Question Video: Converting Cartesian Coordinates into Polar Coordinates Mathematics

Video Transcript

Convert two, three to polar coordinates. Give the angle in degrees and round to one decimal place.

In this question, we’ve been given a pair of Cartesian coordinates and asked to convert to polar coordinates. Remember, polar coordinates are in the form 𝑟, 𝜃. For a point 𝑃 with polar coordinates 𝑟, 𝜃, 𝑟 represents the length of the line segment that joins the point 𝑃 to the origin or pole. 𝜃 represents the angle that this line segment makes with the positive 𝑥-axis. That’s sometimes called the polar axis. It’s important to realize that this is measured in a counterclockwise direction.

By forming a right triangle with 𝑟 as its hypotenuse, we can derive two formula that we can use to convert Cartesian to polar coordinates. We use the Pythagorean theorem to find an equation linking 𝑟, 𝑥, and 𝑦. When we do that, we obtain 𝑟 squared equals 𝑥 squared plus 𝑦 squared. And that’s because in a right triangle, the sum of the square of the smaller two sides is equal to the square of the longest side. Solving for 𝑟, we find that 𝑟 equals the square root of 𝑥 squared plus 𝑦 squared. Remember, 𝑟 is a length. So, we don’t need to worry about taking both the positive and negative square root. We’re only interested in the positive.

To find 𝜃, we recall that the tan of 𝜃 is equal to the opposite over the adjacent side lengths. So, tan 𝜃 is equal to 𝑦 over 𝑥. And 𝜃 is equal to the inverse tan of 𝑦 over 𝑥. Now, we do need to be a little bit careful when calculating the size of the angle. We must always check which quadrant our point lies in. When trying to find values of 𝜃 greater than negative 180 and less than or equal to 180, we use the given results. For points that lie in the first and fourth quadrant, we use the values of 𝜃 that our calculator gives us when we work out the inverse tan of 𝑦 over 𝑥. For points lying in the second quadrant, we add 180 degrees to this value. And for points in the third quadrant, we subtract 180 degrees from this value.

Now, of course, our point two, three has both positive 𝑥- and 𝑦-coordinates. So, it must lie in the first quadrant. We’ll use the value for 𝜃 that our calculator gives us. Substituting our values for 𝑥 and 𝑦, that is, 𝑥 equals two and 𝑦 equals three, into our formula, we get 𝑟 equals square root of two squared plus three squared. And 𝜃 is equal to the inverse tan of three over two. 𝑟 equals the square root of 13, which is 3.605 continuing, or 3.6 when we round to one decimal place. 𝜃 is equal to 56.309 continuing degrees. If your calculator didn’t return this answer, make sure that you are operating in degrees and not in radians. Rounded to one decimal place, 𝜃 becomes 56.3 degrees. This means the polar coordinate form of the point two, three equals 3.6, 56.3 degrees.