Understanding how sine and cosine really work is something that eludes most first year physics students. In my estimation it is in some sense the entire point of first year and certainly the most important thing to grasp, an yet at the same time one of the most difficult. That is why we will do a deep dive on this over the course of several episodes. Included below are several screenshots from the Exploring Mathematics: Sine and Cosines app, available for free above by clicking the link.
To start to tell the story of sin and cosine, one must first visualize a circle, centered at the origin of a cartesian coordinate system. We must also consider the circle to have a radius of one unit, any unit you like, thus leading to the name the unit circle.
Starting with respect to the horizontal, or positive x-axis, we can begin to sweep our radius around the circle. This angle will be referred to as theta.
We also know, since this is a cartesian coordinate system, that we can refer to any point by its coordinates [x,y].
Using the radius as a hypotenuse, we can create a right triangle using our point to create legs parallel to each coordinate axis.
We will now define our x-coordinate to be cos(theta), and our y-coordinate to be sin(theta). Replacing the x and y values on the triangle legs with this definition may start to look familiar from classes past.
One can't help ones self but try out the pythagorean theorem with this new definition...
Which leads to a rather familiar identity.
It is then possible to expand this relation out to any circle or triangle we like, simply by multiplying by the radius, r.
This explains how your physics professors can "magically" break vectors into components and why the x-component is multiplied by cos(theta) and y-component by sin(theta).
More next week.
