All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars
 perplexus dot info

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

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

Comments: ( Back to comment list | You must be logged in to post comments.)
 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

 Search: Search body:
Forums (0)