Pembahasan Soal Lomba Matematika

3. Diketahui sin 18 cos 36 =1/4 , tentukan nilai dari
sin 6 . sin 54 . sin 60=

Penyelesaian
   $sin(6).sin(54).sin(66)=sin(6).sin(66).sin(54)$
   $=-\frac{1}{2}\left ( cos(72)-cos(-60) \right )sin(54)$
   $=-\frac{1}{2}\left ( cos(72)-\frac{1}{2} \right )sin(54)$
   $=-\frac{1}{2}cos(72)sin(54)+\frac{1}{4}sin(54)$
   $=-\frac{1}{2}sin(18)cos(36)+\frac{1}{4}sin(54)$
   $=-\frac{1}{2}.\frac{1}{4}+\frac{1}{4}sin(54)$
   $=-\frac{1}{8}+\frac{1}{4}sin(54)$
$sin(54)=sin(36+18)$
   $=sin(36)cos(18)+cos(36).sin(18)$
$sin(18)= sin(36-18)$
   $= sin(36)cos(18)-cos(36)sin(18)$
Jika kedua sudut dikalikan maka akan diperoleh
   $sin(54).sin(18)=\left ( sin(36)cos(18) \right )^{2}-\left ( cos(36)sin(18) \right )^{2}$
   $\frac{1}{4}=\left ( sin(36)cos(18) \right )^{2}-\frac{1}{16}$
   $\left ( sin(36)cos(18) \right )^{2}=\frac{5}{16}$
   $sin(36)cos(18)=\frac{\sqrt{5}}{4}$
jadi diperoleh

    $sin(54)=sin(36+18)$
   $=sin(36)cos(18)+cos(36).sin(18)$
$=\frac{\sqrt{5}}{4}+\frac{1}{4}$
sehingga dihasilkan
   $-\frac{1}{8}+\frac{1}{4}sin(54)$    $-\frac{1}{8}+\frac{1}{4}\left ( \frac{\sqrt{5}}{4} +\frac{1}{4}\right )$
$= \frac{\sqrt{5}}{16} -\frac{1}{16}\right )$

Daftar Hasil Penyelesaian

Minggu, 22 April 2012

Pembahasan Soal Lomba Matematika

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