Sine and Cosine from -5 to 5

summary:

Quaternions ranging from (-5, -5, -5, -5) to (5, 5, 5, 5) were fed into a sine and cosine function, with these odd looking results.

description:

The linear input is in yellow, the odd sine function is in red, the even cosine in blue. The cosine has only one apparent line diving into the origin because that line is a doublet. I thought I knew what cosines and sines looked like, but it feels like there is a frightening large amount of unknown diversity possible for these formerly familiar friends. The cause of the oddness maybe due to varying t, x, y, and z at the same time.

command:

q_graph -loop 0 -box 10 -dir vp -out sin-cos -command 'q_add_n -10 -10 -10 -10 .01 .01 .01 .01 2000' -color yellow -command 'q_add_n -10 -10 -10 -10 .01 .01 .01 .01 2000 | q_sin' -color red -command 'q_add_n -10 -10 -10 -10 .01 .01 .01 .01 2000 | q_cos' -color blue

iPod downloads:

sin-cos.povray.100.1000.animation.scan_.m4v

math

equation:

$\begin{align*} \sin(t,\vec{R}) &= (\sin(t) \cosh(|R|), \cos(t) \sinh(|R|) \frac{\vec{R}}{|R|})\\ \cos(t,\vec{R}) &= (\cos(t) \cosh(|R|), \sin(t) \sinh(|R|) \frac{\vec{R}}{|R|}) \end{align*}$

sage notebook:

Quaternion Trig Functions

numbers:

sin-cos.povray.100.1000.numbers