Roboguru

cos6A−2cos4A−cos2A+2=...

Pertanyaan

cos space 6 straight A space minus 2 space cos space 4 straight A space minus space cos space 2 straight A plus 2 equals...

  1. 64 space cos squared space straight A space sin to the power of 4 space straight A

  2. 64 space sin squared space straight A space cos to the power of 4 space straight A

  3. 32 space sin squared space straight A space cos to the power of 4 space straight A

  4. 32 space cos squared space straight A space sin to the power of 4 space straight A

  5. 16 space sin squared space straight A space cos to the power of 4 space straight A

Pembahasan:

Ingat:

  • identitas sudut ganda: sin space 2 straight x equals 2 sin space straight x space cos space straight x.
  • identitas Pythagoras: cos squared space x space plus space sin squared space x space equals 1.
  •  bentuk aljabar: a squared minus b squared equals open parentheses a minus b close parentheses left parenthesis a plus b right parenthesis.

Sehingga untuk memudahkan mencari bentuk sederhana dari cos space 6 straight A space minus 2 space cos space 4 straight A space minus space cos space 2 straight A plus 2. Kita akan mencari satu persatu

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell cos space 6 straight A end cell row blank equals cell cos squared space 3 straight A space minus space sin squared space 3 straight A end cell row blank equals cell cos squared space open parentheses 2 straight A plus straight A close parentheses space minus space sin squared space open parentheses 2 straight A plus straight A close parentheses end cell row blank equals cell open parentheses cos space 2 straight A space cos space straight A space minus space sin space 2 straight A space sin space straight A close parentheses squared space minus open parentheses sin space 2 straight A space cos space straight A space plus space cos space 2 straight A space sin space straight A close parentheses squared end cell row blank equals cell open parentheses open parentheses 1 minus 2 sin squared space straight A close parentheses space cos space straight A space minus space 2 space sin squared space straight A space cos space straight A close parentheses squared space minus open parentheses 2 sin space straight A space cos squared space straight A space plus open parentheses 1 minus 2 sin squared space straight A close parentheses space sin space straight A close parentheses squared end cell row blank equals cell left parenthesis 1 minus 4 sin squared space straight A space cos squared straight A right parenthesis squared minus sin space straight A cross times open square brackets 2 open parentheses 1 minus 2 sin squared space straight A close parentheses plus open parentheses 1 minus 2 sin squared space straight A close parentheses close square brackets squared space equals space left parenthesis 1 minus 4 sin squared space straight A space cos squared straight A right parenthesis squared minus sin squared space straight A cross times open parentheses 3 minus 4 sin squared space straight A close parentheses squared end cell row blank equals cell open parentheses 1 minus 8 sin squared space straight A plus 16 space sin to the power of 4 space straight A close parentheses open parentheses 1 minus 2 sin squared space straight A close parentheses minus sin squared space straight A open parentheses 9 minus 24 sin squared space straight A plus 16 space sin to the power of 4 space straight A close parentheses end cell row blank equals cell open parentheses 1 minus 8 sin squared space straight A plus 16 space sin to the power of 4 space straight A close parentheses minus open parentheses sin squared space straight A minus 8 sin to the power of 4 space straight A plus 16 sin to the power of 6 space straight A close parentheses minus open parentheses 9 sin squared space straight A minus 24 sin to the power of 4 space straight A plus 16 sin to the power of 6 space straight A close parentheses end cell row blank equals cell negative 32 sin to the power of 6 space straight A plus 48 sin to the power of 4 space straight A minus 18 sin squared space straight A plus 1 end cell end table

Selanjutnya:

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell cos space 4 straight A end cell row blank equals cell cos squared space 2 straight A minus sin squared space 2 straight A end cell row blank equals cell open parentheses 1 minus 2 sin squared space straight A close parentheses squared minus open parentheses 2 sin space straight A space cos space straight A close parentheses squared end cell row blank equals cell 1 minus 4 sin squared space straight A plus 4 sin to the power of 4 space straight A minus 4 sin squared space straight A space cos squared space straight A end cell row blank equals cell 1 minus 4 sin squared space straight A plus 4 sin to the power of 4 space straight A minus 4 sin squared space straight A open parentheses 1 minus sin squared space straight A close parentheses end cell row blank equals cell 1 minus 4 sin squared space straight A plus 4 sin to the power of 4 space straight A minus 4 sin squared space straight A plus 4 sin squared space straight A end cell row blank equals cell 8 sin to the power of 4 space straight A minus 8 sin squared space straight A plus 1 end cell row blank blank maka row blank blank cell negative 2 cos 4 straight A end cell row blank equals cell negative 16 sin to the power of 4 space straight A plus 16 sin squared space straight A minus 2 end cell end table

Selanjutnya:

table attributes columnalign right center left columnspacing 0px end attributes row cell negative cos space 2 straight A end cell equals cell negative open parentheses 1 minus 2 sin squared space straight A close parentheses end cell row blank equals cell 2 sin squared space straight A minus 1 end cell row blank blank blank end table

Sehingga akan diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell cos space 6 straight A space minus 2 space cos space 4 straight A space minus space cos space 2 straight A plus 2 end cell row blank equals cell negative 32 sin to the power of 6 space straight A plus 48 sin to the power of 4 space straight A minus 18 sin squared space straight A plus 1 minus 16 sin to the power of 4 space straight A plus 16 sin squared space straight A minus 2 plus 2 sin squared space straight A minus 1 plus 2 end cell row blank equals cell negative 32 sin to the power of 6 space straight A plus 32 space sin to the power of 4 straight A end cell row blank equals cell negative 32 sin to the power of 4 space straight A open parentheses 1 minus sin squared space straight A close parentheses end cell row blank equals cell negative 32 sin to the power of 4 space straight A space cos squared space straight A end cell row blank equals cell negative 32 space cos squared space straight A space sin to the power of 4 space straight A end cell end table

Jadi, jawaban yang tepat adalah D.

 

Jawaban terverifikasi

Dijawab oleh:

A. Salim

Mahasiswa/Alumni Universitas Pelita Harapan

Terakhir diupdate 15 Oktober 2021

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