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Himpunan penyelesaian dari 2 sin2 4x−1=0 untuk 0≤x≤2π adalah...

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Himpunan penyelesaian dari 2 space sin squared space 4 x minus 1 equals 0 untuk 0 less or equal than x less or equal than 2 straight pi adalah...

  1. begin mathsize 12px style open curly brackets 1 over 16 straight pi comma space 3 over 16 straight pi comma space 5 over 16 straight pi comma space 7 over 16 straight pi comma space 9 over 16 straight pi comma space 11 over 16 straight pi comma space 13 over 16 straight pi comma space 15 over 16 straight pi comma space close curly brackets open curly brackets 17 over 16 straight pi comma space 19 over 16 straight pi comma space 21 over 16 straight pi comma space 23 over 16 straight pi comma space 25 over 16 straight pi comma space 27 over 16 straight pi comma space 29 over 16 straight pi comma space 31 over 16 straight pi close curly brackets end style 

  2. begin mathsize 12px style open curly brackets 1 over 16 straight pi comma space 1 over 8 straight pi comma space 5 over 16 straight pi comma space 7 over 16 straight pi comma space 9 over 16 straight pi comma space 3 over 4 straight pi comma space 13 over 16 straight pi comma space 15 over 16 straight pi comma space close curly brackets open curly brackets 17 over 16 straight pi comma space 19 over 16 straight pi comma space 21 over 16 straight pi comma space 23 over 16 straight pi comma space 25 over 16 straight pi comma space 27 over 16 straight pi comma space 29 over 16 straight pi comma space 31 over 16 straight pi close curly brackets end style 

  3. begin mathsize 12px style open curly brackets 1 over 16 straight pi comma space 3 over 16 straight pi comma space 4 over 16 straight pi comma space 7 over 16 straight pi comma space 9 over 16 straight pi comma space 10 over 16 straight pi comma space 13 over 16 straight pi comma space 15 over 16 straight pi comma space close curly brackets open curly brackets 17 over 16 straight pi comma space 18 over 16 straight pi comma space 21 over 16 straight pi comma space 23 over 16 straight pi comma space 25 over 16 straight pi comma space 27 over 16 straight pi comma space 29 over 16 straight pi comma space 31 over 16 straight pi close curly brackets end style   

  4. begin mathsize 12px style open curly brackets 1 over 16 straight pi comma space 3 over 16 straight pi comma space 5 over 16 straight pi comma space 8 over 16 straight pi comma space 9 over 16 straight pi comma space 11 over 16 straight pi comma space 12 over 16 straight pi comma space 15 over 16 straight pi comma space close curly brackets open curly brackets 17 over 16 straight pi comma space 19 over 16 straight pi comma space 20 over 16 straight pi comma space 23 over 16 straight pi comma space 24 over 16 straight pi comma space 27 over 16 straight pi comma space 29 over 16 straight pi comma space 31 over 16 straight pi close curly brackets end style   

  5. begin mathsize 12px style open curly brackets 1 over 16 straight pi comma space 2 over 16 straight pi comma space 5 over 16 straight pi comma space 7 over 16 straight pi comma space 10 over 16 straight pi comma space 11 over 16 straight pi comma space 13 over 16 straight pi comma space 15 over 16 straight pi comma space close curly brackets open curly brackets 17 over 16 straight pi comma space 18 over 16 straight pi comma space 21 over 16 straight pi comma space 24 over 16 straight pi comma space 25 over 16 straight pi comma space 27 over 16 straight pi comma space 30 over 16 straight pi comma space 31 over 16 straight pi close curly brackets end style  

Y. Herlanda

Master Teacher

Mahasiswa/Alumni STKIP PGRI Jombang

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah A.

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah A.

Persamaan Trigonometri Dasar

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

Ingat persamaan trigonometri dasar berikut:

Jika sin space x equals sin space alpha, nilai x equals alpha plus k times 2 straight pi atau x equals open parentheses straight pi minus straight alpha close parentheses plus k times 2 straight pi.

Diketahui 2 space sin squared space 4 x minus 1 equals 0 untuk 0 less or equal than x less or equal than 2 straight pi, misal sin space 4 x equals p, maka:

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

Untuk p equals 1 half square root of 2, maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 4 x end cell equals cell 1 half square root of 2 end cell row cell sin space 4 x end cell equals cell sin space 45 degree end cell row cell sin space 4 x end cell equals cell sin space 1 fourth straight pi end cell row cell 4 x end cell equals cell 1 fourth straight pi plus k times 2 straight pi end cell row x equals cell 1 over 16 straight pi plus k times 1 half straight pi end cell end table  

atau 

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 4 x end cell equals cell 1 half square root of 2 end cell row cell sin space 4 x end cell equals cell sin space 45 degree end cell row cell sin space 4 x end cell equals cell sin space 1 fourth straight pi end cell row cell 4 x end cell equals cell open parentheses straight pi minus 1 fourth straight pi close parentheses plus k times 2 straight pi end cell row cell 4 x end cell equals cell 3 over 4 straight pi plus k times 2 straight pi end cell row x equals cell 3 over 16 straight pi plus k times 1 half straight pi end cell end table  

