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Bentuk sederhana dari cos x + cos 2 x + cos 3 x + cos 4 x sin x + sin 2 x + sin 3 x + sin 4 x ​ adalah ....

Bentuk sederhana dari  adalah .... 

  1. begin mathsize 14px style tan invisible function application open parentheses fraction numerator 5 straight x over denominator 2 end fraction close parentheses end style 

  2. begin mathsize 14px style cot invisible function application open parentheses fraction numerator 5 straight x over denominator 2 end fraction close parentheses end style 

  3. begin mathsize 14px style tan invisible function application 5 straight x end style 

  4. begin mathsize 14px style cot invisible function application 5 straight x end style 

  5. begin mathsize 14px style tan invisible function application 10 straight x end style 

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Master Teacher

Mahasiswa/Alumni Universitas Putra Indonesia YPTK Padang

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah A.

jawaban yang tepat adalah A.

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Pembahasan

Perhatikan bahwa bentuk penjumlahan pada pembilang dapat disederhanakan sebagai berikut. Sementara itu, pada bentuk penyebut, kita peroleh bentuk sederhananya dengan perhitungan berikut. Dengan demikian, diperoleh bentuk sederhana pada soal adalah sebagai berikut. Jadi, jawaban yang tepat adalah A.

Perhatikan bahwa bentuk penjumlahan pada pembilang dapat disederhanakan sebagai berikut.

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank sin end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank sin end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank sin end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank sin end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 4 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell left parenthesis sin invisible function application straight x plus sin invisible function application 2 straight x space right parenthesis plus left parenthesis sin invisible function application 3 straight x plus sin invisible function application 4 straight x space right parenthesis end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank sin end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses fraction numerator straight x plus 2 straight x over denominator 2 end fraction close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses fraction numerator straight x minus 2 straight x over denominator 2 end fraction close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank sin end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses fraction numerator 3 straight x plus 4 straight x over denominator 2 end fraction close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses fraction numerator 3 straight x minus 4 straight x over denominator 2 end fraction close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank sin end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses fraction numerator 3 straight x over denominator 2 end fraction close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses negative straight x over 2 close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank sin end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses fraction numerator 7 straight x over denominator 2 end fraction close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses negative straight x over 2 close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses negative straight x over 2 close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses sin open parentheses fraction numerator 3 straight x over denominator 2 end fraction close parentheses plus sin open parentheses fraction numerator 7 straight x over denominator 2 end fraction close parentheses close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank space end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses negative straight x over 2 close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses 2 sin open parentheses fraction numerator begin display style fraction numerator 3 straight x over denominator 2 end fraction plus fraction numerator 7 straight x over denominator 2 end fraction end style over denominator 2 end fraction close parentheses cos open parentheses fraction numerator begin display style fraction numerator 3 straight x over denominator 2 end fraction minus fraction numerator 7 straight x over denominator 2 end fraction end style over denominator 2 end fraction close parentheses close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses negative straight x over 2 close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses 2 sin open parentheses fraction numerator 5 straight x over denominator 2 end fraction close parentheses cos open parentheses negative fraction numerator 2 straight x over denominator 2 end fraction close parentheses close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 4 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses negative straight x over 2 close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank sin end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses fraction numerator 5 straight x over denominator 2 end fraction close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell left parenthesis negative straight x right parenthesis end cell end table end style 

Sementara itu, pada bentuk penyebut, kita peroleh bentuk sederhananya dengan perhitungan berikut.

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 3 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight x end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank invisible function application end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 4 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank straight x end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell left parenthesis cos invisible function application straight x plus cos invisible function application 2 straight x space right parenthesis plus left parenthesis cos invisible function application 3 straight x plus cos invisible function application 4 straight x space right parenthesis end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses fraction numerator straight x plus 2 straight x over denominator 2 end fraction close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses fraction numerator straight x minus 2 straight x over denominator 2 end fraction close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses fraction numerator 3 straight x plus 4 straight x over denominator 2 end fraction close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses fraction numerator 3 straight x minus 4 straight x over denominator 2 end fraction close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses fraction numerator 3 straight x over denominator 2 end fraction close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses negative straight x over 2 close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank plus end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses fraction numerator 7 straight x over denominator 2 end fraction close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses negative straight x over 2 close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses negative straight x over 2 close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses cos open parentheses fraction numerator 3 straight x over denominator 2 end fraction close parentheses plus cos open parentheses fraction numerator 7 straight x over denominator 2 end fraction close parentheses close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses negative straight x over 2 close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses 2 cos open parentheses fraction numerator begin display style fraction numerator 3 straight x over denominator 2 end fraction plus fraction numerator 7 straight x over denominator 2 end fraction end style over denominator 2 end fraction close parentheses cos open parentheses fraction numerator begin display style fraction numerator 3 straight x over denominator 2 end fraction minus fraction numerator 7 straight x over denominator 2 end fraction end style over denominator 2 end fraction close parentheses close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 2 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses negative straight x over 2 close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses 2 cos open parentheses fraction numerator begin display style 5 straight x end style over denominator 2 end fraction close parentheses cos open parentheses negative fraction numerator begin display style 2 straight x end style over denominator 2 end fraction close parentheses close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank equals blank end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank 4 end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses negative straight x over 2 close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses fraction numerator begin display style 5 straight x end style over denominator 2 end fraction close parentheses end cell end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cos end table table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell open parentheses negative straight x close parentheses end cell end table end style 

Dengan demikian, diperoleh bentuk sederhana pada soal adalah sebagai berikut.

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator sin invisible function application straight x plus sin invisible function application 2 straight x plus sin invisible function application 3 straight x plus sin invisible function application 4 straight x over denominator cos invisible function application straight x plus cos invisible function application 2 straight x plus cos invisible function application 3 straight x plus cos invisible function application 4 straight x end fraction end cell row blank equals cell fraction numerator 4 cos open parentheses negative begin display style straight x over 2 end style close parentheses sin open parentheses begin display style fraction numerator 5 straight x over denominator 2 end fraction end style close parentheses cos open parentheses negative straight x close parentheses over denominator 4 cos open parentheses negative begin display style straight x over 2 end style close parentheses cos open parentheses begin display style fraction numerator 5 straight x over denominator 2 end fraction end style close parentheses cos open parentheses negative straight x close parentheses end fraction end cell row blank equals cell fraction numerator sin open parentheses begin display style fraction numerator 5 straight x over denominator 2 end fraction end style close parentheses over denominator cos open parentheses begin display style fraction numerator 5 straight x over denominator 2 end fraction end style close parentheses end fraction end cell row blank equals cell tan open parentheses fraction numerator 5 straight x over denominator 2 end fraction close parentheses end cell end table end style 

Jadi, jawaban yang tepat adalah A.

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