos space 3 x equals p maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 3 x end cell equals p row cell cos space 3 x end cell equals cell 1 half square root of 3 end cell row cell cos space 3 x end cell equals cell cos space 30 degree end cell row cell 3 x end cell equals cell 30 degree plus k times 360 degree end cell row x equals cell fraction numerator up diagonal strike 3 open parentheses 10 degree plus k times 120 degree close parentheses over denominator up diagonal strike 3 end fraction end cell row blank equals cell 10 degree plus k times 120 degree end cell end table 

atau

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 3 x end cell equals p row cell cos space 3 x end cell equals cell 1 half square root of 3 end cell row cell cos space 3 x end cell equals cell cos space 30 degree end cell row cell 3 x end cell equals cell negative 30 degree plus k times 360 degree end cell row x equals cell fraction numerator up diagonal strike 3 open parentheses negative 10 degree plus k times 120 degree close parentheses over denominator up diagonal strike 3 end fraction end cell row blank equals cell negative 10 degree plus k times 120 degree end cell end table 

Diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row k equals cell 0 rightwards arrow x equals 10 degree plus 0 times 120 degree equals 10 degree space left parenthesis straight M right parenthesis space atau end cell row k equals cell 0 rightwards arrow x equals negative 10 degree plus 0 times 120 degree equals negative 10 degree space left parenthesis TM right parenthesis end cell row k equals cell 1 rightwards arrow x equals 10 degree plus 1 times 120 degree equals 130 degree space left parenthesis straight M right parenthesis space atau end cell row k equals cell 1 rightwards arrow x equals negative 10 degree plus 1 times 120 degree equals 110 degree space left parenthesis straight M right parenthesis end cell row k equals cell 2 rightwards arrow x equals 10 degree plus 2 times 120 degree equals 250 degree space left parenthesis straight M right parenthesis space atau end cell row k equals cell 2 rightwards arrow x equals negative 10 degree plus 2 times 120 degree equals 230 degree space left parenthesis straight M right parenthesis end cell row k equals cell 3 rightwards arrow x equals 10 degree plus 3 times 120 degree equals 370 degree space left parenthesis TM right parenthesis space atau end cell row k equals cell 3 rightwards arrow x equals negative 10 degree plus 3 times 120 degree equals 350 degree space left parenthesis straight M right parenthesis end cell end table   

Untuk p equals negative 1 half square root of 3, karena cos space 3 x equals p maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 3 x end cell equals p row cell cos space 3 x end cell equals cell negative 1 half square root of 3 end cell row cell cos space 3 x end cell equals cell cos space 150 degree end cell row cell 3 x end cell equals cell 150 degree plus k times 360 degree end cell row x equals cell fraction numerator up diagonal strike 3 open parentheses 50 degree plus k times 120 degree close parentheses over denominator up diagonal strike 3 end fraction end cell row blank equals cell 50 degree plus k times 120 degree end cell end table 

atau

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 3 x end cell equals p row cell cos space 3 x end cell equals cell negative 1 half square root of 3 end cell row cell cos space 3 x end cell equals cell cos space 150 degree end cell row cell 3 x end cell equals cell negative 150 degree plus k times 360 degree end cell row x equals cell fraction numerator up diagonal strike 3 open parentheses negative 50 degree plus k times 120 degree close parentheses over denominator up diagonal strike 3 end fraction end cell row blank equals cell negative 50 degree plus k times 120 degree end cell end table 

Diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row straight k equals cell 0 rightwards arrow straight x equals 50 degree plus 0 times 120 degree equals 50 degree space left parenthesis straight M right parenthesis space atau end cell row k equals cell 0 rightwards arrow x equals negative 50 degree plus 0 times 120 degree equals negative 50 degree space left parenthesis TM right parenthesis end cell row k equals cell 1 rightwards arrow x equals 50 degree plus 1 times 120 degree equals 170 degree space left parenthesis straight M right parenthesis space atau end cell row k equals cell 1 rightwards arrow x equals negative 50 degree plus 1 times 120 degree equals 70 degree space left parenthesis straight M right parenthesis end cell row k equals cell 2 rightwards arrow x equals 50 degree plus 2 times 120 degree equals 290 degree space left parenthesis straight M right parenthesis space atau end cell row k equals cell 2 rightwards arrow x equals negative 50 degree plus 2 times 120 degree equals 190 degree space left parenthesis straight M right parenthesis end cell row k equals cell 3 rightwards arrow x equals 50 degree plus 3 times 120 degree equals 410 degree space left parenthesis TM right parenthesis space atau end cell row k equals cell 3 rightwards arrow x equals negative 50 degree plus 3 times 120 degree equals 310 degree space left parenthesis straight M right parenthesis end cell end table  

Keterangan:

M (Memenuhi), TM (Tidak Memenuhi)

Dengan demikian, himpunan penyelesaiannya adalah open curly brackets 10 degree comma space 50 degree comma space 70 degree comma space 110 degree comma space 130 degree comma space 170 degree comma space close curly brackets open curly brackets 190 degree comma space 230 degree comma space 250 degree comma space 290 degree comma space 310 degree comma space 350 degree close curly brackets.

Oleh karena itu, jawaban yang benar adalah A.


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