Pertanyaan

Gunakan rumus 2 sin α sin β = cos ( α − β ) − cos ( α + β ) untuk membuktikan: .

Gunakan rumus $2 sin α sin β = cos (α - β) - cos (α + β)$ untuk membuktikan: $\begin{array}{rcl}&&\sin\end{array}\begin{array}{rcl}&&\;\end{array}\begin{array}{rcl}&&54\end{array}\begin{array}{rcl}&&\textdegree\end{array}\begin{array}{rcl}&&\;\end{array}\begin{array}{rcl}&&\sin\end{array}\begin{array}{rcl}&&\;\end{array}\begin{array}{rcl}&&18\end{array}\begin{array}{rcl}&&\textdegree\end{array}\begin{array}{rcl}&=&\end{array}\begin{array}{rcl}&&\frac14\end{array}$ .

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Pembahasan

Ingat bahwa: Mencari : Misalkan: , maka: Mencari : Sehingga didapat: Jadi, terbukti bahwa .

Ingat bahwa:

Mencari sin space 18 degree :

Misalkan: theta equals 18 degree , maka:

$table attributes columnalign right center left columnspacing 0px end attributes row cell 5 theta end cell equals cell 90 degree end cell row cell 2 theta plus 3 theta end cell equals cell 90 degree end cell row cell 2 theta end cell equals cell 90 degree minus 3 theta end cell row cell sin 2 theta end cell equals cell sin left parenthesis 90 minus 3 theta right parenthesis equals cos 3 theta end cell row cell 2 sin theta cos theta end cell equals cell 4 cos cubed theta minus 3 cos theta end cell row cell 2 sin theta cos theta minus 4 cos cubed theta plus 3 cos theta end cell equals 0 row cell cos theta open parentheses 2 sin theta minus 4 cos squared theta plus 3 close parentheses end cell equals 0 row cell 2 sin theta minus 4 open parentheses 1 minus sin squared theta close parentheses plus 3 end cell equals 0 row cell sin theta end cell equals cell fraction numerator negative 2 plus-or-minus square root of 2 squared minus 4 cross times 4 cross times left parenthesis negative 1 right parenthesis end root over denominator 2 cross times 4 end fraction end cell row cell sin theta end cell equals cell fraction numerator negative 2 plus-or-minus square root of 20 over denominator 8 end fraction end cell row cell sin theta end cell equals cell fraction numerator square root of 5 minus 1 over denominator 4 end fraction end cell row cell sin space 18 degree end cell equals cell fraction numerator square root of 5 minus 1 over denominator 4 end fraction end cell end table$

Mencari cos space 36 degree :

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos space 36 degree end cell equals cell cos space open parentheses 2 times 18 degree close parentheses end cell row blank equals cell 1 minus 2 sin squared 18 degree end cell row blank equals cell 1 minus 2 open parentheses fraction numerator square root of 5 minus 1 over denominator 4 end fraction close parentheses squared end cell row blank equals cell 1 minus open parentheses fraction numerator square root of 5 plus 1 minus 2 square root of 5 over denominator 8 end fraction close parentheses end cell row blank equals cell fraction numerator square root of 5 plus 1 over denominator 4 end fraction end cell end table$

Sehingga didapat:

$table attributes columnalign right center left columnspacing 0px end attributes row cell cos open parentheses 36 degree close parentheses sin open parentheses 18 degree close parentheses end cell equals cell open parentheses fraction numerator square root of 5 plus 1 over denominator 4 end fraction close parentheses open parentheses fraction numerator square root of 5 minus 1 over denominator 4 end fraction close parentheses end cell row blank equals cell fraction numerator open parentheses square root of 5 plus 1 close parentheses open parentheses square root of 5 minus 1 close parentheses over denominator 4 times thin space 4 end fraction end cell row blank equals cell fraction numerator open parentheses 1 plus square root of 5 close parentheses open parentheses square root of 5 minus 1 close parentheses over denominator 16 end fraction end cell row blank equals cell 4 over 16 end cell row blank equals cell 1 fourth end cell end table$

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