Uploaded by TheIntegralCALC on 25.02.2011

Transcript:

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.

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.