Session 5 of Course One

What kind of planar curvature is equally distant from its center at all times? A circle. If

                     
the top of a circle were anywhere where you arbitrarily determine it to be at, where

would the circle be maximized at? At that top. A maximum position indicates the

location of its highest value, and the top of something is higher than its bottom. This

depends on whatever you arbitrarily called the “top.” This generalization depends on if

the “top” were the highest point of the circle, and if, depending on your context the

maximization of the circle’s structure were to also be at one with whatever you may also

arbitrarily call the maximization of the circle. For instance, not as a trick question, the

maximum position on the earth would more likely be the magnetic north pole than the

magnetic south pole. In trigonometry, the sine function is maximized at pi over two.

This is at ninety degrees, and is at the top of the circle. If you looked at the circle upside

down, the top of the circle (actually, the bottom) would appear to be 3pi/2. Here is where

the sine function is minimized. If you choose an orientation that is fixed, and pi over two

is at a location mathematically at least, then, by this orientation, that position is always

the top of the circle. Here, we are talking about a unit circle, so when the sine function is

maximized, it equals one. And when it is minimized, it equals negative one. How does

one come up with what the sine and cosine functions are? In a unit circle, what’s the

closest distance from the x-axis to the top of the circle? One. Likewise, what’s the

closest distance from the y-axis to the right side of the circle? One. What’s the closest

distance from the x-axis to the right side of the circle? Zero. What’s the closest distance

from the y-axis to the top of the circle? Zero. Likewise, the sine function is maximized

at the top of the circle (pi/2), and the cosine function is maximized at the right side of the

circle (zero pi). The sine function is zero at 0pi, and the cosine function is zero at pi/2.

What does this indicate? It shows that the sine function is more of an indicator of how

things change in nature. For instance, a toy rocket starts from “scratch”, not accelerating

or decelerating. You shoot it out. It goes from zero to an accelerated speed. Soon, the

rocket slows, rapidly decelerating. (So, once the rocket’s thrust is ended, it is decelerated

by earth’s gravitational force by the same acceleration as when it will fall.) Once the toy

rocket begins to fall, it will accelerate toward the earth until it hits the ground and

crashes. The cosine indicates initial maximization. This is less common, although,

depending on your phase, this may happen more often. You see, as shown above, sine

and cosine are just 90 degrees from being the same thing. So, whether something may be

described by a sine or cosine function depends in part by the phase orientation needed,

proscribed, and/or wanted.