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Selesaikan masing-masing pertidaksamaan trigonometri berikut, untuk 0≤x≤2π. a. sin2x−cos2x−sinx>0

Pertanyaan

Selesaikan masing-masing pertidaksamaan trigonometri berikut, untuk 0 less or equal than x less or equal than 2 straight pi.

a. sin squared x minus cos squared x minus sin space x greater than 0 

Pembahasan Soal:

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 

Langkah penyelesaian pertidaksamaa kuadrat:

  1. Kumpulkan semua suku ke dalam satu ruas, semisal ruas kiri. sehingga ruas kanan bernilai 0.
  2. Selesaikan bentuk kuadrat dengan cara memfaktorkan bentuk tersebut untuk mencari nilai yang memenuhi.
  3. Bentuk himpunan penyelesaian diperoleh dengan menampilkan nilai pada garis bilangan. Nilai ini menjadi pembatas pada interval yang nantinya menjadi himpunan penyelesaian.

table attributes columnalign right center left columnspacing 0px end attributes row cell sin squared x minus cos squared x minus sin space x end cell greater than 0 row cell sin squared x minus left parenthesis 1 minus sin squared x right parenthesis minus sin space x end cell greater than 0 row cell sin squared x minus 1 plus sin squared x minus sin space x end cell greater than 0 row cell 2 space sin squared x minus sin space x minus 1 end cell greater than 0 row cell left parenthesis 2 space sin space x plus 1 right parenthesis left parenthesis sin space x minus 1 right parenthesis end cell greater than 0 end table  

table attributes columnalign right center left columnspacing 0px end attributes row blank blank sin end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank less than blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell negative 1 half end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank atau end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell space sin space end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank greater than blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 1 end table   

table attributes columnalign right center left columnspacing 0px end attributes row blank blank 90 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 less than blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank less than blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 210 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 space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank atau end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 330 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 less than blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank less than blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 360 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table 

Jadi, penyelesaian dari persamaan trigonometri tersebut adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank 90 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 less than blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank less than blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 210 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 space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank atau end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 330 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 less than blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank x end table table attributes columnalign right center left columnspacing 0px end attributes row blank less than blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 360 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank degree end table.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

E. Lestari

Mahasiswa/Alumni Universitas Sebelas Maret

Terakhir diupdate 07 Oktober 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Selesaikan masing-masing pertidaksamaan trigonometri berikut, untuk 0≤x≤2π. b. sin(2x​+43​π)≤−21​

Pembahasan Soal:

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 sin open parentheses x over 2 plus 3 over 4 straight pi close parentheses end cell less or equal than cell negative 1 half end cell end table  

Kita buat terlebih dahulu batasnya dengan mengganti tanda pertidaksamaan menjadi =.

table attributes columnalign right center left columnspacing 0px end attributes row cell sin open parentheses x over 2 plus 3 over 4 straight pi close parentheses end cell equals cell negative 1 half end cell row cell sin open parentheses x over 2 plus 135 degree close parentheses end cell equals cell negative 1 half end cell row cell sin open parentheses x over 2 plus 135 degree close parentheses end cell equals cell sin space 210 degree end cell row cell x over 2 plus 135 degree end cell equals cell 210 degree plus k.360 degree end cell row cell x over 2 end cell equals cell 210 degree minus 135 degree plus k.360 degree end cell row cell x over 2 end cell equals cell 75 degree plus k.360 degree end cell row x equals cell 150 degree plus k.720 degree end cell row blank blank blank row k equals 0 row blank rightwards double arrow cell x equals 150 degree plus 0.720 degree end cell row blank equals cell 150 degree end cell row blank blank blank row cell x over 2 plus 135 degree end cell equals cell left parenthesis 180 degree minus 210 degree right parenthesis plus k.360 degree end cell row cell x over 2 plus 135 degree end cell equals cell negative 30 degree plus k.360 degree end cell row cell x over 2 end cell equals cell negative 30 degree minus 135 degree plus k.360 degree end cell row cell x over 2 end cell equals cell negative 165 degree plus k.360 degree end cell row x equals cell negative 330 degree plus k.720 degree end cell row blank blank blank row k equals 1 row x equals cell negative 330 degree plus 1.720 degree end cell row blank equals cell 390 degree space left parenthesis tidak space memenuhi right parenthesis end cell end table  

Melalui uji garis bilangan, maka:

Sehingga penyelesaiannya adalah 150 degree less or equal than x less or equal than 360 degree atau dapata adiubah ke bentuk radian menjadi 65πx2π.

