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

Pertanyaan

Jika 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 dan 0 degree less than x less than 360 degree, tentukan nilai x yang memenuhi.

Pembahasan Video:

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.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

N. Puspita

Terakhir diupdate 07 Oktober 2021

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Pertanyaan yang serupa

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

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

Pada interval 0∘≤x≤180∘ pernyataan 2sin2x≤1 benar apabila ....

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 sebagai berikut.

Diketahui interval 0 degree less or equal than x less or equal than 180 degree dan 2 space sin squared space x less or equal than 1, akan ditentukan interval nilai x yang memenuhi.

Terlebih dahulu 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 2 space sin squared space x end cell less or equal than 1 row cell 2 space sin squared space x end cell equals 1 row cell sin squared space x end cell equals cell 1 half end cell row cell sin space x end cell equals cell square root of 1 half end root end cell row cell sin space x end cell equals cell 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 cell sin space x end cell equals cell 1 half square root of 2 end cell row cell sin space x end cell equals cell sin space 45 degree end cell 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 45 degree end cell rightwards arrow cell open curly brackets table attributes columnalign left end attributes row cell x equals 45 degree plus k times 360 degree end cell row cell x equals open parentheses 180 degree minus 45 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 45 degree plus k times 360 degree end cell row cell x equals 135 degree plus k times 360 degree end cell end table close end cell end table

Untuk x equals 45 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 45 degree plus open parentheses 0 close parentheses times 360 degree equals 45 degree end cell row cell k equals 1 end cell rightwards arrow cell x equals 45 degree plus open parentheses 1 close parentheses times 360 degree equals 405 degree space open parentheses tidak space memenuhi close parentheses end cell end table

Untuk x equals 135 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 135 degree plus open parentheses 0 close parentheses times 360 degree equals 135 degree end cell row cell k equals 1 end cell rightwards arrow cell x equals 135 degree plus open parentheses 1 close parentheses times 360 degree equals 495 degree space open parentheses tidak space memenuhi close parentheses end cell end table

405 degree dan 495 degree tidak memenuhi syarat karena batas nilai x adalah 0 degree less or equal than x less or equal than 180 degree.

Sehingga nilai x yang memenuhi adalah 45 degree dan 135 degree. Kemudian tetapkan tanda positif atau negatif yang sesuai dengan open parentheses 2 space sin squared space x minus 1 close parentheses pada garis bilangan.
Tanda positif dan negatif pada garis bilangan dapat diperoleh sebagai berikut.

*Menentukan tanda di antara 0 degree dan 45 degree pilih sudut 30 degree.

table attributes columnalign right center left columnspacing 2px end attributes row cell 2 space sin squared space open parentheses 30 degree close parentheses minus 1 end cell equals cell 2 times open parentheses 1 half close parentheses squared minus 1 end cell row blank equals cell 2 times 1 fourth minus 1 end cell row blank equals cell 1 half minus 1 end cell row blank equals cell negative 1 half end cell end table

Karena hasilnya bernilai negatif, maka tanda di antara 0 degree dan 45 degree adalah negatif.

*Menentukan tanda di antara 45 degree dan 135 degree pilih sudut 90 degree.

table attributes columnalign right center left columnspacing 2px end attributes row cell 2 space sin squared space open parentheses 90 degree close parentheses minus 1 end cell equals cell 2 times open parentheses 1 close parentheses squared minus 1 end cell row blank equals cell 2 times 1 minus 1 end cell row blank equals cell 2 minus 1 end cell row blank equals 1 end table

Karena hasilnya bernilai positif, maka tanda di antara 45 degree dan 135 degree adalah positif.

*Menentukan tanda di antara 135 degree dan 180 degree pilih sudut 150 degree.

table attributes columnalign right center left columnspacing 2px end attributes row cell 2 space sin squared space open parentheses 150 degree close parentheses minus 1 end cell equals cell 2 times open parentheses 1 half close parentheses squared minus 1 end cell row blank equals cell 2 times 1 fourth minus 1 end cell row blank equals cell 1 half minus 1 end cell row blank equals cell negative 1 half end cell end table

Karena hasilnya bernilai negatif, maka tanda di antara 135 degree dan 180 degree adalah negatif.
Sehingga diperoleh gambar garis bilangan seperti berikut.
 


