**POLAR EQUATIONS**

Example 1:

Change the rectangular equation to polar form. Write the polar equation as

in terms of .rKnowing that and , we can convert as follows:

Now, we must writerin terms of .In this case, this is done by factoring.

Wewill factorout of both expressions to getrUsing the

Zero Product Principlewe getor .

The graph of is the pole.

Since the pole is already included in (Hint: Let = 0), we don't need .

Therefore, all we need to do is write in terms of .

That is, .Thus, the polar form of is .

WARNING:

It is incorrect to divide both sides of the equation byrto arrive at the solution !!! From algebra you should remember that we cannot divide by a variable UNLESS we are certain that this variable will NEVER take on the value of 0. In our case, we find thatractually can take on this value!

Example 2:

Change the rectangular equation to polar form. Write the polar equation as

in terms of .rKnowing that and , we can convert as follows:

Now, we must writerin terms of .In this case, this is done by factoring.Wewill factorout of both expressions to getr

Next, we will factor

out of both expressions as follows:rUsing the

Zero Product Principlewe getor

The graph of is the pole. Since the pole is already included in the graph of(Hint: Let = 0), we don't need.

Therefore, all we need to do is writein terms of .

That is,.Thus, the polar form of is .

Example 3:

Change the rectangular equation to polar form. Write the polar equation as

in terms ofr.Knowing that , we can convert as follows:

It may seem to you that we have accomplished our goal, that is, we converted the rectangular equation to polar form. However, it is standard procedure to writerin terms of .

Now, we must writerin terms of .In this case, this is done by factoring as follows:and (Notice that we are dealing with a

Difference of Squares).Using the

Zero Product Principlewe getor

Next, we will rewrite the polar equation as

in terms ofr.That is, or

Either of the two equations is correct. However, it is accepted mathematical practice to let the positive value,, represent in polar form.

Remember that the variables in a polar equation arerand! Only having anr-value means that it is held constant while the anglecan take on any value.

Example 4:

Change the rectangular equation to polar form. Write the polar equation as

in terms of .rKnowing that , we can convert as follows:

Now, we must write the polar equation as

in terms of . All we have to do is divide both sides of the equation byr.sinor (Both solutions are acceptable!)

Thus, the polar form of

is either or .

Example 5:

Change the polar equation to rectangular form.

In this case, we will multiply both sides of the equation by

to getr

Knowing that and , we can convert as follows:

Thus, the rectangular form of is .

Example 6:

Change the polar equation to rectangular form.

Since we do not have a formula that converts

to rectangular form, we will raise both sides to the second power to getrNow we can use the formula to get

.Thus, the rectangular form of is .

Example 7:

Change the polar equation to rectangular form.

Since we do not have a formula that converts

to rectangular form, we will place the "tan" in front of both sides of the equation as follows:

We can use the formula to get

and or (Both solutions are acceptable!)

Thus, the rectangular form of is or .

Example 8:

Change the polar equation to rectangular form.

Since we do not have a formula that converts secant to to rectangular form, we will use a

Reciprocal Identityas follows:

Next, let's multiply both sides by , so that we get

Knowing that , we can convert as follows:

Thus, the rectangular form of is .

Example 9:

Convert the polar equation to a rectangular equation.

Since we do not have a formula that converts

to rectangular form, we'll multiply both sides of the equation by the denominator of the term on the right of the equal sign to getr

r(3 3cos) = 1then

3r 3rcos = 1and

3r = 1+ 3cosNext, we'll square both sides to get

and given that and , we can now write

Thus, the rectangular form of is

.9y^{2}= 6x + 1