Roboguru

Selesaikanlah tiap persamaan berikut untuk 0≤x≤2π. f. 2cosxcos(x+3π​)=1

Pertanyaan

Selesaikanlah tiap persamaan berikut untuk 0 less or equal than x less or equal than 2 straight pi.

f. 2 space cos space x space cos space open parentheses x plus straight pi over 3 close parentheses equals 1 

Pembahasan Soal:

Ingat bahwa:

  • cos space left parenthesis a plus b right parenthesis equals cos space a space cos space b minus sin space a space sin space b 
  • sin squared x plus cos squared x equals 1 

Penyelesaiannya adalah sebagai berikut:

table attributes columnalign right center left columnspacing 0px end attributes row cell 2 cos open parentheses x close parentheses cos open parentheses x plus pi over 3 close parentheses end cell equals 1 row cell 2 cos open parentheses x close parentheses open parentheses 1 half cos open parentheses x close parentheses minus fraction numerator square root of 3 over denominator 2 end fraction sin open parentheses x close parentheses close parentheses end cell equals 1 row cell 2 cos open parentheses x close parentheses 1 half cos open parentheses x close parentheses minus 2 cos open parentheses x close parentheses fraction numerator square root of 3 over denominator 2 end fraction sin open parentheses x close parentheses end cell equals 1 row cell cos squared open parentheses x close parentheses minus square root of 3 cos open parentheses x close parentheses sin open parentheses x close parentheses end cell equals 1 row cell cos squared open parentheses x close parentheses minus square root of 3 cos open parentheses x close parentheses sin open parentheses x close parentheses minus 1 end cell equals 0 row cell negative sin squared open parentheses x close parentheses minus cos open parentheses x close parentheses sin open parentheses x close parentheses square root of 3 end cell equals 0 row cell negative sin open parentheses x close parentheses open parentheses sin open parentheses x close parentheses plus square root of 3 cos open parentheses x close parentheses close parentheses end cell equals 0 end table  

table attributes columnalign right center left columnspacing 0px end attributes row blank blank blank row blank blank blank row blank blank blank row blank blank cell table row cell rightwards arrow sin open parentheses x close parentheses equals 0 x equals 2 pi n comma thin space x equals pi plus 2 pi n end cell row blank row cell rightwards arrow sin open parentheses x close parentheses plus square root of 3 cos open parentheses x close parentheses equals 0 fraction numerator sin open parentheses x close parentheses plus square root of 3 cos open parentheses x close parentheses over denominator cos open parentheses x close parentheses end fraction equals fraction numerator 0 over denominator cos open parentheses x close parentheses end fraction tan open parentheses x close parentheses plus square root of 3 equals 0 tan open parentheses x close parentheses equals negative square root of 3 x equals fraction numerator 2 pi over denominator 3 end fraction plus pi n end cell row blank end table end cell end table 

Dari penyelesaian tersebut, yang memenuhi 0 less or equal than x less or equal than 2 straight pi adalah

HP equals open curly brackets 0 comma thin space pi comma thin space 2 pi comma thin space fraction numerator 2 pi over denominator 3 end fraction comma thin space fraction numerator 5 pi over denominator 3 end fraction close curly brackets 

Jadi, penyelesaiannya adalah HP equals open curly brackets 0 comma thin space pi comma thin space 2 pi comma thin space fraction numerator 2 pi over denominator 3 end fraction comma thin space fraction numerator 5 pi over denominator 3 end fraction close curly brackets.

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

S. Yoga

Mahasiswa/Alumni Universitas Pendidikan Indonesia

Terakhir diupdate 06 Oktober 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Tentukan himpunan penyelesaian tiap persamaan berikut untuk 0∘≤x≤360∘. e. 2cos(2x+30∘)cos(2x−60∘)=21​3​

Pembahasan Soal:

Ingat bahwa:

  • cos space left parenthesis a plus b right parenthesis equals cos space a space cos space b minus sin space a space sin space b cos space left parenthesis a minus b right parenthesis equals cos space a space cos space b plus sin space a space sin space b
  • cos squared open parentheses x close parentheses equals 1 minus sin squared open parentheses x close parentheses  

Mencari himpunan penyelesaian:

