The diagram below shows a pentagram (a five-pointed star) inscribed inside a regular pentagon. there are several golden ratios in this figure. for example:
and .
this means that if length a = 2, then length b = 3.236. in this case, what is length c?
a.2.618
b.1.618
c.0.618
d.5.236
e.4.236

Let the angle between a and b be x then,
x = arccos((a^2 + b^2 - b^2)/2ab = arccos(a/2b)
x = arccos(2/2(3.236)) = arccos(1/3.236) = arccos(0.3090) = 80°

Thus, the angle opposite c = 180 - 80 = 100°

Therefore, c = sqrt(a^2 + b^2 - 2abcos C) = sqrt(2^2 + 3.236^2 - 2(2)(3.236)cos 100) = sqrt(14.471696 + 2.247702) = sqrt(16.719398) = 4.089
c = 4.089

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