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Tunjukkan bahwa: cos 15 π cos 15 2 π cos 15 3 π cos 15 4 π cos 15 5 π cos 15 6 π cos 15 7 π = 2 − 7

Tunjukkan bahwa:

$cos \frac{π}{15} cos \frac{2 π}{15} cos \frac{3 π}{15} cos \frac{4 π}{15} cos \frac{5 π}{15} cos \frac{6 π}{15} cos \frac{7 π}{15} = 2^{- 7}$

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S. Solehuzain

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Jawaban terverifikasi

Pembahasan

ingat Rumus sudut ganda sinus sin 2 α = 2 sin α cos α Sehingga soal tersebut dapat diuraikan sebagai berikut Dengan demikian, terbukti bahwa cos 15 π cos 15 2 π cos 15 3 π cos 15 4 π cos 15 5 π cos 15 6 π cos 15 7 π = 2 − 7

ingat
Rumus sudut ganda sinus
$sin 2 α = 2 sin α cos α$

Sehingga soal tersebut dapat diuraikan sebagai berikut

$begin mathsize 12px style cos pi over 15 space cos fraction numerator 2 pi over denominator 15 end fraction space cos fraction numerator 3 pi over denominator 15 end fraction space cos fraction numerator 4 pi over denominator 15 end fraction space cos fraction numerator 5 pi over denominator 15 end fraction space cos fraction numerator 6 pi over denominator 15 end fraction space cos fraction numerator 7 pi over denominator 15 end fraction equals fraction numerator 2 space sin pi over 15 over denominator 2 space sin pi over 15 end fraction cross times cos pi over 15 space cos fraction numerator 2 pi over denominator 15 end fraction space cos fraction numerator 3 pi over denominator 15 end fraction space cos fraction numerator 4 pi over denominator 15 end fraction space cos fraction numerator 5 pi over denominator 15 end fraction space cos fraction numerator 6 pi over denominator 15 end fraction space cos fraction numerator 7 pi over denominator 15 end fraction equals fraction numerator 1 over denominator 2 space sin pi over 15 end fraction cross times sin fraction numerator 2 pi over denominator 15 end fraction space cos fraction numerator 2 pi over denominator 15 end fraction space cos fraction numerator 3 pi over denominator 15 end fraction space cos fraction numerator 4 pi over denominator 15 end fraction space cos fraction numerator 5 pi over denominator 15 end fraction space cos fraction numerator 6 pi over denominator 15 end fraction space cos fraction numerator 7 pi over denominator 15 end fraction equals fraction numerator 1 over denominator 2.2 space sin pi over 15 end fraction cross times 2 space sin fraction numerator 2 pi over denominator 15 end fraction space cos fraction numerator 2 pi over denominator 15 end fraction space cos fraction numerator 3 pi over denominator 15 end fraction space cos fraction numerator 4 pi over denominator 15 end fraction space cos fraction numerator 5 pi over denominator 15 end fraction space cos fraction numerator 6 pi over denominator 15 end fraction space cos fraction numerator 7 pi over denominator 15 end fraction equals fraction numerator 1 over denominator 4 space sin pi over 15 end fraction cross times space sin fraction numerator 4 pi over denominator 15 end fraction space cos fraction numerator 3 pi over denominator 15 end fraction space cos fraction numerator 4 pi over denominator 15 end fraction space cos fraction numerator 5 pi over denominator 15 end fraction space cos fraction numerator 6 pi over denominator 15 end fraction space cos fraction numerator 7 pi over denominator 15 end fraction equals fraction numerator 1 over denominator 2.4 space sin pi over 15 end fraction cross times 2. space sin fraction numerator 4 pi over denominator 15 end fraction space cos fraction numerator 3 pi over denominator 15 end fraction space cos fraction numerator 4 pi over denominator 15 end fraction space cos fraction numerator 5 pi over denominator 15 end fraction space cos fraction numerator 6 pi over denominator 15 end fraction space cos fraction numerator 7 pi over denominator 15 end fraction equals fraction numerator 1 over denominator 8 space sin pi over 15 end fraction cross times sin fraction numerator 8 pi over denominator 15 end fraction space cos fraction numerator 3 pi over denominator 15 end fraction space cos fraction numerator 5 pi over denominator 15 end fraction space cos fraction numerator 6 pi over denominator 15 end fraction space cos fraction numerator 7 pi over denominator 15 end fraction text ingat cos end text left parenthesis text π-α end text right parenthesis text =-cosα end text equals fraction numerator 1 over denominator 8 space sin pi over 15 end fraction cross times sin fraction numerator 8 pi over denominator 15 end fraction