Roboguru

Himpunan penyelesaian dari sin2x∘=sinx∘ pada interval 0∘≤x≤180∘  adalah ….

Pertanyaan

Himpunan penyelesaian dari sin invisible function application 2 x degree equals sin invisible function application x degree pada interval 0 degree less or equal than x less or equal than 180 degree  adalah ….

  1. open curly brackets 0 degree comma 180 degree close curly brackets

  2. open curly brackets 0 degree comma 60 degree comma space 180 degree close curly brackets

  3. open curly brackets 60 degree comma 180 degree close curly brackets

  4. open curly brackets 0 degree comma 60 degree close curly brackets

  5. open curly brackets 180 degree close curly brackets

Pembahasan Soal:

Dengan uji titik maka

x equals 0 rightwards arrow sin invisible function application 0 degree equals sin invisible function application 0 degree equals 0  x equals 60 degree rightwards arrow sin invisible function application 120 degree equals sin invisible function application 60 degree equals 1 half square root of 3  x equals 180 degree rightwards arrow sin invisible function application 360 degree equals sin invisible function application 180 degree equals 0

Jadi, himpunan penyelesaiannya adalah  open curly brackets 0 degree comma 60 degree comma space 180 degree close curly brackets

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

N. Ayu

Mahasiswa/Alumni Universitas Negeri Padang

Terakhir diupdate 04 Oktober 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Tentukan penyelesaian dari setiap persamaan trigonometri berikut. b. sin31​x=sin9π​,0≤x≤2π

Pembahasan Soal:

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 alpha plus 2 straight pi k end cell row cell x equals open parentheses straight pi minus straight alpha close parentheses plus 2 straight pi k end cell end table close 

Diketahui persamaan trigonometri sin space 1 third x equals sin space straight pi over 9 comma space 0 less or equal than x less or equal than 2 straight pi. 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 1 third x end cell equals cell sin space straight pi over 9 end cell end table 

Sehingga:

- Persamaan 1 

table attributes columnalign right center left columnspacing 0px end attributes row x equals cell alpha plus 2 straight pi k end cell row cell 1 third x end cell equals cell straight pi over 9 plus 2 straight pi k end cell row x equals cell fraction numerator straight pi over 9 plus 2 straight pi k over denominator begin display style 1 third end style end fraction end cell row x equals cell open square brackets straight pi over 9 plus 2 straight pi k close square brackets cross times 3 over 1 end cell row x equals cell straight pi over 3 plus 6 straight pi k end cell end table 

begin mathsize 12px style k equals negative 1 rightwards arrow x equals straight pi over 3 plus 6 straight pi k equals straight pi over 3 plus 6 straight pi open parentheses negative 1 close parentheses equals negative 17 over 3 straight pi space open parentheses TM close parentheses k equals 0 rightwards arrow x equals straight pi over 3 plus 6 straight pi k equals straight pi over 3 plus 6 straight pi left parenthesis 0 right parenthesis equals straight pi over 3 space space open parentheses straight M close parentheses k equals 1 rightwards arrow x equals straight pi over 3 plus 6 straight pi k equals straight pi over 3 plus 6 straight pi left parenthesis 1 right parenthesis equals 19 over 3 straight pi 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 straight pi minus straight alpha close parentheses plus 2 straight pi k end cell row cell 1 third x end cell equals cell open parentheses straight pi minus straight pi over 9 close parentheses plus 2 straight pi k end cell row cell 1 third x end cell equals cell 8 over 9 straight pi plus 2 straight pi k end cell row x equals cell fraction numerator 8 over 9 straight pi plus 2 straight pi k over denominator begin display style 1 third end style end fraction end cell row x equals cell open parentheses 8 over 9 straight pi plus 2 straight pi k close parentheses cross times 3 over 1 end cell row x equals cell 8 over 3 straight pi plus 6 straight pi k end cell end table 

begin mathsize 12px style k equals negative 1 rightwards arrow x equals 8 over 3 straight pi plus 6 straight pi k equals 8 over 3 straight pi plus 6 straight pi open parentheses negative 1 close parentheses equals negative 10 over 3 straight pi space open parentheses TM close parentheses k equals 0 rightwards arrow x equals 8 over 3 straight pi plus 6 straight pi k equals 8 over 3 straight pi plus 6 straight pi open parentheses 0 close parentheses equals 8 over 3 straight pi space space open parentheses TM close parentheses k equals 1 rightwards arrow x equals 8 over 3 straight pi plus 6 straight pi k equals 8 over 3 straight pi plus 6 straight pi open parentheses 1 close parentheses equals 26 over 3 straight pi space open parentheses TM close parentheses end style 

