Cubic and Quartic Formulae

[LCieParagon]

(08-06-2015 07:19 PM)Tim Wessman Wrote:  Very interesting! Thanks for the comments.

(08-06-2015 01:42 PM)LCieParagon Wrote:  For instance (if we are talking about degrees), one can find sin(36), sin (18), sin(54), and sin(72) exactly. This angles are incredibly useful for architecture or engineering. One can apply half-angle, addition/subtraction, and double angle formulae to receive every whole number degree divisible by 3.

If I am remembering my math from ages ago correctly, you can always find an exact angle if you go out far enough... how would you decide at which point it is useful to find an exact angle or not? Would something like this be implemented by generating a collection of "interesting" values of finer and finer resolution until you were happy with the capability?


Speaking of the nspire, why would doing something like sin(73) in degrees helpfully convert it to "cos(17)"? Simply moving to a smaller angle in the trig function, or is that a common thing?

You are both correct and incorrect at the same time. You can never get to sin(1) this way. You can get to sin (3/2) by utilizing the half-angle formula. sin(1) is to be calculated using the cubic formula. At first glance, it would seem that sin(1) has an imaginary component, but the imaginary part is exactly 0.

Additionally, you wrote sin(73) which does in fact = cos(17). It really just depends on which you prefer. Once again there is an exact answer to this. It's not pretty, and must be calculated using the cubic formula.