Saturday, September 26, 2009

cos(20) * cos(40) * cos(80) = ?

Simplify cos(20°) * cos(40°) * cos(80°) without using a calculator.
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  1. DavidSeptember 27, 2009 at 11:25 AM

    Answer: 1/8

    The trick is to multiply by sin(20°) and repeatedly apply the double-angle identity sin(2x)=2sin(x)cos(x).

    Let x = cos(20°)cos(40°)cos(80°).

    x sin(20°) = sin(20°)cos(20°)cos(40°)cos(80°)
    = 1/2 sin(40°)cos(40°)cos(80°)
    = 1/4 sin(80°)cos(80°)
    = 1/8 sin(160°)
    = 1/8 sin(20°).

    Therefore, x = 1/8.

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