Jadi, penyelesaian pertidaksamaan trigonometri tersebut untuk 0 less or equal than x less or equal than 2 straight pi adalah 65πx2π.

0

Roboguru

Penyelesaian 3tanx+cotx>5cscx; pada 0∘<x<180∘, adalah ....

Pembahasan Soal:

Gunakan konsep penyelesaian pertidaksamaan trigonometri dengan garis bilangan dan persamaan trigonometri.

cos space x equals cos space alpha rightwards arrow open curly brackets table attributes columnalign left end attributes row cell x equals alpha plus k times 360 degree end cell row cell x equals negative alpha plus k times 360 degree end cell end table close

Ingat kembali nilai trigonometri pada sudut istimewa dengan menggunakan tabel trigonometri, perbandingan dan identitas trigonometri sebagai berikut.
 


 

table attributes columnalign right center left columnspacing 2px end attributes row cell sin squared space x plus cos squared space x end cell equals 1 row cell tan space x end cell equals cell fraction numerator sin space x over denominator cos space x end fraction end cell row cell cot space x end cell equals cell fraction numerator cos space x over denominator sin space x end fraction end cell row cell csc space x end cell equals cell fraction numerator 1 over denominator sin space x end fraction end cell end table

Diketahui interval 0 degree less or equal than x less or equal than 180 degree dan 3 space tan space x plus cot space x greater than 5 csc space x, akan ditentukan interval nilai x yang memenuhi.

Terlebih dahulu ubahlah bentuk pertidaksamaan menjadi bentuk pertidaksamaan trigonometri sederhana.

table attributes columnalign right center left columnspacing 2px end attributes row cell 3 space tan space x plus cot space x end cell greater than cell 5 csc space x end cell row cell 3 space tan space x plus cot space x minus 5 csc space x end cell greater than 0 row cell 3 fraction numerator sin space x over denominator cos space x end fraction plus fraction numerator cos space x over denominator sin space x end fraction minus 5 fraction numerator 1 over denominator sin space x end fraction end cell greater than 0 row cell fraction numerator 3 space sin space x over denominator cos space x end fraction plus fraction numerator cos space x over denominator sin space x end fraction minus fraction numerator 5 over denominator sin space x end fraction end cell greater than 0 row cell fraction numerator 3 space sin squared space x plus cos squared space x minus 5 cos space x over denominator cos space x times sin space x end fraction end cell greater than 0 row cell fraction numerator 2 space sin squared space x plus left parenthesis sin squared x plus cos squared space x right parenthesis minus 5 cos space x over denominator cos space x times sin space x end fraction end cell greater than 0 row cell fraction numerator 2 space sin squared space x plus 1 minus 5 cos space x over denominator cos space x times sin space x end fraction end cell greater than 0 row cell fraction numerator 2 space left parenthesis 1 minus cos squared space x right parenthesis plus 1 minus 5 cos space x over denominator cos space x times sin space x end fraction end cell greater than 0 row cell fraction numerator negative 2 cos squared space x minus 5 cos space x plus 3 over denominator cos space x times sin space x end fraction end cell greater than 0 row cell fraction numerator 2 cos squared space x plus 5 cos space x minus 3 over denominator cos space x times sin space x end fraction end cell less than 0 end table

tentukan nilai x yang memenuhi dengan menggunakan cara mengubah ke pertidaksamaan trigonometri menjadi persamaan trigonometri.

table attributes columnalign right center left columnspacing 2px 2px 2px end attributes row cell fraction numerator 2 cos squared space x plus 5 cos space x minus 3 over denominator cos space x times sin space x end fraction end cell equals 0 row cell 2 cos squared x plus 5 cos space x minus 3 end cell equals 0 row cell misal space t equals cos space x end cell blank blank row cell 2 t squared plus 5 t minus 3 end cell equals 0 row cell 1 half left parenthesis 2 t plus 6 t right parenthesis left parenthesis 2 t minus 1 right parenthesis end cell equals 0 row cell open parentheses t plus 3 close parentheses open parentheses 2 t minus 1 close parentheses end cell equals 0 row cell t plus 3 equals 0 end cell blank cell 2 t minus 1 equals 0 end cell row cell t equals negative 3 left parenthesis T M right parenthesis end cell blank cell t equals 1 half end cell row cell t equals 1 half end cell blank blank row cell cos space x equals 1 half end cell blank blank row cell cos space x equals cos space 60 degree end cell blank blank end table
Dengan menggunakan persamaan trigonometri diperoleh sebagai berikut.