Jadi, solusi dan himpunan penyelesaian persamaan trigonometri sin space 1 third x equals sin space straight pi over 9 comma space 0 less or equal than x less or equal than 2 straight pi adalah HP equals open curly brackets straight pi over 3 close curly brackets.

0

Roboguru

Bila sin z = sin a, maka z adalah... (1)  (180∘−a)+k.360∘ (2) −a+k.360o (3) a+k.360o (4) a+k.180o

Pembahasan Soal:

sin z =sin a
z = (180° - a) + k.360
z =a+ k.360
Pernyataan 1 dan 3 benar

0

Roboguru

Nilai x yang memenuhi persamaan 2sin(2x−30∘)+3​ untuk 0≤x≤180∘

Pembahasan Soal:

 Ingat kembali konsep dasar:

sin space x equals sin space a rightwards arrow x equals a plus k times 360 rightwards arrow x equals open parentheses 180 minus a close parentheses plus k times 360

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space open parentheses 180 plus a close parentheses end cell equals cell negative sin space a end cell row cell sin space 60 degree end cell equals cell 1 half square root of 3 end cell end table

Sehingga diperoleh perhitungan:

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 space sin space open parentheses 2 x minus 30 degree close parentheses plus square root of 3 end cell equals 0 row cell 2 sin space open parentheses 2 x minus 30 degree close parentheses end cell equals cell negative square root of 3 end cell row cell sin space open parentheses 2 x minus 30 degree close parentheses end cell equals cell fraction numerator negative square root of 3 over denominator 2 end fraction end cell row cell sin space open parentheses 2 x minus 30 degree close parentheses end cell equals cell negative sin space 60 degree space end cell row cell sin space open parentheses 2 x minus 30 degree close parentheses end cell equals cell sin space open parentheses 180 plus 60 close parentheses end cell row cell sin space open parentheses 2 x minus 30 degree close parentheses end cell equals cell sin space 240 degree end cell end table

Solusi I

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x minus 30 end cell equals cell 240 plus k times 360 end cell row cell 2 x end cell equals cell 270 plus k times 360 end cell row x equals cell 135 plus k times 180 end cell row blank blank blank row blank rightwards arrow cell K equals 0 end cell row cell x subscript 1 end cell equals cell 135 plus 0 end cell row cell x subscript 1 end cell equals cell 135 degree end cell end table 

Solusi II

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 x minus 30 end cell equals cell open parentheses 180 minus 240 close parentheses plus k times 360 end cell row cell 2 x end cell equals cell negative 60 plus 30 plus k times 360 end cell row x equals cell negative 15 plus k times 180 end cell row blank blank blank row blank rightwards arrow cell K equals 1 end cell row cell x subscript 2 end cell equals cell negative 15 plus 180 end cell row cell x subscript 2 end cell equals cell 165 degree end cell end table

Dengan mikian, nilai x yang memenuhi persamaan tersebut adalah open curly brackets 135 degree comma space 165 degree close curly brackets  

Jadi,  jawaban yang tepat adalah E

0

Roboguru

Himpunan penyelesaian yang memenuhi persamaan sinx=21​3​ untuk 0∘<x<360∘ adalah ....

Pembahasan Soal:

Diketahui:

Persamaan begin mathsize 14px style sin space straight x equals 1 half square root of 3 end style.