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell 2 cos open parentheses 2 x plus 30 to the power of ring operator thin space end exponent close parentheses cos open parentheses 2 x minus 60 to the power of ring operator thin space end exponent close parentheses end cell equals cell 1 half square root of 3 end cell row cell fraction numerator square root of 3 over denominator 2 end fraction cos squared open parentheses 2 x close parentheses plus sin open parentheses 2 x close parentheses cos open parentheses 2 x close parentheses minus fraction numerator square root of 3 over denominator 2 end fraction sin squared open parentheses 2 x close parentheses end cell equals cell fraction numerator square root of 3 over denominator 2 end fraction end cell row cell fraction numerator square root of 3 over denominator 2 end fraction cos squared open parentheses 2 x close parentheses plus sin open parentheses 2 x close parentheses cos open parentheses 2 x close parentheses minus fraction numerator square root of 3 over denominator 2 end fraction sin squared open parentheses 2 x close parentheses minus fraction numerator square root of 3 over denominator 2 end fraction end cell equals 0 row cell fraction numerator square root of 3 cos squared open parentheses 2 x close parentheses minus square root of 3 sin squared open parentheses 2 x close parentheses minus square root of 3 plus 2 sin open parentheses 2 x close parentheses cos open parentheses 2 x close parentheses over denominator 2 end fraction end cell equals 0 row cell fraction numerator square root of 3 cos squared open parentheses 2 x close parentheses minus square root of 3 sin squared open parentheses 2 x close parentheses minus square root of 3 plus 2 sin open parentheses 2 x close parentheses cos open parentheses 2 x close parentheses over denominator 2 end fraction end cell equals 0 row cell 2 cos open parentheses 2 x close parentheses sin open parentheses 2 x close parentheses minus 2 sin squared open parentheses 2 x close parentheses square root of 3 end cell equals 0 row cell 2 sin open parentheses 2 x close parentheses open parentheses cos open parentheses 2 x close parentheses minus square root of 3 sin open parentheses 2 x close parentheses close parentheses end cell equals 0 end table end style  

sin open parentheses 2 x close parentheses equals 0 space atau space cos open parentheses 2 x close parentheses minus square root of 3 sin open parentheses 2 x close parentheses equals 0 x equals 180 degree n comma thin space x equals 90 degree plus 180 degree n comma thin space x equals 15 degree plus fraction numerator 180 degree n over denominator 2 end fraction  

Dari penyelesaian tersebut, yang memenuhi 0 degree less or equal than x less or equal than 360 degree adalah:

table attributes columnalign center left center end attributes row blank cell x equals 180 degree n end cell blank row cell x equals 0 rightwards arrow end cell cell x equals 0 degree end cell blank row cell x equals 1 rightwards arrow end cell cell x equals 180 degree end cell blank row cell x equals 2 rightwards arrow end cell cell x equals 360 degree end cell blank row blank cell x equals 90 degree plus 180 degree n end cell blank row cell x equals 0 rightwards arrow end cell cell x equals 90 degree end cell blank row cell x equals 1 rightwards arrow end cell cell x equals 270 degree end cell blank row blank cell x equals 15 degree plus fraction numerator 180 degree n over denominator 2 end fraction end cell blank row cell x equals 0 rightwards arrow end cell cell x equals 15 degree end cell blank row cell x equals 1 rightwards arrow end cell cell x equals 105 degree end cell blank row cell x equals 2 rightwards arrow end cell cell x equals 195 degree end cell blank row cell x equals 3 rightwards arrow end cell cell x equals 285 degree end cell blank end table 

Jadi, himpunan penyelesaiannya adalah  HP equals left curly bracket 0 degree comma space 15 degree comma space 90 degree comma space 105 degree comma space 180 degree comma space 195 degree comma space 270 degree comma space 285 degree comma space 360 degree right curly bracket.

0

Roboguru

Tentukan nilai dari: sin35∘cos5∘−cos35∘cos85∘.

Pembahasan Soal:

Gunakan konsep rumus perkalian trigonometri dan sudut relasi open parentheses 90 degree minus x close parentheses.