 

Karena tanda pertidaksamaan adalah less or equal than, maka daerah yang memenuhi pertidaksamaan 2 space sin squared space x less or equal than 1 adalah yang bertanda negatif, yaitu 0 degree less or equal than x less or equal than 45 degree atau 135 degree less or equal than x less or equal than 180 degree.

Jadi, diperoleh jawaban yang tepat adalah A.

0

Roboguru

Pada interval 0∘≤x≤180∘ pernyataan 4cos2x−3≤0 benar apabila ....

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 sebagai berikut.

Diketahui interval 0 degree less or equal than x less or equal than 180 degree dan 4 space cos squared space x minus 3 less or equal than 0, akan ditentukan interval nilai x yang memenuhi.

Terlebih dahulu 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 4 space cos squared space x minus 3 end cell less or equal than 0 row cell 4 space cos squared space x minus 3 end cell equals 0 row cell 4 space cos squared space x end cell equals 3 row cell cos squared space x end cell equals cell 3 over 4 end cell row cell cos space x end cell equals cell plus-or-minus square root of 3 over 4 end root end cell row cell cos space x end cell equals cell plus-or-minus fraction numerator square root of 3 over denominator square root of 4 end fraction end cell row cell cos space x end cell equals cell plus-or-minus fraction numerator square root of 3 over denominator 2 end fraction end cell row cell cos space x equals fraction numerator square root of 3 over denominator 2 end fraction end cell blank cell cos space x equals negative fraction numerator square root of 3 over denominator 2 end fraction end cell row cell cos space x equals cos space 30 degree end cell blank cell cos space x equals cos space 150 degree end cell 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 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 negative 30 degree plus k times 360 degree end cell end table close end cell row cell cos space x equals cos space 150 degree end cell rightwards arrow cell open curly brackets table attributes columnalign left end attributes row cell x equals 150 degree plus k times 360 degree end cell row cell x equals negative 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 negative 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 negative 30 degree plus open parentheses 0 close parentheses times 360 degree equals negative 30 degree space open parentheses tidak space memenuhi close parentheses end cell row cell k equals 1 end cell rightwards arrow cell x equals negative 30 degree plus open parentheses 1 close parentheses times 360 degree equals 330 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

Untuk x equals negative 150 degree plus k times 360 degree.

begin mathsize 12px style table attributes columnalign right center left columnspacing 2px end attributes row cell k equals 0 end cell rightwards arrow cell x equals negative 150 degree plus open parentheses 0 close parentheses times 360 degree equals negative 150 degree space open parentheses tidak space memenuhi close parentheses end cell row cell k equals 1 end cell rightwards arrow cell x equals negative 150 degree plus open parentheses 1 close parentheses times 360 degree equals 210 degree space open parentheses tidak space memenuhi close parentheses end cell end table end style

Sehingga nilai xyang memenuhi syarat 0 degree less or equal than x less or equal than 180 degree adalah 30 degree dan 150 degree.

Kemudian tetapkan tanda positif atau negatif yang sesuai dengan open parentheses cos squared space x minus 3 over 4 close parentheses pada garis bilangan.
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).

table attributes columnalign right center left columnspacing 2px end attributes row cell cos squared space open parentheses 15 degree close parentheses minus 3 over 4 end cell equals cell open parentheses 0 comma 97 close parentheses squared minus 3 over 4 end cell row blank equals cell open parentheses 0 comma 94 close parentheses minus 0 comma 75 end cell row blank equals cell 0 comma 19 end cell end table

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

*Menentukan tanda di antara 30 degree dan 150 degree pilih sudut 60 degree.

table attributes columnalign right center left columnspacing 2px end attributes row cell cos squared space open parentheses 90 degree close parentheses minus 3 over 4 end cell equals cell open parentheses 0 close parentheses squared minus 3 over 4 end cell row blank equals cell 0 minus 3 over 4 end cell row blank equals cell negative 0 comma 75 end cell end table

Karena hasilnya bernilai negatif, maka tanda di antara 30 degree dan 150 degree adalah negatif.

