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Lamar Mercer
19 July, 05:47
Value of cos^2 (48) - sin^2 (12) ?
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William
19 July, 08:45
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The value of that (√5 + 1) / 8 and We can prove it like this:
cos 2A = 2cos^2 A-1=1-2sin^2 A
so cos^2 48 - sin^2 12
= (1/2) (1+cos 96) - (1/2) (1-cos24)
= (1/2) (cos 96 + cos 24) and using cos (A) + cosB) = 2cos ((A+B) / 2) cos ((A-B) / 2)
=cos60cos36
= (1/2) cos36
and you have to find cos36.
Suppose 5x=180, then
cos (5x) = cos (180) then x=-180,-108, - 36, 36, 108
Also 3x=180-2x
so cos (3x) = cos (180-2x) = - cos2x
so 4cos^3 (x) - 3cos (x) = 1 - 2cos^2 (x)
giving 4c^3+2c^2 - 3c-1=0 where c=cosx
c=-1 is a root and factorizing gives
(c+1) (4c^2-2c-1) = 0
so 4c^-2c-1=0 giving
c = (2±√20) / 8 = (1±√5) / 4 and these have values cos (36) and cos (108)
the positive root is therefore cos (36) = (1+√5) / 4
and the required value (1/2) cos (36) = (1+√5) / 8
I think this can be very useful
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