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Pertanyaan

9. Jika A plus B plus C equals 180 degree, tunjukkan bahwa:

c. cos space 2 A plus cos space 2 B minus cos space 2 C equals 1 minus 4 space sin space A times sin space B times cos space C  space 

I. Ridha

Master Teacher

Mahasiswa/Alumni Universitas Negeri Surabaya

Jawaban terverifikasi

Jawaban

telah ditunjukan bahwa cos space 2 A plus cos space 2 B minus cos space 2 C equals 1 minus 4 times sin space A times sin space B times cos space C.space 

Pembahasan

Ingat kembali:

  • sudut berelasi: cos space open parentheses 180 minus alpha close parentheses equals negative cos space alpha
  • rumus cosinus untuk penjumlahan dua sudut: cos space open parentheses alpha plus beta close parentheses equals cos space alpha times cos space beta minus sin space alpha times sin space beta 
  • rumus cosinus untuk sudut ganda: cos space 2 alpha equals 1 minus 2 times sin squared alpha equals 2 cos squared alpha minus 1 
  • identitas trigonometri: cos squared alpha equals 1 minus sin squared alphasin squared alpha equals 1 minus cos squared alpha, dan sin squared alpha plus cos squared alpha equals 1 

Oleh karena itu, dengan menggunakan sudut berelasi dapat diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell cos space C end cell equals cell cos space open parentheses 180 degree minus left parenthesis A plus B right parenthesis close parentheses end cell row blank equals cell negative cos space left parenthesis A plus B right parenthesis end cell row blank equals cell negative open parentheses cos space A times cos space B minus sin space A times sin space B close parentheses end cell row blank equals cell negative cos space A times cos space B plus sin space A times sin space B end cell row blank equals cell sin space A times sin space B minus cos space A times cos space B end cell end table 

sehingga diperoleh:

begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell cos squared C end cell equals cell open parentheses cos space C close parentheses squared end cell row blank equals cell open parentheses sin space A times sin space B minus cos space A times cos space B close parentheses squared end cell row blank equals cell open parentheses sin space A times sin space B close parentheses squared plus open parentheses cos space A times cos space B close parentheses squared end cell row blank blank cell negative 2 times sin space A times sin space B times cos space A times cos space B end cell row blank equals cell sin squared A times sin squared B plus cos squared A times cos squared B end cell row blank blank cell negative 2 times sin space A times sin space B times cos space A times cos space B end cell row blank equals cell sin squared A times sin squared B plus open parentheses 1 minus sin squared A close parentheses times open parentheses 1 minus sin squared B close parentheses end cell row blank blank cell negative 2 times sin space A times sin space B times cos space A times cos space B end cell row blank equals cell sin squared A times sin squared B plus 1 minus sin squared A minus sin squared B plus sin squared A times sin squared B end cell row blank blank cell negative 2 times sin space A times sin space B times cos space A times cos space B end cell row blank equals cell 1 minus sin squared A minus sin squared B plus sin squared A times sin squared B plus sin squared A times sin squared B end cell row blank blank cell negative 2 times sin space A times sin space B times cos space A times cos space B end cell row blank equals cell 1 minus sin squared A minus sin squared B end cell row blank blank cell plus 2 times sin squared A times sin squared B minus 2 times sin space A times sin space B times cos space A times cos space B end cell row blank equals cell 1 minus sin squared A minus sin squared B end cell row blank blank cell plus 2 times sin space A times sin space B left parenthesis sin space A times sin space B minus c os space A times cos space B right parenthesis end cell row blank equals cell 1 minus sin squared A minus sin squared B end cell row blank blank cell plus 2 times sin space A times sin space B times cos space C end cell end table end style 

Selanjutnya, dengan menggunakan rumus sudut ganda dan kemudian menyubtitusi bentuk cos squared C yang diperoleh di atas, maka diperoleh: 

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell cos space 2 A plus cos space 2 B minus cos space 2 C end cell row blank equals cell 2 times cos squared A minus 1 plus 2 times cos squared B minus 1 minus open parentheses 2 times cos squared C minus 1 close parentheses end cell row blank equals cell 2 times cos squared A minus 1 plus 2 times cos squared B minus 1 minus 2 times cos squared C plus 1 end cell row blank equals cell 2 times cos squared A plus 2 times cos squared B minus 2 times cos squared C minus 1 minus 1 plus 1 end cell row blank equals cell 2 times cos squared A plus 2 times cos squared B minus 2 times cos squared C minus 1 end cell row blank equals cell 2 times cos squared A plus 2 times cos squared B end cell row blank blank cell negative 2 times open parentheses 1 minus sin squared A minus sin squared B plus 2 times sin space A times sin space B times cos space C close parentheses minus 1 end cell row blank equals cell 2 times cos squared A plus 2 times cos squared B end cell row blank blank cell negative 2 plus 2 times sin squared A plus 2 times sin squared B minus 4 times sin space A times sin space B times cos space C minus 1 end cell row blank equals cell negative 2 minus 1 plus 2 times cos squared A plus 2 times sin squared A plus 2 times cos squared B plus 2 times sin squared B end cell row blank blank cell negative 4 times sin space A times sin space B times cos space C end cell row blank equals cell negative 3 plus 2 times open parentheses cos squared A plus sin squared A close parentheses plus 2 times open parentheses cos squared B plus sin squared B close parentheses end cell row blank blank cell negative 4 times sin space A times sin space B times cos space C end cell row blank equals cell negative 3 plus 2 times open parentheses 1 close parentheses plus 2 times open parentheses 1 close parentheses minus 4 times sin space A times sin space B times cos space C end cell row blank equals cell negative 3 plus 2 plus 2 minus 4 times sin space A times sin space B times cos space C end cell row blank equals cell negative 3 plus 4 minus 4 times sin space A times sin space B times cos space C end cell row blank equals cell 1 minus 4 times sin space A times sin space B times cos space C end cell end table     

Dengan demikian, telah ditunjukan bahwa cos space 2 A plus cos space 2 B minus cos space 2 C equals 1 minus 4 times sin space A times sin space B times cos space C.space 

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