*Menentukan tanda di antara 150 degree dan 180 degree pilih sudut 165 degree (gunakan kalkulator).

table attributes columnalign right center left columnspacing 2px end attributes row cell cos squared space open parentheses 165 degree close parentheses minus 3 over 4 end cell equals cell open parentheses negative 0 comma 97 close parentheses squared minus 3 over 4 end cell row blank equals cell open parentheses 0 comma 94 close parentheses minus 0 comma 75 end cell row blank equals cell 0 comma 19 end cell end table

Karena hasilnya bernilai positif, maka tanda di antara 150 degree dan 180 degree adalah positif.
Sehingga diperoleh gambar garis bilangan seperti berikut.
 


 

Karena tanda pertidaksamaan adalah less or equal than, maka daerah yang memenuhi pertidaksamaan 4 space cos squared space x minus 3 less or equal than 0 adalah yang bertanda negatif, yaitu H P equals left curly bracket x vertical line 30 degree less or equal than x less or equal than 150 degree right curly bracket.

Jadi, diperoleh jawaban yang tepat adalah C.

0

Roboguru

Pada interval 0∘≤x≤180∘ pernyataan 3tan(2x+120∘)≤3​ benar apabila ....

Pembahasan Soal:

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

tan space x equals tan space alpha rightwards arrow x equals alpha plus k times 180 degree

Ingat kembali nilai trigonometri pada sudut istimewa dengan menggunakan tabel trigonometri sebagai berikut.
 


 

Diketahui interval 0 degree less or equal than x less or equal than 180 degree dan 3 space tan space open parentheses 2 x plus 120 degree close parentheses less or equal than square root of 3akan ditentukan interval nilai x yang memenuhi.

Terlebih dahulu 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 3 space tan space open parentheses 2 x plus 120 degree close parentheses end cell less or equal than cell square root of 3 end cell row cell 3 space tan space open parentheses 2 x plus 120 degree close parentheses end cell equals cell square root of 3 end cell row cell tan space open parentheses 2 x plus 120 degree close parentheses end cell equals cell 1 third square root of 3 end cell row cell tan space open parentheses 2 x plus 120 degree close parentheses end cell equals cell tan space 30 degree end cell end table

Dengan menggunakan persamaan trigonometri diperoleh sebagai berikut.

table attributes columnalign right center left columnspacing 2px end attributes row cell tan space x equals tan space alpha end cell rightwards arrow cell x equals alpha plus k times 180 degree end cell row cell tan space open parentheses 2 x plus 120 degree close parentheses end cell equals cell tan space 30 degree end cell row cell 2 x plus 120 degree end cell equals cell 30 degree plus k times 180 degree end cell row cell 2 x end cell equals cell 30 degree plus k times 180 degree minus 120 degree end cell row cell 2 x end cell equals cell negative 90 degree plus k times 180 degree end cell row x equals cell fraction numerator negative 90 degree plus k times 180 degree over denominator 2 end fraction end cell row x equals cell negative 45 degree plus k times 90 degree end cell end table

Menentukan nilai x

table attributes columnalign right center left columnspacing 2px end attributes row cell k equals 0 end cell rightwards arrow cell x equals negative 45 degree plus open parentheses 0 close parentheses times 90 degree equals negative 45 degree space open parentheses tidak space memenuhi close parentheses end cell row cell k equals 1 end cell rightwards arrow cell x equals negative 45 degree plus open parentheses 1 close parentheses times 90 degree equals 45 degree end cell row cell k equals 2 end cell rightwards arrow cell x equals negative 45 degree plus open parentheses 2 close parentheses times 90 degree equals 135 degree end cell row cell k equals 3 end cell rightwards arrow cell x equals negative 45 degree plus open parentheses 3 close parentheses times 90 degree equals 225 degree open parentheses tidak space memenuhi close parentheses end cell end table

negative 45 degree dan 225 degree tidak memenuhi syarat karena batas nilai x adalah 0 degree less or equal than x less or equal than 180 degree.

Sehingga nilai x yang memenuhi adalah 45 degree dan 135 degree. Kemudian tetapkan tanda positif atau negatif yang sesuai dengan open parentheses 3 space tan space open parentheses 2 x plus 120 degree close parentheses minus square root of 3 close parentheses pada garis bilangan.

