HP-35: Sin and cos function formulas, another point of view

[Albert Chan]

this is my guess of why they pick the first set of equations. It is not as slow as we think.

say, we want cos(0.1)

N = 1 + tan(0.1)^2 = 1.010067046
cos(x) = 1/sqrt(N)

Newton's method for 1/sqrt(N) avoided the expensive division

1/sqrt(N) => x += 0.5 x * (1 - N x^2), until converge

if guess = 1, next iteration = 1.5 - N/2, so use that for x0

x0 = 0.994966476 -- guess
x1 = 0.995004163
x2 = 0.995004165
x3 = 0.995004165 -- value of cos(0.1)