table attributes columnalign right center left columnspacing 2px end attributes row cell cos space x equals cos space alpha end cell rightwards arrow cell open curly brackets table attributes columnalign left end attributes row cell x equals alpha plus k times 360 degree end cell row cell x equals negative alpha plus k times 360 degree end cell end table close end cell row cell cos space x equals cos space 60 degree end cell rightwards arrow cell open curly brackets table attributes columnalign left end attributes row cell x equals 60 degree plus k times 360 degree end cell row cell x equals negative 60 degree plus k times 360 degree end cell end table close end cell end table

Untuk x equals 60 degree plus k times 360 degree.

table attributes columnalign right center left columnspacing 2px end attributes row cell k equals 0 end cell rightwards arrow cell x equals 60 degree plus open parentheses 0 close parentheses times 360 degree equals 60 degree end cell row cell k equals 1 end cell rightwards arrow cell x equals 60 degree plus open parentheses 1 close parentheses times 360 degree equals 420 degree space open parentheses tidak space memenuhi close parentheses end cell end table

Untuk x equals negative 60 degree plus k times 360 degree.

table attributes columnalign right center left columnspacing 2px end attributes row cell k equals 0 end cell rightwards arrow cell x equals negative 60 degree plus open parentheses 0 close parentheses times 360 degree equals negative 60 degree space open parentheses tidak space memenuhi close parentheses end cell row cell k equals 1 end cell rightwards arrow cell x equals negative 60 degree plus open parentheses 1 close parentheses times 360 degree equals 300 degree space open parentheses tidak space memenuhi close parentheses end cell end table

Sehingga nilai xyang memenuhi syarat 0 degree less than x less than 180 degree adalah 60 degree.

Kemudian tetapkan tanda positif atau negatif yang sesuai dengan mensubstitusikan nilai x pada setiap daerah interval pada fungsi fraction numerator 2 cos squared space x plus 5 cos space x minus 3 over denominator cos space x times sin space x end fraction pada garis bilangan.
Karena penyebut tidak boleh nol, maka x not equal to left curly bracket 90 to the power of 0 right curly bracket. Sehingga batas batas pada garis bilangan adalah 0 to the power of 0 comma 60 to the power of 0 comma 90 to the power of 0 comma space dan space 180 to the power of 0.
Tanda positif dan negatif pada garis bilangan dapat diperoleh sebagai berikut.

*Menentukan tanda di antara 0 degree dan 60 degree pilih sudut 30 degree.

table attributes columnalign right center left columnspacing 2px end attributes row cell fraction numerator 2 cos squared space x plus 5 cos space x minus 3 over denominator cos space x times sin space x end fraction end cell equals cell fraction numerator 2 cos squared space left parenthesis 30 to the power of 0 right parenthesis plus 5 cos space left parenthesis 30 to the power of 0 right parenthesis minus 3 over denominator cos space left parenthesis 30 to the power of 0 right parenthesis cross times sin space left parenthesis 30 to the power of 0 right parenthesis end fraction end cell row blank equals cell fraction numerator begin display style 3 over 2 end style plus begin display style fraction numerator 5 square root of 3 over denominator 2 end fraction end style minus 3 over denominator left parenthesis plus right parenthesis end fraction end cell row blank equals cell fraction numerator left parenthesis plus right parenthesis over denominator left parenthesis plus right parenthesis end fraction equals left parenthesis plus right parenthesis end cell end table

Karena hasilnya bernilai positif, maka tanda di antara 0 degree dan 60 degree adalah positif.

*Menentukan tanda di antara 60 degree dan 90 degree pilih sudut 75 degree (gunakan kalkulator).

table attributes columnalign right center left columnspacing 2px end attributes row cell fraction numerator 2 cos squared space x plus 5 cos space x minus 3 over denominator cos space x times sin space x end fraction end cell equals cell fraction numerator 2 cos squared space left parenthesis 75 to the power of 0 right parenthesis plus 5 cos space left parenthesis 75 to the power of 0 right parenthesis minus 3 over denominator cos space left parenthesis 75 to the power of 0 right parenthesis cross times sin space left parenthesis 75 to the power of 0 right parenthesis end fraction end cell row blank equals cell fraction numerator 0 comma 13 plus begin display style 1 comma 29 end style minus 3 over denominator left parenthesis plus right parenthesis end fraction end cell row blank equals cell fraction numerator left parenthesis negative right parenthesis over denominator left parenthesis plus right parenthesis end fraction equals left parenthesis negative right parenthesis end cell end table

Karena hasilnya bernilai negatif, maka tanda di antara 60 degree dan 90 degree adalah negatif.