Diperoleh

table attributes columnalign right center left columnspacing 0px end attributes row k equals cell 0 rightwards arrow x equals 1 over 16 straight pi plus 0 times 1 half straight pi equals 1 over 16 straight pi space open parentheses straight M close parentheses space atau end cell row k equals cell 0 rightwards arrow x equals 3 over 16 straight pi plus 0 times 1 half straight pi equals 3 over 16 straight pi space open parentheses straight M close parentheses end cell row k equals cell 1 rightwards arrow x equals 1 over 16 straight pi plus 1 times 1 half straight pi equals 9 over 16 straight pi space open parentheses straight M close parentheses space atau end cell row k equals cell 1 rightwards arrow x equals 3 over 16 straight pi plus 1 times 1 half straight pi equals 11 over 16 straight pi space open parentheses straight M close parentheses end cell row k equals cell 2 rightwards arrow x equals 1 over 16 straight pi plus 2 times 1 half straight pi equals 17 over 16 straight pi space open parentheses straight M close parentheses space atau end cell row k equals cell 2 rightwards arrow x equals 3 over 16 straight pi plus 2 times 1 half straight pi equals 19 over 16 straight pi space open parentheses straight M close parentheses end cell row k equals cell 3 rightwards arrow x equals 1 over 16 straight pi plus 3 times 1 half straight pi equals 25 over 16 straight pi space open parentheses straight M close parentheses space atau end cell row k equals cell 3 rightwards arrow x equals 3 over 16 straight pi plus 3 times 1 half straight pi equals 27 over 16 straight pi space open parentheses straight M close parentheses end cell end table  

Untuk p equals negative 1 half square root of 2, maka:

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 4 x end cell equals cell negative 1 half square root of 2 end cell row cell sin space 4 x end cell equals cell sin space 225 degree end cell row cell sin space 4 x end cell equals cell sin space 5 over 4 straight pi end cell row cell 4 x end cell equals cell 5 over 4 straight pi plus k times 2 straight pi end cell row x equals cell 5 over 16 straight pi plus k times 1 half straight pi end cell end table   

atau 

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 4 x end cell equals cell negative 1 half square root of 2 end cell row cell sin space 4 x end cell equals cell sin space 225 degree end cell row cell sin space 4 x end cell equals cell sin space 5 over 4 straight pi end cell row cell 4 x end cell equals cell open parentheses straight pi minus 5 over 4 straight pi close parentheses plus k times 2 straight pi end cell row cell 4 x end cell equals cell negative 1 fourth straight pi plus k times 2 straight pi end cell row x equals cell negative 1 over 16 straight pi plus k times 1 half straight pi end cell end table 

Diperoleh

k equals 0 rightwards arrow x equals 5 over 16 straight pi plus 0 times 1 half straight pi equals 5 over 16 straight pi space left parenthesis straight M right parenthesis space atau k equals 0 rightwards arrow x equals negative 1 over 16 straight pi plus 0 times 1 half straight pi equals negative 1 over 16 straight pi space left parenthesis TM right parenthesis k equals 1 rightwards arrow x equals 5 over 16 straight pi plus 1 times 1 half straight pi equals 13 over 16 straight pi space left parenthesis straight M right parenthesis space atau k equals 1 rightwards arrow x equals negative 1 over 16 straight pi plus 1 times 1 half straight pi equals 7 over 16 straight pi space left parenthesis straight M right parenthesis k equals 2 rightwards arrow x equals 5 over 16 straight pi plus 2 times 1 half straight pi equals 21 over 16 straight pi space left parenthesis straight M right parenthesis space atau k equals 2 rightwards arrow x equals negative 1 over 16 straight pi plus 2 times 1 half straight pi equals 31 over 16 straight pi space left parenthesis straight M right parenthesis k equals 3 rightwards arrow x equals 5 over 16 straight pi plus 3 times 1 half straight pi equals 29 over 16 straight pi space left parenthesis straight M right parenthesis space atau k equals 3 rightwards arrow x equals negative 1 over 16 straight pi plus 3 times 1 half straight pi equals 23 over 16 straight pi space left parenthesis straight M right parenthesis    

Keterangan:

M (Memenuhi), TM (Tidak Memenuhi)

Dengan demikian, himpunan penyelesaiannya adalah begin mathsize 12px style open curly brackets 1 over 16 straight pi comma space 3 over 16 straight pi comma space 5 over 16 straight pi comma space 7 over 16 straight pi comma space 9 over 16 straight pi comma space 11 over 16 straight pi comma space 13 over 16 straight pi comma space 15 over 16 straight pi comma space close curly brackets open curly brackets 17 over 16 straight pi comma space 19 over 16 straight pi comma space 21 over 16 straight pi comma space 23 over 16 straight pi comma space 25 over 16 straight pi comma space 27 over 16 straight pi comma space 29 over 16 straight pi comma space 31 over 16 straight pi close curly brackets end style.

Oleh karena itu, jawaban yang benar adalah A.

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