The Lemon Fool

How's your trig?  This puzzle is to evaluate the following expression:

6 * ( (sine 1 degree)^2 + (sine 2 degrees)^2 + (sine 3 degrees)^2 + ... + (sine 720 degrees)^2 – 23)

Happy new year to all readers.  

Cinelli

cinelli wrote:How's your trig?

Unused since school, but a vague ahhh-wait-a-minute came up from the depths of my memory, and ... spoiler

cinelli wrote:How's your trig?  This puzzle is to evaluate the following expression:

6 * ( (sine 1 degree)^2 + (sine 2 degrees)^2 + (sine 3 degrees)^2 + ... + (sine 720 degrees)^2 – 23)

Happy new year to all readers.  

Cinelli

Oh dear. Neither the 'puter nor anything more obscure than Πῡθαγόρᾱς required.

That's 720 squares. Twice round the circle.

But sin(x) = cos(90-x). We can express the sum as one circle of sines and t'other of cosines. And sin2(x) + cos2(x) = 1. So the sum of all your trig is 360.

I guess 6*(360-23) must be 2022? Hmm, yes, arithmetic confirms it.

Edit: damn, I thought [sup] worked as markup here for those squares, but evidently not. I'll leave it as sin2(x)/etc.