*Menentukan tanda di antara 90 degree dan 180 degree pilih sudut 120 degree (gunakan kalkulator).

table attributes columnalign right center left columnspacing 2px end attributes row cell fraction numerator 2 cos squared space x plus 5 cos space x minus 3 over denominator cos space x times sin space x end fraction end cell equals cell fraction numerator 2 cos squared space left parenthesis 120 to the power of 0 right parenthesis plus 5 cos space left parenthesis 120 to the power of 0 right parenthesis minus 3 over denominator cos space left parenthesis 120 to the power of 0 right parenthesis cross times sin space left parenthesis 120 to the power of 0 right parenthesis end fraction end cell row blank equals cell fraction numerator begin display style 1 half end style minus begin display style 5 over 2 end style minus 3 over denominator left parenthesis negative right parenthesis end fraction end cell row blank equals cell fraction numerator left parenthesis negative right parenthesis over denominator left parenthesis negative right parenthesis end fraction equals left parenthesis plus right parenthesis end cell end table

Karena hasilnya bernilai positif, maka tanda di antara 90 degree dan 180 degree adalah positf.


Sehingga diperoleh gambar garis bilangan seperti berikut.


Karena tanda pertidaksamaan adalah less than, maka daerah yang memenuhi pertidaksamaan fraction numerator 2 cos squared space x plus 5 cos space x minus 3 over denominator cos space x times sin space x end fraction less than 0 adalah yang bertanda negatif, yaitu 60 degree less than x less than 90 degree.

Jadi, jawaban yang tepat adalah D.

0

Roboguru

Tentukanlah penyelesaian dari 3cotx+tanx≥5secx pada interval 0∘<x<360∘.

Pembahasan Soal:

Gunakan konsep penyelesaian pertidaksamaan trigonometri dengan garis bilangan dan persamaan trigonometri.

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

Ingat kembali nilai trigonometri pada sudut istimewa dengan menggunakan tabel trigonometri, perbandingan dan identitas trigonometri sebagai berikut.
 


 

table attributes columnalign right center left columnspacing 2px end attributes row cell sin squared space x plus cos squared space x end cell equals 1 row cell tan space x end cell equals cell fraction numerator sin space x over denominator cos space x end fraction end cell row cell cot space x end cell equals cell fraction numerator cos space x over denominator sin space x end fraction end cell row cell sec space x end cell equals cell fraction numerator 1 over denominator cos space x end fraction end cell end table

Diketahui interval 0 degree less than x less than 360 degree dan 3 space cot space x plus tan space x greater or equal than 5 sec space x, akan ditentukan interval nilai x yang memenuhi.

Terlebih dahulu ubahlah 3 space cot space x plus tan space x greater or equal than 5 sec space x dalam bentuk pertidaksamaan sederhana. 

table attributes columnalign right center left columnspacing 2px end attributes row cell 3 space cot space x plus tan space x end cell greater or equal than cell 5 sec space x end cell row cell 3 space cot space x plus greater or equal than space x end cell greater or equal than cell 5 sec space x end cell row cell 3 space cot space x plus tan space x minus 5 sec space x end cell greater or equal than 0 row cell 3 fraction numerator cos space x over denominator sin space x end fraction plus fraction numerator sin space x over denominator cos space x end fraction minus 5 fraction numerator 1 over denominator cos space x end fraction end cell greater or equal than 0 row cell fraction numerator 3 cos space x over denominator sin space x end fraction plus fraction numerator sin space x over denominator cos space x end fraction minus fraction numerator 5 over denominator cos space x end fraction end cell greater or equal than 0 row cell fraction numerator cos space x over denominator sin space x end fraction plus fraction numerator sin space x minus 5 over denominator cos space x end fraction end cell greater or equal than 0 row cell fraction numerator 3 cos space x times cos space x plus open parentheses sin space x minus 5 close parentheses times sin space x over denominator sin space x times cos space x end fraction end cell greater or equal than 0 row cell fraction numerator 3 cos squared space x plus sin squared space x minus 5 sin space x over denominator sin space x times cos space x end fraction end cell greater or equal than 0 row cell fraction numerator 2 cos squared space x plus left parenthesis cos squared x plus sin squared space x right parenthesis minus 5 sin space x over denominator sin space x times cos space x end fraction end cell greater or equal than 0 row cell fraction numerator 2 cos squared space x plus 1 minus 5 sin space x over denominator sin space x times cos space x end fraction end cell greater or equal than 0 row cell fraction numerator 2 left parenthesis 1 minus sin squared x right parenthesis minus 5 sin space x plus 1 over denominator sin space x times cos space x end fraction end cell greater or equal than 0 row cell fraction numerator negative 2 sin squared x minus 5 sin space x plus 3 over denominator sin space x times cos space x end fraction end cell greater or equal than 0 row cell fraction numerator 2 sin squared x plus 5 sin space x minus 3 over denominator sin space x times cos space x end fraction end cell less or equal than 0 end table
tentukan nilai x yang memenuhi dengan menggunakan cara mengubah ke pertidaksamaan trigonometri menjadi persamaan trigonometri.