Batas-batas x pada garis bilangan yaitu 0 to the power of 0 comma 45 to the power of 0 comma 75 to the power of 0 comma 135 to the power of 0 space dan space 180 to the power of 0
batas 75 to the power of 0 diperoleh dari 2 x plus 120 to the power of 0 equals 270 to the power of 0 left right arrow 2 x equals 150 to the power of 0 left right arrow x equals 75 to the power of 0

Tanda positif dan negatif pada garis bilangan dapat diperoleh sebagai berikut.

*Menentukan tanda di antara 0 degree dan 45 degree pilih sudut 30 degree.

table attributes columnalign right center left columnspacing 2px end attributes row cell 3 space tan space open parentheses 2 times 30 degree plus 120 degree close parentheses minus square root of 3 end cell equals cell 3 space tan space open parentheses 60 degree plus 120 degree close parentheses minus square root of 3 end cell row blank equals cell 3 space tan space 180 degree minus square root of 3 end cell row blank equals cell 3 times 0 minus square root of 3 end cell row blank equals cell negative square root of 3 end cell end table

Karena hasilnya bernilai negatif, maka tanda di antara 0 degree dan 45 degree adalah negatif.

*Menentukan tanda di antara 45 degree dan 75 degree pilih sudut 60 degree.

table attributes columnalign right center left columnspacing 2px end attributes row cell 3 space tan space open parentheses 2 times 60 degree plus 120 degree close parentheses minus square root of 3 end cell equals cell 3 space tan space open parentheses 120 degree plus 120 degree close parentheses minus square root of 3 end cell row blank equals cell 3 space tan space 240 degree minus square root of 3 end cell row blank equals cell 3 times open parentheses square root of 3 close parentheses minus square root of 3 end cell row blank equals cell 3 square root of 3 minus square root of 3 end cell row blank equals cell 2 square root of 3 end cell end table

Karena hasilnya bernilai positif, maka tanda di antara 45 degree dan 135 degree adalah positif.

*Menentukan tanda di antara 75 degree dan 135 degree pilih sudut 90 degree.

table attributes columnalign right center left columnspacing 2px end attributes row cell 3 space tan space open parentheses 2 times 90 degree plus 120 degree close parentheses minus square root of 3 end cell equals cell 3 space tan space open parentheses 180 degree plus 120 degree close parentheses minus square root of 3 end cell row blank equals cell 3 space tan space 300 degree minus square root of 3 end cell row blank equals cell 3 times open parentheses negative square root of 3 close parentheses minus square root of 3 end cell row blank equals cell negative 3 square root of 3 minus square root of 3 end cell row blank equals cell negative 4 square root of 3 end cell end table

Karena hasilnya bernilai negatif, maka tanda di antara 75 degree dan 135 degree adalah negatif.

*Menentukan tanda di antara 135 degree dan 180 degree pilih sudut 150 degree.

table attributes columnalign right center left columnspacing 2px end attributes row cell 3 space tan space open parentheses 2 times 150 degree plus 120 degree close parentheses minus square root of 3 end cell equals cell 3 space tan space open parentheses 300 degree plus 120 degree close parentheses minus square root of 3 end cell row blank equals cell 3 space tan space 420 degree minus square root of 3 end cell row blank equals cell 3 times open parentheses square root of 3 close parentheses minus square root of 3 end cell row blank equals cell 3 square root of 3 minus square root of 3 end cell row blank equals cell 2 square root of 3 end cell end table

Karena hasilnya bernilai positif, maka tanda di antara 135 degree dan 180 degree adalah positif.
Sehingga diperoleh gambar garis bilangan seperti berikut.
 

Karena tanda pertidaksamaan adalah less or equal than, maka penyelesaian dari pertidaksamaan 3 space tan space open parentheses 2 x plus 120 degree close parentheses less or equal than square root of 3 adalah yang bertanda negatif, yaitu open curly brackets x vertical line 0 degree less or equal than x less or equal than 45 degree space atau space 75 degree less than straight x less or equal than 180 degree close curly brackets.

Jadi, diperoleh jawaban yang tepat adalah C.

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