SESSION 5 OF COURSE 1

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 rather 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 over two. 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 closes 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 (0pi). 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. Once it falls, it will accelerate toward the earth (the earth's acceleration upon this toy rocket being constant)