What even is "cos"?

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 short).

x = cos θ

y = sin θ

Now we see that we can calculate cos & sin for some angles just by reading them off the diagram:

cos 0 = 1, sin 0 = 0

cos 90 = 0, sin 90 = 1

cos 180 = -1, sin 180 = 0

cos 270 = 0, sin 270 = -1

A few other angles are fairly easy to calculate.  For θ = 45 degrees, x = y.  A simple application of Pythagoras' theorem shows that:

 

It's also possible to calculate the values for 60 degrees and 30 degrees by constructing an equilateral triangle (with sides of length 1) and then cutting it in half to make a triangle with angles 90, 60 and 30:

After that it gets a lot harder, so before calculators people usually relied on tables made by people who almost certainly weren't paid enough.  But you really shouldn't go to your calculator to get cos 90.  Just think about what you're being asked, and you'll know.