Sine and Cosine for a Fixed Point in Space is a Circle

summary: 

When a position in space is fixed, the sine and cosine functions circle that point, the angle depending on the exact values of x, y, and z.

description: 

Sines and cosines have to do with circles. By fixing x, y, and z, the circle stays fixed. What direction the line in space points to is arbitrary.

The line in yellow is the input for the The length of the line in space is the amplitude.

command: 
q_graph -loop 0 -box 25 -dir vp -out sin-cos_xyz_constant -command 'q_add_n -50 1 2 1 .05 0 0 0 2000' -color yellow -command 'q_add_n -50 1 2 1 .05 0 0 0 2000 | q_sin' -color red -command 'q_add_n -50 2 1 2 .01 0 0 0 10000 | q_cos | q_x_scalar 2' -color blue
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*}

tags
Physics Tag: 
simple harmonic oscillator
Math Tag: 
trig functions
sine
cosine
Programming Tag: 
command line quaternions
q_add_n
q_sin
q_cos