Quandaries
and Queries
Show that if x= 18 degrees, then cos2x =sin 3x. HENCE find the exact value of sin 18 degrees, and prove that cos 36 - sin 18 =1/2.
The first part is trivial, but how does one use this first part to get to the second part. Note that this DOES NOT involve looking up in tables! I have found the correct answer by using the golden ratio, but how to get the answer from the first part I do not know. I thought that this would be a simple problem, since in the 1882 entrance exam for Sandhurst Military Academy, one was asked to find the exact value of sin 18 degrees. It is also interesting that sin18 (cos 72), sin36, (cos 54), sin54 (cos 36) and sin72 (cos 18) all come out as exact and easily represented ratios, just like sin30, sin45 and sin60, and is this due to 18 degrees =pi/10 and so on?
cos(2x) = sin(3x)
so
cos(2x) = sin(x)cos(2x) + cos(x)sin(2x)
or
cos(2x)- sin(x)cos(2x) = cos(x)(2sin(x)cos(x))
cos(2x)[1 - sin(x)] = 2cos2(x)sin(x) = 2[1 - sin2(x)] sin(x)
Hence
cos(2x)[1 - sin(x)] = 2[1 - sin(x)][1 + sin(x)] sin(x)
and thus
cos(2x) = 2 [1 + sin(x)] sin(x)
Note: You need to check that sin(x) = 1 is not a solution to the original problem.
Finally
1 - 2 sin2(x) = 2 sin(x) + 2 sin2(x)
or
4 sin2(x) +2 sin(x) - 1 = 0
Solve for sin(x).
Andrei, Claude and Penny