table attributes columnalign right center left columnspacing 2px end attributes row cell 2 space sin space straight A space cos space straight B end cell equals cell sin space open parentheses straight A plus straight B close parentheses plus sin space open parentheses straight A minus straight B close parentheses end cell row atau blank blank row cell sin space straight A space cos space straight B end cell equals cell 1 half open parentheses sin space open parentheses straight A plus straight B close parentheses plus sin space open parentheses straight A minus straight B close parentheses close parentheses end cell row blank blank blank row cell 2 space cos space straight A space cos space straight B end cell equals cell cos space open parentheses straight A plus straight B close parentheses plus cos space open parentheses straight A minus straight B close parentheses end cell row atau blank blank row cell cos space straight A space cos space straight B end cell equals cell 1 half open parentheses cos space open parentheses straight A plus straight B close parentheses plus cos space open parentheses straight A minus straight B close parentheses close parentheses end cell row blank blank blank row cell sin space open parentheses 90 degree minus alpha close parentheses end cell equals cell cos space alpha end cell row cell cos space open parentheses negative alpha close parentheses end cell equals cell cos space alpha end cell end table

Ingat kembali nilai trigonometri sudut istimewa 30 degree dan sudut istimewa di kuadran II 120 degree.

table attributes columnalign right center left columnspacing 2px end attributes row cell sin space 30 degree end cell equals cell 1 half end cell row cell cos space 120 degree end cell equals cell negative 1 half end cell end table

Perhatikan perhitungan berikut.

table attributes columnalign right center left columnspacing 2px end attributes row blank blank cell sin space 35 degree space cos space 5 degree minus cos space 35 degree space cos space 85 degree end cell row blank equals cell 1 half open parentheses sin space open parentheses 35 degree plus 5 degree close parentheses plus sin space open parentheses 35 degree minus 5 degree close parentheses close parentheses minus end cell row blank blank cell 1 half open parentheses cos space open parentheses 35 degree plus 85 degree close parentheses plus cos space open parentheses 35 degree minus 85 degree close parentheses close parentheses end cell row blank equals cell 1 half open parentheses sin space 40 degree plus sin space 30 degree close parentheses minus 1 half open parentheses cos space 120 degree plus cos space open parentheses negative 50 degree close parentheses close parentheses end cell row blank equals cell 1 half open parentheses sin space open parentheses 90 degree minus 50 degree close parentheses plus 1 half close parentheses minus 1 half open parentheses negative 1 half plus cos space 50 degree close parentheses end cell row blank equals cell 1 half open parentheses cos space 50 degree plus 1 half close parentheses minus 1 half open parentheses negative 1 half plus cos space 50 degree close parentheses end cell row blank equals cell fraction numerator cos space 50 degree over denominator 2 end fraction plus 1 fourth plus 1 fourth minus fraction numerator cos space 50 degree over denominator 2 end fraction end cell row blank equals cell open parentheses 1 fourth plus 1 fourth close parentheses plus open parentheses fraction numerator cos space 50 degree over denominator 2 end fraction minus fraction numerator cos space 50 degree over denominator 2 end fraction close parentheses end cell row blank equals cell 2 over 4 plus 0 end cell row blank equals cell 1 half end cell end table 

Jadi, diperoleh nilai sin space 35 degree space cos space 5 degree minus cos space 35 degree space cos space 85 degree equals 1 half.

0

Roboguru

Jika sinAsinB=p dan cos(A−B)=q maka nyatakan bentuk cosAcosB dalam p dan q.

Pembahasan Soal:

Ingat bahwa rumus perkalian sinus sebagai berikut:

2sinAsinB=cos(A+B)cos(A-B) 

Kemudian rumus perkalian cosinus sebagai berikut:

2cosAcosB=cos(A+B)+cos(A-B) 

Berdasarkan kedua rumus di atas, maka persoalan tersebut dapat diselesaikan sebagai berikut:

sinAsinBppp21qq2p=====21(cos(A+B)cos(A-B))21(cos(A+B)q)21(cos(A+B))+21q21(cos(A+B))cos(A+B) 

Sehingga penyelesaian persoalan tersebut dapat dijabarkan sebagai berikut:

cosAcosB====2cos(A+B)+cos(A-B)2q2p+q22q2p(p-q) 

Dengan demikian, bentuk cosAcosB adalah (p-q).

0

Roboguru

Hasil dari tan165∘+tan75∘tan52,5∘−tan7,5∘​ adalah ...

