Convert 𝑟 equals two sec 𝜃 into
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.