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 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:
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:
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,
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.