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Not So Simple Sines (Posted on 2009-12-24) Difficulty: 3 of 5
Analytically prove sin(54)-sin(18) = 1/2

See The Solution Submitted by Brian Smith    
Rating: 4.0000 (2 votes)

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Solution an analytical proof | Comment 2 of 4 |
It may be a little roundabout, but here goes:

Using the product sum identity,
sin(x) - sin(y) = 2*cos((x+y)/2)*sin((x-y)/2),
we have for x=54 and y=18:
sin(54) - sin(18) = 2*cos(36)*sin(18)

Using the double angle formula,
sin(2t) = 2*sin(t)*cos(t),
we have for t=18:
sin(36) = 2*sin(18)cos(18), therefore
sin(18) = sin(36)/(2*cos(18))

Substituting back into the earlier equation and simplifying:
sin(54) - sin(18) = cos(36)*sin(36)/cos(18)

Using the complement of the trig function,
cos(t) = sin(90 - t),
we have for t=18:
sin(54) - sin(18) = cos(36)*sin(36)/sin(72)

Reapplying the double angle formula and substituting, with t=36, we have:
sin(54) - sin(18) = cos(36)*sin(36)/[2*sin(36)cos(36)]

Simplifying:    
sin(54) - sin(18) = 1/2

  Posted by Dej Mar on 2009-12-25 07:24:35
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