table attributes columnalign right center left columnspacing 2px end attributes row cell fraction numerator 2 sin squared x plus 5 sin space x minus 3 over denominator sin space x space. space cos space x end fraction end cell equals 0 row cell 2 sin squared x plus 5 sin space x minus 3 end cell equals 0 row cell misal space t equals sin space x end cell blank blank row cell 2 t squared plus 5 t minus 3 end cell equals 0 row cell 1 half left parenthesis 2 t plus 6 right parenthesis left parenthesis 2 t minus 1 right parenthesis end cell equals 0 row cell open parentheses t plus 3 close parentheses open parentheses 2 t minus 1 close parentheses end cell equals 0 row cell t plus 3 equals 0 end cell blank cell 2 t minus 1 equals 0 end cell row cell t equals negative 3 space left parenthesis T M right parenthesis end cell blank cell t equals 1 half left parenthesis M e m e n u h i right parenthesis end cell row cell sin space x equals 1 half end cell blank blank row cell sin space x equals sin space 30 degree end cell blank blank end table

Dengan menggunakan persamaan trigonometri diperoleh sebagai berikut.

table attributes columnalign right center left columnspacing 2px end attributes row cell sin space x equals sin space alpha end cell rightwards arrow cell open curly brackets table attributes columnalign left end attributes row cell x equals alpha plus k times 360 degree end cell row cell x equals open parentheses 180 degree minus alpha close parentheses plus k times 360 degree end cell end table close end cell row cell sin space x equals sin space 30 degree end cell rightwards arrow cell open curly brackets table attributes columnalign left end attributes row cell x equals 30 degree plus k times 360 degree end cell row cell x equals open parentheses 180 degree minus 30 degree close parentheses plus k times 360 degree end cell end table close end cell row blank rightwards arrow cell open curly brackets table attributes columnalign left end attributes row cell x equals 30 degree plus k times 360 degree end cell row cell x equals 150 degree plus k times 360 degree end cell end table close end cell end table

Untuk x equals 30 degree plus k times 360 degree.

table attributes columnalign right center left columnspacing 2px end attributes row cell k equals 0 end cell rightwards arrow cell x equals 30 degree plus open parentheses 0 close parentheses times 360 degree equals 30 degree end cell row cell k equals 1 end cell rightwards arrow cell x equals 30 degree plus open parentheses 1 close parentheses times 360 degree equals 390 degree space open parentheses tidak space memenuhi close parentheses end cell end table

Untuk x equals 150 degree plus k times 360 degree.

table attributes columnalign right center left columnspacing 2px end attributes row cell k equals 0 end cell rightwards arrow cell x equals 150 degree plus open parentheses 0 close parentheses times 360 degree equals 150 degree end cell row cell k equals 1 end cell rightwards arrow cell x equals 150 degree plus open parentheses 1 close parentheses times 360 degree equals 510 degree space open parentheses tidak space memenuhi close parentheses end cell end table

Sehingga nilai xyang memenuhi syarat 0 degree less than x less than 360 degree adalah 30 degree dan 150 degree.

Kemudian tetapkan tanda positif atau negatif yang sesuai dengan fraction numerator 2 sin squared x plus 5 sin space x minus 3 over denominator sin space x space. space cos space x end fraction pada garis bilangan.
Karena penyebutnya tidak boleh nol maka x not equal to left curly bracket 90 to the power of 0 comma 180 to the power of 0 comma 270 to the power of 0 right curly bracket, sehingga batas-batas pada garis bilangan adalah 0 to the power of 0 comma 30 to the power of 0 comma 90 to the power of 0 comma 150 to the power of 0 comma 180 to the power of 0 comma 270 to the power of 0 space dan space 360 to the power of 0.
Tanda positif dan negatif pada garis bilangan dapat diperoleh sebagai berikut.

