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Coordinate Systems - Example 1

In mathematics, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element on a manifold such as Euclidean space. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the "x"-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The most common coordinate system in modern mathematics is the Cartesian coordinate system.

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Video Transcript

Welcome to the video on converting from Cartesian to polar coordinates and then back from polar to Cartesian coordinates Again, Um, let's jump right in here with an example. Say we have a Cartesian position. Remember, Cartesian has written is X comma. Why? We're gonna say that we have the position Five comma three So counting that we have 12345123 We're gonna look right here. It's in the first quadrant So we do this. What we want to do is we want to use the formulas that I gave previously where X is equal to the square root of X are rather is equal to the square root of X squared plus y squared and Satya is going to be equal to the arc Tangent of why over X. Remember, you can also write this as 10 in the negative one. This is how it'll pillar on your calculator over X. Okay, so just plugging these in very quickly we're going to see that our is equal to the square root of five squared plus three squared. So that's gonna be squared of 34 is our radius and then r theta. It's gonna be equal to in verse 10 of 3/5. So when we plugged that in, we get approximately 31 degrees. Now, looking at this, that that's pretty simple. But what if we tried a different quarter? What if we twitched it to negative 512345 123 So if we did negative 53 coming back. And here we have negative five. Well, the negative five is gonna get squared. So negative five times negative. Five. That's 25. Again, we're gonna end up with squared of 34 just like we did before. On the other hand, when we look at the data calculation, we're gonna end up with three over negative five. And it turns out that the inverse tangent function has this property where a tangent of negative X is equal to negative inverse tangent of X. So this cause a little bit of problem. It means that we're going to get a value of negative 31 degrees here. So, looking at that, we need to interpret what that means, because before I told you always, always, always, we wanna measure theta and r theta format as being as coming from the positive X axis going in the counter clockwise direction. So we got that correct when we were in first Quadrant. But now we're getting negative. 31 degrees. Well, what does that mean? Well, what is telling us is that this angle is 31 degrees, which means that we're actually gonna need 180 minus 31 in order to get the correct answer. So that's going to be 149 degrees is the correct answer from here to here. But you're never gonna get your calculator to say that, because the way that Arc Tangent marks So what we need to do then, is keep track of which quadrant you are in. And what is your Why and what is your ex, right? So if we are taking the arc tangent of an angle and we say why over X, then it means we're always going to be measuring the angle from the x axis. Okay, so make sure make sure make sure that you always realize whatever quadrant in your amazing from the x axis. So if you were in the third quadrant, it gives you this angle If you're in the fourth quadrant, it gives you this angle and then convert so that your theta that you report to your teacher is always being measured from the positive X axis in the counter clockwise direction. As long as you do that, you'll be fine. I recommend that you practice that a few times to make sure that you can do it correctly. So let's go do a quick example in the opposite direction. Say that we have our that is three and a tha tha that is 140 degrees. So if I were to look at that here, it would be in this quadrant somewhere around here. So what we wanna do is convert to X and y. So remember, our conversions are R cosine theta and R sine theta. Luckily, because we're using cosine and sine, we don't have the same issue that we did with the inverse tangent function. So X then is going to be equal to three times to co sign of 140 degrees. Why will be equal to three times to sign of 140 degrees, which gives us answers of approximately negative 2.3 and 1.9. Uh, try this a few times. Practice going back and forth. It's a pretty simple exercise.