Converting Rectangular Equations to Polar Equations Example 3

Hi everyone. Welcome back to integralCALC.com. Today we’re going to be doing another example
about how to convert a rectangular equation to a polar equation. The rectangular equation
that we’re given is x times y equals 1 and remember that with rectangular equations,
the equation will be in terms of x and y variables. With the polar equation, it’s going to be
in terms of r and theta.
The two formulas that we’re going to use to convert our rectangular equation to a polar
equation are x equals r cosine of theta and y equals r sine of theta and what we will
do is plug in r cosine of theta for x here and r sine of theta for y here, for y.
So when we make those two substitutions, we’re left with r cosine of theta times r sine of
theta equals 1 and what we want to do with this is simplify it so that we get to the
point where we have the equation in the form r equals and then something in terms of theta.
Remember that with the rectangular equation, you usually see y equals and then something
in terms of x. With a polar equation, we’re looking for r equals and then something in
terms of theta. So that’s what we’re working toward. We’re going to try to get r by itself
on the left side and then have theta on the right side.
So we will go ahead and we will multiply these two together and we will end up with r squared
times cosine of theta times sine of theta equals 1. Remember in order to get r squared
by itself, we will divide both sides by cosine of theta times sine of theta. We will just
be left with r squared on the left side and cosine of theta times sine of theta becomes
our denominator on the right side.
Now to simplify the right side here, what we will do is separate cosine of theta and
sine of theta into separate fractions. So these two fractions are multiplied together,
1 over cosine theta and 1 over sine of theta. It’s equal to 1 over cosine of theta times
sine of theta.
So the reason that we do that is because 1 over cosine theta is equal to secant of theta
and 1 over sine of theta is equal to cosecant of theta. So our final answer is r squared
equals secant of theta times cosecant of theta and in this case, it’s simpler to leave
r squared on the left hand side than to take the square root of r and take the square root
of the right side.
If you can, you want to solve for r. If it’s simpler to leave the polar equation in terms
of r squared, that’s fine. We often find polar equations in terms of r squared. If
you end up canceling out r during your simplification process, you can also solve for theta and
that’s acceptable in terms of a polar equation unless obviously your professor or the textbook
indicates that you should do otherwise. But in this case, our final answer is r squared
equals secant of theta times cosecant of theta and this is exactly equal to our rectangular
equation, x times y equals 1.
So I hope that helped you guys and I will see you in the next video. Bye.