Convert two, three to polar coordinates. Give the angle in degrees and round to three significant figures throughout.
In this question, we’ve been given a pair of Cartesian coordinates and asked to convert to polar coordinates. Remember, polar coordinates are of the form 𝑟, 𝜃. For a point 𝑝 with polar coordinates 𝑟, 𝜃, 𝑟 presents 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 realise that this is measured in a counterclockwise direction.
By forming a right-angled triangle with 𝑟 as its hypotenuse, we can derive two formulae 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, we obtain 𝑟 squared equals 𝑥 squared plus 𝑦 squared. And that’s because, remember, in a right-angled triangle, the sum of the square of the smaller two sides is equal to the square of the longest side.
And so, we obtain 𝑟 to be equal to 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 use the fact that tan 𝜃 is equal to opposite over adjacent. 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 value 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, and 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. So, we’ll use the value of 𝜃 that our calculator gives us.
Substituting our values for 𝑥 and 𝑦, that is 𝑥 equals two and 𝑦 equals three, into our formulae, and 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 is equal to the square root of 13, which is 3.61, correct to three significant figures. The inverse tan of three over two, correct to three significant figures, is 56.3. And we’re done. We’ve found that two, three in polar coordinate form is given by 3.61, 56.3 degrees.