Deriving x = A cos(ωt) for simple harmonic motion

[Oz Alikhan]

Short story:

How does one go about to derive x = a cos(wt)? The way it is derived in my book is from the "SHM Diagram" file that I have uploaded but it seems that the diagram is incorrect as it does not correspond to the expression. Also, why is the Amplitude in the expression constant when the radius of a pendulum is also constant which implies that x has to be also constant?More of the story:

I have seen the expression, x = a cos(wt) derive other expressions through calculus but I have not seen how this expression itself is derived from.

In the file "SHM Text" that I have uploaded, I understand up to the part of x = rcos(wt) and where that corresponds to the diagram. However shortly after that, the book states that for pendulums r = A therefore the expression becomes x = A coswt. I cannot see why or how or even where that corresponds to the diagram. I have tried drawing diagrams from that expression but it never seems to match the book's diagram.

Thanks for the help

 

[HallsofIvy]

The notation is unfortunate. In the picture they appear to have labeled two points "A" and "B" but, in the text, then use "A" as if it were a number. It looks like they are thinking of the point "A" as corresponding to the point (A, 0) where A is now a number, the distance from the origin to the point "A". In that case, number, A, is the radius of the circle: r= A.

 

[Oz Alikhan]

Makes sense now. Thanks a lot