space cos fraction numerator 3 pi over denominator 15 end fraction space cos fraction numerator 5 pi over denominator 15 end fraction space cos fraction numerator 6 pi over denominator 15 end fraction text cos end text left parenthesis straight pi minus fraction numerator 7 pi over denominator 15 end fraction right parenthesis equals fraction numerator 1 over denominator 8 space sin pi over 15 end fraction cross times sin fraction numerator 8 pi over denominator 15 end fraction minus text cos end text fraction numerator 8 pi over denominator 15 end fraction space cos fraction numerator 3 pi over denominator 15 end fraction space cos fraction numerator 5 pi over denominator 15 end fraction space cos fraction numerator 6 pi over denominator 15 end fraction space equals fraction numerator 1 over denominator 2.8 space sin pi over 15 end fraction cross times 2 space sin fraction numerator 8 pi over denominator 15 end fraction minus cos fraction numerator 8 pi over denominator 15 end fraction space cos fraction numerator 3 pi over denominator 15 end fraction space cos fraction numerator 5 pi over denominator 15 end fraction space cos fraction numerator 6 pi over denominator 15 end fraction space equals fraction numerator negative 1 over denominator 16 space sin pi over 15 end fraction cross times space sin fraction numerator 16 pi over denominator 15 end fraction space cos fraction numerator 3 pi over denominator 15 end fraction space cos fraction numerator 5 pi over denominator 15 end fraction space cos fraction numerator 6 pi over denominator 15 end fraction space text ingat sin end text left parenthesis text π+α end text right parenthesis text =-sinα end text equals fraction numerator negative 1 over denominator 16 space sin pi over 15 end fraction cross times space sin left parenthesis straight pi plus pi over 15 right parenthesis space cos fraction numerator 3 pi over denominator 15 end fraction space cos fraction numerator 5 pi over denominator 15 end fraction space cos fraction numerator 6 pi over denominator 15 end fraction space equals fraction numerator negative 1 over denominator 16 up diagonal strike space sin pi over 15 end strike end fraction cross times space minus up diagonal strike sin pi over 15 end strike space cos fraction numerator 3 pi over denominator 15 end fraction space cos fraction numerator 5 pi over denominator 15 end fraction space cos fraction numerator 6 pi over denominator 15 end fraction space equals 1 over 16 cos fraction numerator 3 pi over denominator 15 end fraction.1 half cos fraction numerator 6 pi over denominator 15 end fraction space equals 1 over 32 cos fraction numerator 3 pi over denominator 15 end fraction cos fraction numerator 6 pi over denominator 15 end fraction space cross times fraction numerator 2 space sin fraction numerator 3 pi over denominator 15 end fraction over denominator 2 space sin fraction numerator 3 pi over denominator 15 end fraction end fraction equals fraction numerator 1 over denominator 64 space sin fraction numerator 3 pi over denominator 15 end fraction end fraction cross times sin fraction numerator 6 pi over denominator 15 end fraction cos fraction numerator 6 pi over denominator 15 end fraction equals fraction numerator 1 over denominator 2.64 space sin fraction numerator 3 pi over denominator 15 end fraction end fraction cross times 2 space sin fraction numerator 6 pi over denominator 15 end fraction cos fraction numerator 6 pi over denominator 15 end fraction equals fraction numerator 1 over denominator 128 space sin fraction numerator 3 pi over denominator 15 end fraction end fraction sin fraction numerator 12 pi over denominator 15 end fraction text ingat end text space sin left parenthesis straight pi minus straight alpha right parenthesis equals sin alpha equals fraction numerator 1 over denominator 128 space sin fraction numerator 3 pi over denominator 15 end fraction end fraction sin left parenthesis pi minus fraction numerator 12 pi over denominator 15 end fraction right parenthesis equals fraction numerator 1 over denominator 128 space up diagonal strike sin fraction numerator 3 pi over denominator 15 end fraction end strike end fraction s up diagonal strike in fraction numerator 3 pi over denominator 15 end fraction end strike equals 1 over 128 equals 1 over 2 to the power of 7 equals 2 to the power of negative 7 end exponent end style$

Dengan demikian, terbukti bahwa
$cos \frac{π}{15} cos \frac{2 π}{15} cos \frac{3 π}{15} cos \frac{4 π}{15} cos \frac{5 π}{15} cos \frac{6 π}{15} cos \frac{7 π}{15} = 2^{- 7}$