*Menentukan tanda di antara 0 degree dan 30 degree pilih sudut 15 degree (gunakan kalkulator).

fraction numerator 2 sin squared left parenthesis 15 to the power of 0 right parenthesis plus 5 sin left parenthesis 15 to the power of 0 right parenthesis minus 3 over denominator sin space left parenthesis 15 to the power of 0 right parenthesis cross times cos space left parenthesis 15 to the power of 0 right parenthesis end fraction equals fraction numerator 0 comma 13 plus 1 comma 29 minus 3 over denominator left parenthesis plus right parenthesis end fraction equals fraction numerator left parenthesis negative right parenthesis over denominator left parenthesis plus right parenthesis end fraction equals left parenthesis negative right parenthesis

Karena hasilnya bernilai negatif, maka tanda di antara 0 degree dan 30 degree adalah negatif.

*Menentukan tanda di antara 30 degree dan 90 degree pilih sudut 60 degree.

fraction numerator 2 sin squared left parenthesis 60 to the power of 0 right parenthesis plus 5 sin left parenthesis 60 to the power of 0 right parenthesis minus 3 over denominator sin space left parenthesis 60 to the power of 0 right parenthesis cross times cos space left parenthesis 60 to the power of 0 right parenthesis end fraction equals fraction numerator begin display style 3 over 2 end style plus begin display style fraction numerator 5 square root of 3 over denominator 2 end fraction end style minus 3 over denominator left parenthesis plus right parenthesis end fraction equals fraction numerator left parenthesis plus right parenthesis over denominator left parenthesis plus right parenthesis end fraction equals left parenthesis plus right parenthesis

Karena hasilnya bernilai positif, maka tanda di antara 30 degree dan 90 degree adalah positif.

*Menentukan tanda di antara 90 degree dan 150 degree pilih sudut 120 degree.

fraction numerator 2 sin squared left parenthesis 120 to the power of 0 right parenthesis plus 5 sin left parenthesis 120 to the power of 0 right parenthesis minus 3 over denominator sin space left parenthesis 120 to the power of 0 right parenthesis cross times cos space left parenthesis 120 to the power of 0 right parenthesis end fraction equals fraction numerator begin display style 3 over 2 plus fraction numerator 5 square root of 3 over denominator 2 end fraction minus 3 end style over denominator left parenthesis negative right parenthesis end fraction equals fraction numerator left parenthesis plus right parenthesis over denominator left parenthesis negative right parenthesis end fraction equals left parenthesis negative right parenthesis

Karena hasilnya bernilai negatif, maka tanda di antara 90 degree dan 150 degree adalah negatif.
 

*Menentukan tanda di antara 150 degree dan 180 degree pilih sudut 160 degree (gunakan kalkulator).

fraction numerator 2 sin squared left parenthesis 160 to the power of 0 right parenthesis plus 5 sin left parenthesis 160 to the power of 0 right parenthesis minus 3 over denominator sin space left parenthesis 160 to the power of 0 right parenthesis cross times cos space left parenthesis 160 to the power of 0 right parenthesis end fraction equals fraction numerator begin display style 0 comma 23 plus 1 comma 71 minus 3 end style over denominator left parenthesis negative right parenthesis end fraction equals fraction numerator left parenthesis negative right parenthesis over denominator left parenthesis negative right parenthesis end fraction equals left parenthesis plus right parenthesis

Karena hasilnya bernilai positif, maka tanda di antara 150 degree dan 180 degree adalah positif.

*Menentukan tanda di antara 180 degree dan 270 degree pilih sudut 210 degree .

fraction numerator 2 sin squared left parenthesis 210 to the power of 0 right parenthesis plus 5 sin left parenthesis 210 to the power of 0 right parenthesis minus 3 over denominator sin space left parenthesis 210 to the power of 0 right parenthesis cross times cos space left parenthesis 210 to the power of 0 right parenthesis end fraction equals fraction numerator begin display style 1 half minus 5 over 2 minus 3 end style over denominator left parenthesis negative right parenthesis end fraction equals fraction numerator left parenthesis negative right parenthesis over denominator left parenthesis plus right parenthesis end fraction equals left parenthesis negative right parenthesis

Karena hasilnya bernilai negatif, maka tanda di antara 90 degree dan 150 degree adalah negatif.

*Menentukan tanda di antara 270 degree dan 360 degree pilih sudut 330 degree .

fraction numerator 2 sin squared left parenthesis 330 to the power of 0 right parenthesis plus 5 sin left parenthesis 330 to the power of 0 right parenthesis minus 3 over denominator sin space left parenthesis 330 to the power of 0 right parenthesis cross times cos space left parenthesis 330 to the power of 0 right parenthesis end fraction equals fraction numerator begin display style 1 half minus 5 over 2 minus 3 end style over denominator left parenthesis negative right parenthesis end fraction equals fraction numerator left parenthesis negative right parenthesis over denominator left parenthesis negative right parenthesis end fraction equals left parenthesis plus right parenthesis

Karena hasilnya bernilai positif, maka tanda di antara 270 degree dan 360 degree adalah positif.

