Theorem 5.4c: The area of the region in the figure above is
2r2cos2(180k/n)sin2(180/n)sec(180j/n)tan(180(j-1)/n)sec(180(j-2)/n)
proof 1: If we substitute
sj =
rcos(180k/n)sin(180/n)sec(180j/n)sec(180(j-1)/n)
and
sj =
rcos(180k/n)sin(180/n)sec(180(j-1)/n)sec(180(j-2)/n)
by Theorem 4.4b into the
formula from Theorem 5.4a, we get
that the area is
r2cos2(180k/n)sin2(180/n)sec2(180j/n)sec2(180(j-1)/n)sin(180j/n)cos(180j/n)
+
r2cos2(180k/n)sin2(180/n)sec2(180(j-1)/n)sec2(180(j-2)/n)sin(180(j-2)/n)cos(180(j-2)/n)
=
r2cos2(180k/n)sin2(180/n)sec2(180(j-1)/n)[sec2(180j/n)
sin(180j/n)cos(180j/n) +
sec2(180(j-2)/n)sin(180(j-2)/n)cos(180(j-2)/n) ]
=
r2cos2(180k/n)sin2(180/n)sec2(180(j-1)/n)
[tan(180j/n) + tan(180(j-2)/n)]
=
r2cos2(180k/n)sin2(180/n)sec2(180(j-1)/n)
[sin(180j/n)cos(180(j-2)/n)+
sin(180(j-2)/n)cos(180j/n)]sec(180j/n)sec(180(j-2)/n)
=
r2cos2(180k/n)sin2(180/n)sec2(180(j-1)/n)
sin(360(j-1)/n)sec(180j/n)sec(180(j-2)/n)
=
r2cos2(180k/n)sin2(180/n)sec2(180(j-1)/n)
2sin(180(j-1)/n)cos(180(j-1)/n)sec(180j/n)sec(180(j-2)/n)
= 2
r2cos2(180k/n)sin2(180/n)
tan(180(j-1)/n)sec(180j/n)sec(180(j-2)/n)
proof 2
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