Ditanya:

Himpunan penyelesaian untuk begin mathsize 14px style 0 degree less than straight x less than 360 degree end style

Jawaban:

Dengan menerapkan konsep persamaan bentuk begin mathsize 14px style sin space x equals sin space theta end style maka begin mathsize 14px style straight x equals straight theta plus 360 degree times straight k end style atau begin mathsize 14px style straight x equals left parenthesis 180 minus straight theta right parenthesis plus 360 degree times straight k end style, maka diperoleh: 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell sin space straight x end cell equals cell 1 half square root of 3 end cell row cell sin space straight x end cell equals cell sin space 60 degree end cell row straight x equals cell 60 degree plus 360 degree times straight k space atau end cell row straight x equals cell left parenthesis 180 degree minus 60 degree right parenthesis plus 360 degree times straight k end cell row cell untuk space straight k end cell equals cell 0 comma space maka colon end cell row straight x equals cell 60 degree plus 360 degree times 0 end cell row blank equals cell 60 degree space space atau end cell row straight x equals cell left parenthesis 180 degree minus 60 degree right parenthesis plus 360 degree times 0 end cell row blank equals cell 120 degree end cell row cell untuk space straight k end cell equals cell 1 comma space maka colon end cell row straight x equals cell 60 degree plus 360 degree times 1 end cell row blank equals cell 420 degree space atau end cell row straight x equals cell left parenthesis 180 degree minus 60 degree right parenthesis plus 360 degree times 1 end cell row blank equals cell 480 degree end cell end table end style

untuk begin mathsize 14px style k equals 2 space end style sampai seterusnya tidak perlu di uji coba karena nilai begin mathsize 14px style x end style jika begin mathsize 14px style k equals 1 end style sudah melebihi batas begin mathsize 14px style 0 degree less than straight x less than 360 degree end style.

Maka nilai begin mathsize 14px style x end style yang memenuhi untuk begin mathsize 14px style 0 degree less than straight x less than 360 degree end style adalah begin mathsize 14px style straight x equals 60 degree space dan space straight x equals 120 degree end style.

Jadi, Himpunan penyelesaian yang memenuhi persamaan begin mathsize 14px style sin space straight x equals 1 half square root of 3 end style untuk begin mathsize 14px style 0 degree less than straight x less than 360 degree end style adalah begin mathsize 14px style open curly brackets 60 degree comma space 120 degree close curly brackets end style.

1

Roboguru

Jika sinx=−21​3​ dan 0∘≤x≤360∘, maka x=...

Pembahasan Soal:

Ingat konsep persamaan trigonometri jika sin space x equals sin space alpha maka:

x subscript 1 equals alpha plus k times 360 degree space x subscript 2 equals open parentheses 180 degree minus alpha close parentheses plus k times 360 degree 

Jika sin space x equals negative 1 half square root of 3 dan 0 degree less or equal than x less or equal than 360 degree, maka

sin space x equals negative 1 half square root of 3 sin space x equals sin space 240 degree comma space sin space 300 degree 

untuk sin space x equals sin space 240 degree maka:

 table row cell k equals 0 end cell rightwards arrow cell x subscript 1 equals 240 degree plus 0 times 360 degree end cell row blank blank cell x subscript 1 equals 240 degree end cell row cell k equals 0 end cell rightwards arrow cell x subscript 2 equals open parentheses 180 degree minus 240 close parentheses plus 0 times 360 degree end cell row blank blank cell x subscript 2 equals negative 60 degree space open parentheses Tidak space memenuhi close parentheses end cell row cell k equals 0 end cell rightwards arrow cell x subscript 2 equals open parentheses 180 degree minus 240 close parentheses plus 1 times 360 degree end cell row blank blank cell equals negative 60 degree plus 360 degree end cell row blank blank cell equals 300 degree end cell end table

untuk sin space x equals sin space 300 degree maka

table row cell k equals 0 end cell rightwards arrow cell x subscript 1 equals 300 degree plus 0 times 360 degree end cell row blank blank cell x subscript 1 equals 300 degree end cell row cell k equals 0 end cell rightwards arrow cell x subscript 2 equals open parentheses 180 degree minus 300 close parentheses plus 0 times 360 degree end cell row blank blank cell x subscript 2 equals negative 120 degree space open parentheses Tidak space memenuhi close parentheses end cell row cell k equals 0 end cell rightwards arrow cell x subscript 2 equals open parentheses 180 degree minus 240 close parentheses plus 1 times 360 degree end cell row blank blank cell equals negative 120 degree plus 360 degree end cell row blank blank cell equals 240 degree end cell end table 

Jadi nilai x yang memenuhi adalah 240 degree atau 300 degree 

Oleh karena itu, jawaban yang benar adalah D.

1

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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