Pembahasan Soal:

Ingat: sin space open parentheses A plus-or-minus B close parentheses equals sin space A space cos space B space plus-or-minus space cos space A space sin space B 

          cos space A space cos space B equals 1 half cos space open parentheses A plus B close parentheses plus 1 half cos space open parentheses A minus B close parentheses

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator tan space 52 comma 5 degree minus tan space 7 comma 5 degree over denominator tan space 165 degree plus tan space 75 degree end fraction end cell equals cell fraction numerator begin display style fraction numerator sin space 52 comma 5 degree over denominator cos space 52 comma 5 degree end fraction end style minus begin display style fraction numerator sin space 7 comma 5 degree over denominator cos space 7 comma 5 degree end fraction end style over denominator begin display style fraction numerator sin space 165 degree over denominator cos space 165 degree end fraction end style plus begin display style fraction numerator sin space 75 degree over denominator cos space 75 degree end fraction end style end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator sin space 52 comma 5 degree space cos space 7 comma 5 degree minus cos space 52 comma 5 degree sin space 7 comma 5 degree over denominator cos space 52 comma 5 degree space cos space 7 comma 5 degree end fraction end style over denominator begin display style fraction numerator sin space 165 degree space cos space 75 degree plus cos space 165 degree sin space 75 degree over denominator cos space 165 degree cos space 75 degree end fraction end style end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator sin space open parentheses 52 comma 5 degree minus 7 comma 5 degree close parentheses over denominator begin display style 1 half end style cos space open parentheses 52 comma 5 degree plus 7 comma 5 degree close parentheses plus begin display style 1 half end style cos space open parentheses 52 comma 5 degree minus 7 comma 5 degree close parentheses end fraction end style over denominator begin display style fraction numerator sin space open parentheses 165 degree plus 75 degree close parentheses over denominator 1 half cos space open parentheses 165 degree plus 75 degree close parentheses plus 1 half cos space open parentheses 165 degree minus 75 degree close parentheses end fraction end style end fraction cross times fraction numerator begin display style 1 half end style over denominator begin display style 1 half end style end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator sin space 45 degree over denominator cos space 60 degree plus cos space 45 degree end fraction end style space over denominator begin display style fraction numerator sin space 240 degree over denominator cos space 240 degree plus cos space 90 degree end fraction end style end fraction end cell end table

Dengan sudut berelasi

table attributes columnalign right center left columnspacing 0px end attributes row cell sin space 240 degree end cell equals cell sin space open parentheses 180 degree plus 60 degree close parentheses end cell row blank equals cell negative sin space 60 degree end cell row blank equals cell negative 1 half square root of 3 end cell end table

dan 

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 240 degree end cell equals cell cos space open parentheses 180 degree plus 60 degree close parentheses end cell row blank equals cell negative cos space 60 degree end cell row blank equals cell negative 1 half end cell end table

maka menjadi

table attributes columnalign right center left columnspacing 0px end attributes row cell fraction numerator tan space 52 comma 5 degree minus tan space 7 comma 5 degree over denominator tan space 165 degree plus tan space 75 degree end fraction end cell equals cell fraction numerator begin display style 1 half end style square root of 2 over denominator begin display style 1 half end style plus begin display style 1 half end style square root of 2 end fraction cross times fraction numerator negative begin display style 1 half end style plus 0 over denominator negative begin display style 1 half end style square root of 3 end fraction cross times 4 over 4 end cell row blank equals cell fraction numerator square root of 2 over denominator square root of 6 minus square root of 3 end fraction cross times fraction numerator square root of 6 minus square root of 3 over denominator square root of 6 minus square root of 3 end fraction end cell row blank equals cell fraction numerator 2 square root of 3 minus square root of 6 over denominator 6 minus 3 end fraction end cell row blank equals cell 1 third open parentheses 2 square root of 3 minus square root of 6 close parentheses end cell end table

Jadi, jawaban yang tepat adalah A

0

Roboguru

Jika α+β=6π​ dan cosα+cosβ=43​, maka cos(α+β)=....

Pembahasan Soal:

alpha plus beta equals straight pi over 6 dan cos space alpha space cos space beta equals 3 over 4

alpha plus beta equals straight pi over 6  cos space left parenthesis alpha plus beta right parenthesis equals cos straight pi over 6 equals 1 half square root of 3  cos space alpha space cos space beta equals 3 over 4  2 space cos space alpha space cos space beta space equals 3 over 2  2 space cos space alpha space cos space beta space equals cos space left parenthesis alpha plus beta right parenthesis plus space cos space left parenthesis alpha minus beta right parenthesis  3 over 2 equals 1 half square root of 3 plus cos left parenthesis alpha minus beta right parenthesis  cos space left parenthesis alpha minus beta right parenthesis equals 3 over 2 minus 1 half square root of 3

1

Roboguru

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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