Help With Trig Functions

Neocities
1.)

Imagine the point P being spun around a circle




2.)

Now we will connect P to the center with a line. You can see that the line's length doesnt change. It is the radius of the circle, or it's Amplitude




3.1)

While this line can be plotted just using magniude, the length of the line, and angle, the direction that the line points in, humans like to work in grids, so why dont we find the coordinates of the point P on a graph.




4.)

First we will draw a triangle, with sides along the X and Y axis, the hypotenuse of which was the line we drew earlier.




5.)

Now lets color in the adjacent side of the triangle. This length of this line represents the function cos(Θ). We will name this function "f(Θ)", and note that here it is a function of theta, Θ.




5.1)

Before the next step, notice that every 360 degrees, the point P is back where it was before you noticed that, which is pretty neat. This angle, 360 degrees, can also be measured in radians.



A radian doesnt mean anything by itself, like how an inch doesnt mean anything by itself, but given that there are 12 inches in a foot, an inch has meaning. Like that, we know that there are 2π radians in a circle(360 degrees), which is ~6.28 radians.




6.)

If we graph f(Θ) using x instead of Θ(because our smooth little human brains like x much more than Θ) we get a graph you are maybe familiar with:




6.1)

There are a couple things to remember before we move on. The graph above is f(x), and it represents the length of the adjacent side of our triangle we drew as x increases. Also remember that x has no units.




The default Cosine function

There is a default way that you can write your cos function f(x):

f(x) = Acos(ωx+Φ)+C

The only things here that change in the above expression is f(x) and x. Everything else is a constant, and you dont need to know their names, but I willgive them anyways.


What are these constants?!?

A - This is the amplitude. If you think back to "3", the length of the hypotenuse is the amplitde. This is what varying it does:

ω - This is the angular frequency, it dictates as how frequently the function f(x) repeats. This is what varying it does:

Φ - This is the phase constant, and it will move the function left and right, or along the x axis. This is what varying it does:

C - This is the vertical offset, and it dictates where the middle, or midline, of the function f(x) is. This is what varying it does:

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