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Nyatakan bentuk di bawah ini sebagai jumlah atau selisih kosinus. cos21​πcos41​π

Pertanyaan

Nyatakan bentuk di bawah ini sebagai jumlah atau selisih kosinus.

cos21πcos41π

Pembahasan Soal:

Ingat rumus perkalian trigonometri 2cosαcosβ=cos(α+β)+cos(αβ).

Sehingga diperoleh 

cos21πcos41π====21(2cos21πcos41π)21(cos(21π+41π)+cos(21π41π))21(cos(43π)+cos(41π))21cos(43π)+21cos(41π)

Dengan demikian, bentuk cos21πcos41π sebagai jumlah atau selisih kosinus menjadi 21cos(43π)+21cos(41π).

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

E. Lestari

Mahasiswa/Alumni Universitas Sebelas Maret

Terakhir diupdate 12 September 2021

Roboguru sudah bisa jawab 91.4% pertanyaan dengan benar

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Pertanyaan yang serupa

Buktikan:

Pembahasan Soal:

Ingat kembali:

2 space cos space straight alpha space cos space straight beta equals cos space open parentheses straight alpha plus straight beta close parentheses plus cos space open parentheses straight alpha minus straight beta close parentheses 2 space sin space straight alpha space sin space straight beta equals negative cos space open parentheses straight alpha plus straight beta close parentheses plus cos space open parentheses straight alpha minus straight beta close parentheses 

Akan dibuktikan 2 space sin space 3 x space sin space 4 x plus 2 space cos space 5 x space cos space 2 x minus cos space 3 x equals cos space x. Maka:

  begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell Ruas space kiri end cell equals cell 2 space sin space 3 x space sin space 4 x plus 2 space cos space 5 x space cos space 2 x minus cos space 3 x end cell row blank equals cell open square brackets cos space open parentheses 3 x plus 4 x close parentheses plus cos space open parentheses 3 x minus 4 x close parentheses close square brackets plus open square brackets negative cos space open parentheses 5 x plus 2 x close parentheses plus cos space open parentheses 5 x minus 2 x close parentheses close square brackets minus cos space 3 x end cell row blank equals cell open parentheses cos space 7 x plus cos space open parentheses negative x close parentheses close parentheses plus open parentheses negative cos space 7 x plus cos space 3 x close parentheses minus cos space 3 x end cell row blank equals cell open parentheses cos space 7 x plus cos space x close parentheses plus open parentheses negative cos space 7 x plus cos space 3 x close parentheses minus cos space 3 x end cell row blank equals cell up diagonal strike cos space 7 x end strike plus cos space x up diagonal strike negative cos space 7 x end strike up diagonal strike plus cos space 3 x end strike up diagonal strike negative cos space 3 x end strike end cell row blank equals cell cos space x end cell end table end style 

Jadi, terbukti bahwa 2 space sin space 3 x space sin space 4 x plus 2 space cos space 5 x space cos space 2 x minus cos space 3 x equals cos space x.

0

Roboguru

Selesaikanlah tiap persamaan berikut untuk . f.

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.

0

Roboguru

Jika k=sin10∘sin20cos25∘cos35∘​, maka nilai cos10∘ sama dengan ....

Pembahasan Soal:

Ingat rumus perkalian trigonometri berikut ini:

cosAcosB=21(cos(A+B)+cos(AB)) dan sinAsinB=21(cos(A+B)cos(AB))

Dengan menggunakan konsep di atas, diperoleh hasil:

kk(213cos10)213kkcos10213kkcos10kcos10+cos10cos10(k+1)cos10==========sin10sin20cos25cos3521(cos30cos(10))21(cos60+cos(10))(213cos10)21+cos1021+cos10(21+cos10)21cos1021+213k21(1+3k)2(k+1)k3+12k+2k3+1

Jadi, cos10=2k+2k3+1.

Oleh karena itu, jawaban yang benar adalah C.

0

Roboguru

Tentukan himpunan penyelesaian tiap persamaan berikut untuk . g.

Pembahasan Soal:

Mencari himpunan penyelesaian:

5 space cos squared space x plus 3 space cos space x minus 2 equals 0

table attributes columnalign right center left columnspacing 0px end attributes row cell Misal space cos open parentheses x close parentheses end cell equals u row cell 5 u squared plus 3 u minus 2 end cell equals 0 row u equals cell 2 over 5 comma thin space u equals negative 1 end cell row cell cos open parentheses x close parentheses end cell equals cell 2 over 5 comma thin space cos open parentheses x close parentheses equals negative 1 end cell end table 
 

table attributes columnalign right center left columnspacing 0px end attributes row blank rightwards arrow cell x equals arccos open parentheses 2 over 5 close parentheses plus 360 degree n comma thin space end cell row x equals cell 66 comma 42182 degree plus 360 degree n end cell row blank blank blank row blank rightwards arrow cell x equals 360 degree minus arccos open parentheses 2 over 5 close parentheses plus 360 degree n comma end cell row x equals cell 293 comma 58 degree plus 360 degree n comma thin space end cell row blank blank blank row blank rightwards arrow cell x equals 180 degree plus 360 degree n end cell row x equals cell 180 degree plus 360 degree n end cell end table    

Dari penyelesaian diatas, yang memenuhi 0 degree less or equal than x less or equal than 360 degree adalah HP equals left curly bracket 66 comma 42182 degree semicolon space 180 degree semicolon space 293 comma 58 degree right curly bracket 

Jadi, himpunan penyelesaiannya adalah HP equals left curly bracket 66 comma 42182 degree semicolon space 180 degree semicolon space 293 comma 58 degree right curly bracket.

0

Roboguru

Hasil dari  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

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