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crash_matrix

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I stumbled on an interesting and unexpected identity when looking for a simpler summation technique for inverse kinementics. Basically, I was trying to find a simpler way of summing IK vectors for some particular armature (like a tentacle or multi-branch armature on a robot). This question doesn't have to do with IK, but I accidentally calculated an identity rule I've not been able to attach to any other identity rule in trigonometry. This is the XPlore code that graphs the identity:

[The trig set is in degrees, not radians]

f(x)=sum(sin(t),t=0 to x step 1)

(Period,C)=(180,f(180))

g(x)=C*sin(x/2)^2+sin(x)/2

graph((f(x),g(x)),x=0 to Period)

When running those lines in XPlore (or any graphing calculator), you'll find that as the "step" value in the first line approaches zero, the two curves become equal. I've tested this rule across hundreds of different data sets and it always diverges perfectly, regarless of the precision of the calculator.

My question is: Is there an existing trig identity or rule (or set of iedntities or rules) that explains why the above functional set works?

Basically the "law" that it asserts is:

sum[0 to x](sin(x))=180 * sin(x/2)*sin(x/2) + sin(x)/2

where x is in degrees

I've looked up every trig and calculus identity and "rule" that I can find and I can't match it to anything.

It's not an important question, just something that's been bugging me for a while.

Thanks for looking,

--CM