I will post a challenging math problem each day. The level of difficulty will vary, but most problems should not require any specialized knowledge beyond calculus. Problems are also posted on Twitter using the hashtag #mathpotd.
Saturday, September 26, 2009
cos(20) * cos(40) * cos(80) = ?
Simplify cos(20°) * cos(40°) * cos(80°) without using a calculator.
Answer: 1/8
ReplyDeleteThe 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.