Patrick Teaches Physics

The relativity principle says that we get to choose the coordinate system we use when describing physical problems, and that includes deciding which way is up.  To show how useful that can be I'm going to use the inclined plane. We have an object sitting on a surface which is inclined at some angle: a box on a ramp or some such.  We want to know the object's acceleration.  Drawing the free-body diagram is straight-forward: there is weight, a normal force, and usually there is friction as well. We can write down the equations of motion and solve them, but it won't be pretty!  The acceleration has horizontal and vertical components, and because the friction depends on the normal force, which depends on the weight and the angle, the trig functions are going to build up.  Also, it's not so easy to tell what the normal force should be. And so on.... A much better approach is to rotate the coordinate system so that motion of the object is along the x-axis. We can immediately

What are "cos" and "sin"?  They're buttons on a calculator -- right?  You enter an angle, push the button, and some number appears.  Somehow -- if you remember your trig rules (SOH, CAH, TOA) -- you can use this number get the side lengths of triangles. I think the above pretty much sums up the typical student's understanding, and it's adequate to complete most of the tasks a physics student has to with trig functions.  But it doesn't have to be so mysterious, and it can be very helpful if it isn't.  So in this post I'll explain what these functions are and how they are related. Consider trying to specify a point on the unit circle: We could use the Cartesian coordinates x and y, or we could use the angle θ. Both have their merits, and in practice we might want to go back and forth between them.  So, given an angle θ, what are x and y?  We call the functions that answer that question sine and cosine (usually written as sin and cos for sho