Sehingga diperoleh gambar garis bilangan seperti berikut.


 

Karena tanda pertidaksamaan adalah less or equal than, maka daerah yang memenuhi pertidaksamaan fraction numerator 2 sin squared x plus 5 sin space x minus 3 over denominator sin space x times cos space x end fraction less or equal than 0 adalah yang bertanda negatif , yaitu 0 degree less than x less or equal than 30 degree atau 90 degree less than x less or equal than 150 degree atau 180 to the power of 0 less than x less than 270 to the power of 0.

Jadi, diperoleh himpunan penyelesaiannya dari pertidaksamaan table attributes columnalign right center left columnspacing 2px end attributes row cell 3 space cot space x plus tan space x end cell greater or equal than cell 5 sec space x end cell end table adalah open curly brackets straight x vertical line 0 degree less than straight x less or equal than 30 degree space atau space 90 degree less than straight x less or equal than 150 degree space atau space 180 degree less than straight x less than 270 degree close curly brackets.

1

Roboguru

Jika secx−1tanx​+secx+1tanx​<0 dan 0∘<x<360∘, tentukan nilai x yang memenuhi.

Pembahasan Soal:

Gunakan konsep penyelesaian pertidaksamaan trigonometri dengan garis bilangan.

Ingat kembali nilai trigonometri pada sudut istimewa dengan menggunakan tabel trigonometri, perbandingan dan identitas trigonometri sebagai berikut.
 


 

table attributes columnalign right center left columnspacing 2px end attributes row cell sin squared space x plus cos squared space x end cell equals 1 row cell tan space x end cell equals cell fraction numerator sin space x over denominator cos space x end fraction end cell row cell csc space x end cell equals cell fraction numerator 1 over denominator sin space x end fraction end cell row cell sec space x end cell equals cell fraction numerator 1 over denominator cos space x end fraction end cell end table

Diketahui interval 0 degree less than x less than 360 degree dan fraction numerator tan space x over denominator sec space x minus 1 end fraction plus fraction numerator tan space x over denominator sec space x plus 1 end fraction less than 0, akan ditentukan interval nilai x yang memenuhi.

Sederhanakan pertidaksamaan trigonometri menjadi seperti berikut.

table attributes columnalign right center left columnspacing 2px end attributes row cell fraction numerator tan space x over denominator sec space x minus 1 end fraction plus fraction numerator tan space x over denominator sec space x plus 1 end fraction end cell less than 0 row cell fraction numerator tan space x over denominator sec space x minus 1 end fraction plus fraction numerator tan space x over denominator sec space x plus 1 end fraction end cell less than 0 row cell fraction numerator begin display style fraction numerator sin space x over denominator cos space x end fraction end style over denominator begin display style fraction numerator 1 over denominator cos space x end fraction end style minus 1 end fraction plus fraction numerator begin display style fraction numerator sin space x over denominator cos space x end fraction end style over denominator begin display style fraction numerator 1 over denominator cos space x end fraction end style plus 1 end fraction end cell less than 0 row cell fraction numerator begin display style fraction numerator sin space x over denominator up diagonal strike cos space x end strike end fraction end style over denominator begin display style fraction numerator 1 minus cos space x over denominator up diagonal strike cos space x end strike end fraction end style end fraction plus fraction numerator begin display style fraction numerator sin space x over denominator up diagonal strike cos space x end strike end fraction end style over denominator begin display style fraction numerator 1 plus cos space x over denominator up diagonal strike cos space x end strike end fraction end style end fraction end cell less than 0 row cell fraction numerator sin space x over denominator 1 minus cos space x end fraction plus fraction numerator sin space x over denominator 1 plus cos space x end fraction end cell less than 0 row cell fraction numerator open parentheses sin space x close parentheses open parentheses 1 plus cos space x close parentheses plus open parentheses sin space x close parentheses open parentheses 1 minus cos space x close parentheses over denominator open parentheses 1 minus cos space x close parentheses times open parentheses 1 plus cos space x close parentheses end fraction end cell less than 0 row cell fraction numerator sin space x plus sin space x space cos space x plus sin space x minus sin space x space cos space x over denominator 1 minus cos squared space x end fraction end cell less than 0 row cell fraction numerator 2 space sin space x over denominator sin squared space x end fraction end cell less than 0 row cell fraction numerator 2 over denominator sin space x end fraction end cell less than 0 row cell 2 space csc space x end cell less than 0 row cell csc space x end cell less than 0 end table


Berdasarkan tabel trigonometri diperoleh bahwa nilai csc space x bernilai positif pada interval 0 degree less than x less than 180 degree dan bernilai negatif pada interval 180 degree less than x less than 360 degree.

