Question Video: Converting Polar Equations to Cartesian Form Mathematics

Video Transcript

Convert 𝑟 equals two sec 𝜃 into cartesian form.

To answer this type of question, we’ve got to do some substitution and some manipulation in most cases. But it’s always helpful to start by writing out the rectangular polar relationships. 𝑥 equals 𝑟 cos 𝜃 and 𝑦 equals 𝑟 sin 𝜃.

A good place to start with this question is by writing sec 𝜃 in terms of something else. So, we remember that sec 𝜃 is equal to one over cos 𝜃. So, we can rewrite 𝑟 equals two sec 𝜃 as 𝑟 equals two over cos 𝜃. Now, we need to use these relationships to rewrite what we’ve got in terms of only 𝑥 and 𝑦.

So, if we now multiply both sides by cos 𝜃, we get that 𝑟 cos 𝜃 equals two. And then, we can use our rectangular polar relationship, 𝑥 equals 𝑟 cos 𝜃, and replace 𝑟 cos of 𝜃 with 𝑥 to give us 𝑥 equals two. You might be wondering why we would ever need to convert a polar equation to cartesian form. This becomes apparent if we draw the graphs of both functions. We can see that even though the grids are different, the graphs are actually the same.