Thread: Interesting Identity View Single Post
P: 688
 Quote by crash_matrix sum[0 to x](sin(x))=180 * sin(x/2)*sin(x/2) + sin(x)/2 where x is in degrees
Didn't you mean f(180), instead of 180, above? f(180) being the sum[t=0 to 180](sin(t)).

Also, to avoid confusing the two meanings of 'x', you may want to express the left-hand side as sum[t=0 to x](sin(t)) ; 'x' is the limit of the sum, not really the argument to sin().

The x's in the right-hand side are correct, as they mean the same value of 'x' as the limit of the sum in the left-hand side.

So, if I get you right, the conjecture is$$\sum_{t=0}^x \sin t = \left( \sum_{t=0}^{180} \sin t \right) \sin^2(x/2) + (\sin x)/2$$with the arguments in degrees, not radians.

The two overlapping curves (left-hand side, right-hand side) looking like this, for x=0 to 180: