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

Pertanyaan

Tentukanlah penyelesaian dari 3 space cot space x plus tan space x greater or equal than 5 sec space x pada interval 0 degree less than x less than 360 degree.

Pembahasan Video:

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.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

N. Puspita

Terakhir diupdate 07 Oktober 2021

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