Jika digambarkan pada garis bilangan tanda positif dan negatif dapat diperoleh sebagai berikut.
 


 

Karena tanda pertidaksamaan adalah less than, maka daerah yang memenuhi pertidaksamaan fraction numerator tan space x over denominator sec space x minus 1 end fraction plus fraction numerator tan space x over denominator sec space x plus 1 end fraction less than 0 adalah yang bertanda negatif, yaitu 180 degree less than x less than 360 degree.

Jadi, diperoleh himpunan penyelesaiannya adalah open curly brackets x vertical line 180 degree less than x less than 360 degree close curly brackets.

1

Roboguru

Selesaikan persamaan trigonometri di bawah ini, untuk 0∘≤x≤360∘. d. 7secanx=9cosx−3tanx

Pembahasan Soal:

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:

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell 7 space secan space x end cell equals cell 9 space cos space x minus 3 space tan space x end cell row cell fraction numerator 7 over denominator cos space x end fraction end cell equals cell 9 space cos space x minus 3 cross times fraction numerator sin space x over denominator cos space x end fraction space left parenthesis kedua space ruas cross times cos space x right parenthesis end cell row 7 equals cell 9 space cos squared x minus 3 space sin space x end cell row 0 equals cell 9 space cos squared x minus 3 space sin space x minus 7 end cell row 0 equals cell 9 left parenthesis 1 minus sin squared x right parenthesis minus 3 space sin space x minus 7 end cell row 0 equals cell 9 minus 9 space sin squared x minus 3 space sin space x minus 7 end cell row 0 equals cell negative 9 space sin squared x minus 3 space sin space x plus 2 end cell row 0 equals cell 9 space sin squared x plus 3 space sin space x minus 2 end cell row 0 equals cell left parenthesis 3 space sin space x plus 2 right parenthesis left parenthesis 3 space sin space x minus 1 right parenthesis end cell end table end style  

begin mathsize 12px style sin space x equals negative 2 over 3 space atau space sin space x equals 1 third end style   

  • sin space x equals negative 2 over 3  

  table attributes columnalign right center left columnspacing 0px end attributes row cell sin space x end cell equals cell negative 2 over 3 end cell row cell sin space x end cell equals cell sin space left parenthesis negative 41 comma 81 degree right parenthesis end cell row x equals cell negative 41 comma 81 degree plus k.360 degree end cell row blank blank blank row k equals 1 row blank rightwards double arrow cell x equals negative 41 comma 81 degree plus 1.360 degree end cell row blank equals cell 318 comma 19 degree end cell row blank blank blank row x equals cell left parenthesis 180 degree minus left parenthesis negative 41 comma 81 degree right parenthesis right parenthesis plus k.360 degree end cell row blank equals cell 221 comma 81 degree plus k.360 degree end cell row blank blank blank row k equals 0 row blank rightwards double arrow cell x equals 221 comma 81 degree plus k.360 degree end cell row blank equals cell 221 comma 81 degree plus 0.360 degree end cell row blank equals cell 221 comma 81 degree end cell end table  

  • sin space x equals 1 third  

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell sin space x equals 1 third sin space x equals sin space 19 comma 47 degree end cell row x equals cell 19 comma 47 degree plus k.360 degree end cell row blank blank blank row k equals 0 row blank rightwards double arrow cell x equals 19 comma 47 degree plus 0.360 degree end cell row blank equals cell 19 comma 47 degree end cell row blank blank blank row x equals cell left parenthesis 180 degree minus 19 comma 47 degree right parenthesis plus k.360 degree end cell row blank equals cell 160 comma 53 degree plus k.360 degree end cell row blank blank blank row k equals 0 row x equals cell 160 comma 53 degree plus k.360 degree end cell row blank equals cell 160 comma 53 degree plus 0.360 degree end cell row blank equals cell 160 comma 53 degree end cell end table 

Jadi, himpunan penyelesaian dari persamaan trigonometri tersebut adalah open curly brackets 19 comma 47 degree space semicolon space 160 comma 53 degree space semicolon space 221 comma 81 degree space semicolon space 318 comma 19 degree close curly brackets.   

0

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