*crow flies*, to continue the map comparison. Thus:

*θ*), and a distance, which is the length of our red line (we'll call this

*r,*for radius), and it's written as (

*r,*

*θ*).

Pretty simple, yes? By now it should be clear how to find these values given what we already know. We'll start by finding

*c*using the trusty Pythagorean theorem:

*a^2 + b^2 = r^2*

*2^2 + 3^2 = r^2*

*4 + 9 = r^2*

*13 = r^2*

*√13 = r*

So there's one of our two coordinates. Let's find the value of the angle

So there's one of our two coordinates. Let's find the value of the angle

*θ*now, shall we? Seems the easiest way, or at least the way that involves all integers, is to find the arctangent of 2/3. Plunking*tan^-1(2/3)*into our trusty calculator in radian mode, we see that the arctangent of 2/3 is approximately 0.59 radians, which coverts nicely to 33.69*°, so our polar coordinates would be expressed (*

*√13,33.69°). Thus:*

*atan2 (y,x)*. It's a simple way to express ALL THIS:

Now, converting from polar coordinates to cartesian coordinates is pretty simple too, and I won't go through the process of proving these formulae, but here they are:

*x=r cosθ*

*y=r sinθ*

Let's work it out given our same problem from above.

*x=√13 cos(.59)*We rounded that angle to the nearest hundredth, so our answer is a little off, but plugging that in a calculator does give us approximately 3. Similarly,

*y=*

*√13 sin(.59)*

Yields a decimal that's pretty close to 2. So - our calculations were a success!

That's all for now - soon we begin writing about pre-calc, so stay tuned for that.