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4. Buktikan bahwa: sin3θ⋅sin3θ+cos3θ⋅cos3θ=cos32θ

Pertanyaan

4. Buktikan bahwa:

sin space 3 theta times sin cubed space theta plus cos space 3 theta times cos cubed space theta equals cos cubed space 2 theta  

Pembahasan Soal:

Ingat kembali:

  • rumus sinus untuk penjumlahan dua sudut: sin space open parentheses alpha plus beta close parentheses space equals sin space alpha times cos beta plus cos alpha times sin beta
  • rumus cosinus untuk penjumlahan dua sudut: cos space open parentheses alpha plus beta close parentheses space equals cos space alpha times cos beta plus sin space alpha times sin space beta
  • rumus sinus untuk sudut ganda: sin space 2 alpha equals 2 times sin space alpha times cos space alpha
  • rumus cosinus untuk sudut ganda: cos space 2 alpha equals cos squared alpha minus sin squared alpha
  • identitas trigonometri: sin squared alpha plus cos squared alpha equals 1cos squared alpha equals 1 minus sin squared alpha, dan sin squared alpha equals 1 minus cos squared alpha
  • rumus pemfaktoran pangkat tiga: a squared minus b squared equals open parentheses a minus b close parentheses open parentheses a plus b close parentheses dan a cubed minus b cubed equals open parentheses a minus b close parentheses open parentheses a squared plus b squared plus a b close parentheses

Oleh karena itu, dapat diperoleh rumus berikut:

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

dan

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 

sehingga

 begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell sin space 3 theta times sin cubed theta plus cos space 3 theta times cos cubed theta end cell row blank equals cell open parentheses 3 times sin space theta minus 4 times sin cubed theta close parentheses times sin cubed space theta plus open parentheses 4 times cos cubed theta minus 3 times cos space theta close parentheses times cos cubed theta end cell row blank equals cell 3 times sin to the power of 4 theta minus 4 times sin to the power of 6 theta plus 4 times cos to the power of 6 theta minus 3 times cos to the power of 4 theta end cell row blank equals cell 4 times cos to the power of 6 theta minus 4 times sin to the power of 6 theta minus 3 times cos to the power of 4 theta plus 3 times sin to the power of 4 theta end cell row blank equals cell 4 times open parentheses cos to the power of 6 theta minus sin to the power of 6 theta close parentheses minus 3 times open parentheses cos to the power of 4 theta minus sin to the power of 4 theta close parentheses end cell row blank equals cell 4 times open parentheses open parentheses cos squared theta close parentheses cubed minus open parentheses sin squared theta close parentheses cubed close parentheses minus 3 times open parentheses open parentheses cos squared theta close parentheses squared minus open parentheses sin squared theta close parentheses squared close parentheses end cell row blank equals cell 4 times open parentheses cos squared theta minus sin squared theta close parentheses times open parentheses cos to the power of 4 theta plus sin to the power of 4 theta plus cos squared theta times sin squared theta close parentheses end cell row blank blank cell negative 3 times open parentheses cos squared theta minus sin squared theta close parentheses times open parentheses cos squared theta plus sin squared theta close parentheses end cell row blank equals cell 4 times left parenthesis cos squared theta minus sin squared theta right parenthesis times left parenthesis cos to the power of 4 theta plus sin to the power of 4 theta plus cos squared theta times sin squared theta right parenthesis end cell row blank blank cell negative 3 times open parentheses cos squared theta minus sin squared theta close parentheses times 1 end cell row blank equals cell 4 times left parenthesis cos squared theta minus sin squared theta right parenthesis times left parenthesis cos to the power of 4 theta plus sin to the power of 4 theta plus cos squared theta times sin squared theta right parenthesis end cell row blank blank cell negative 3 times open parentheses cos squared theta minus sin squared theta close parentheses times left parenthesis cos squared theta plus sin squared theta right parenthesis squared end cell row blank equals cell 4 times left parenthesis cos squared theta minus sin squared theta right parenthesis times left parenthesis cos to the power of 4 theta plus sin to the power of 4 theta plus cos squared theta times sin squared theta right parenthesis end cell row blank blank cell negative 3 times open parentheses cos squared theta minus sin squared theta close parentheses times left parenthesis cos to the power of 4 theta plus sin to the power of 4 theta plus 2 times cos squared theta times sin squared theta right parenthesis end cell row blank equals cell left parenthesis cos squared theta minus sin squared theta right parenthesis times left parenthesis 4 times cos to the power of 4 theta plus 4 times sin to the power of 4 theta plus 4 times cos squared theta times sin squared theta right parenthesis end cell row blank blank cell plus open parentheses cos squared theta minus sin squared theta close parentheses times left parenthesis negative 3 times cos to the power of 4 theta minus 3 times sin to the power of 4 theta minus 6 times cos squared theta times sin squared theta right parenthesis end cell row blank equals cell left parenthesis cos squared theta minus sin squared theta right parenthesis times open parentheses cos to the power of 4 theta plus sin to the power of 4 theta minus 2 times cos squared theta times sin squared theta close parentheses end cell row blank equals cell left parenthesis cos squared theta minus sin squared theta right parenthesis times left parenthesis cos squared theta minus sin squared theta right parenthesis squared end cell row blank equals cell left parenthesis cos squared theta minus sin squared theta right parenthesis cubed end cell row blank equals cell left parenthesis cos space 2 theta right parenthesis cubed end cell row blank equals cell cos cubed 2 theta end cell end table end style

Dengan demikian, terbukti bahwa sin space 3 theta times sin cubed space theta plus cos space 3 theta times cos cubed space theta equals cos cubed space 2 theta.space 

Pembahasan terverifikasi oleh Roboguru

Dijawab oleh:

I. Ridha

Mahasiswa/Alumni Universitas Negeri Surabaya

Terakhir diupdate 04 September 2021

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