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Pertanyaan

6. Jika cos space x equals 1 half, maka cos space 3 x sama dengan ....space 

  1. 3 over 2  

  2. 1 

  3. 1 half square root of 3  

  4. negative 1 half   

  5. negative 1  

I. Ridha

Master Teacher

Mahasiswa/Alumni Universitas Negeri Surabaya

Jawaban terverifikasi

Jawaban

jawaban yang benar adalah E.space 

Pembahasan

Ingat kembali rumus penjumlahan dua sudut:

  • cos open parentheses alpha plus b close parentheses equals cos alpha times cos beta minus sin alpha times sin beta
  • cos space 2 alpha equals 2 times cos squared alpha minus 1
  • sin space 2 alpha equals 2 times sin space alpha times cos space alpha

dan identitas trigonometri sin squared alpha equals 1 minus cos squared alpha sehingga dapat diperoleh rumus berikut:

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 3 alpha end cell equals cell cos open parentheses 2 alpha plus alpha close parentheses end cell row blank equals cell cos 2 alpha times cos alpha minus sin 2 alpha times sin alpha end cell row blank equals cell open parentheses 2 times cos squared alpha minus 1 close parentheses times co s alpha minus open parentheses 2 times sin alpha times cos alpha close parentheses times sin alpha end cell row blank equals cell 2 times cos cubed alpha minus co s alpha minus 2 times sin squared alpha times cos alpha end cell row blank equals cell 2 times cos cubed alpha minus co s alpha minus 2 times open parentheses 1 minus cos squared alpha close parentheses times cos alpha end cell row blank equals cell 2 times cos cubed alpha minus co s alpha minus 2 times open parentheses 1 minus cos squared alpha close parentheses times cos alpha end cell row blank equals cell 2 times cos cubed alpha minus co s alpha minus 2 times 1 times cos alpha plus 2 times cos squared alpha times cos alpha end cell row blank equals cell 2 times cos cubed alpha minus co s alpha minus 2 times cos alpha plus 2 times cos cubed alpha end cell row blank equals cell 2 times cos cubed alpha plus 2 times cos cubed alpha minus co s alpha minus 2 times cos alpha end cell row cell cos space 3 alpha end cell equals cell 4 times cos cubed alpha minus 3 times cos space alpha end cell end table 

Oleh karena cos space x equals 1 half, maka dengan menggunakan rumus di atas, diperoleh:

 table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 3 x end cell equals cell 4 times cos to the power of 3 space end exponent x minus 3 times cos space x end cell row blank equals cell 4 cross times open parentheses 1 half close parentheses cubed minus 3 cross times open parentheses 1 half close parentheses end cell row blank equals cell 4 cross times 1 cubed over 2 cubed minus 3 cross times open parentheses 1 half close parentheses end cell row blank equals cell 4 cross times 1 over 8 minus 3 cross times 1 half end cell row blank equals cell fraction numerator 4 cross times 1 over denominator 8 end fraction minus fraction numerator 3 cross times 1 over denominator 2 end fraction end cell row blank equals cell 4 over 8 minus 3 over 2 end cell row blank equals cell fraction numerator 4 divided by 4 over denominator 8 divided by 4 end fraction minus 3 over 2 end cell row blank equals cell 1 half minus 3 over 2 end cell row blank equals cell fraction numerator 1 minus 3 over denominator 2 end fraction end cell row blank equals cell fraction numerator negative 2 over denominator 2 end fraction end cell row blank equals cell negative 1 end cell end table 

Dengan demikian, diperoleh  cos space 3 x sama dengan negative 1.

Oleh karena itu, jawaban yang benar adalah E.space 

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