Convert Polar Equations To Rectangular Form and vice versa - Pre-Calculus
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Convert
to polar coordinates.
Convert to polar coordinates.
Write the Cartesian to polar conversion formulas.


Substitute the coordinate point to the equations and solve for
.



Since
is located in between the first and second quadrant, this is the correct angle.
Therefore, the answer is
.
Write the Cartesian to polar conversion formulas.
Substitute the coordinate point to the equations and solve for .
Since is located in between the first and second quadrant, this is the correct angle.
Therefore, the answer is .
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Convert
to polar form.
Convert to polar form.
Write the Cartesian to polar conversion formulas.


Substitute the coordinate point to the equations to find
.



Since
is not located in between the first quadrant, this is not the correct angle. The correct location of this coordinate is in the third quadrant. Add
radians to get the correct angle.

Therefore, the answer is
.
Write the Cartesian to polar conversion formulas.
Substitute the coordinate point to the equations to find .
Since is not located in between the first quadrant, this is not the correct angle. The correct location of this coordinate is in the third quadrant. Add
radians to get the correct angle.
Therefore, the answer is .
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Convert from polar form to rectangular form:

Convert from polar form to rectangular form:
Start by multiplying both sides by
.


Keep in mind that 

Remember that 
So then,


Now, complete the square.


Start by multiplying both sides by .
Keep in mind that
Remember that
So then,
Now, complete the square.
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Convert the polar equation to rectangular form:

Convert the polar equation to rectangular form:
Start by multiplying both sides by
.


Remember that 

Keep in mind that 
So then,


Now, complete the square.


This is a graph of a circle with a radius of
and a center at 
Start by multiplying both sides by .
Remember that
Keep in mind that
So then,
Now, complete the square.
This is a graph of a circle with a radius of and a center at
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Convert the polar equation into rectangular form:

Convert the polar equation into rectangular form:
Remember that 
So then
becomes

Now, multiply both sides by
to get rid of the fraction.

Since
the rectangular form of this equation is

Remember that
So then becomes
Now, multiply both sides by to get rid of the fraction.
Since the rectangular form of this equation is
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Convert the polar equation into rectangular form:

Convert the polar equation into rectangular form:
Recall that 
Now, substitute in that value into the given equation.

Multiply both sides by
to get rid of the fraction.

Remember that 
The rectangular form of this equation is then 
Recall that
Now, substitute in that value into the given equation.
Multiply both sides by to get rid of the fraction.
Remember that
The rectangular form of this equation is then
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Convert the polar equation into rectangular form:

Convert the polar equation into rectangular form:
Start by using the double angle formula for
.

Substitute that into the equation gives the following:


Because we need
and
to get
and
respectively, multiply both sides by
.

Now, recall that
.


Start by using the double angle formula for .
Substitute that into the equation gives the following:
Because we need and
to get
and
respectively, multiply both sides by
.
Now, recall that .
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Convert to polar form: 
Convert to polar form:
Convert by making the substitutions
and
:
subtract
from both sides
factor out r
divide

Convert by making the substitutions and
:
subtract
from both sides
factor out r
divide
Compare your answer with the correct one above
Convert the polar equation to rectangular form:

Convert the polar equation to rectangular form:
Start by taking the tangent.


Recall that 


Start by taking the tangent.
Recall that
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Convert the polar equation to rectangular form:

Convert the polar equation to rectangular form:
Start by taking the tangent.


Recall that 


Start by taking the tangent.
Recall that
Compare your answer with the correct one above
Convert the polar equation to rectangular form:

Convert the polar equation to rectangular form:
Recall that 
Substituting this into the given equation gives

Multiply both sides by
to get rid of the fraction.

Recall that
and that 



Recall that
Substituting this into the given equation gives
Multiply both sides by to get rid of the fraction.
Recall that and that
Compare your answer with the correct one above
Convert the polar equation into rectangular form:

Convert the polar equation into rectangular form:
Recall that
and 

Multiply both sides by 

Recall that
and 




Recall that and
Multiply both sides by
Recall that and
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Convert the polar equation into rectangular form:

Convert the polar equation into rectangular form:
Recall that 
Plugging this into the equation gives us

Multiply both sides by
to get rid of the fraction.

Recall that 
So then the rectangular form of the equation is 
Recall that
Plugging this into the equation gives us
Multiply both sides by to get rid of the fraction.
Recall that
So then the rectangular form of the equation is
Compare your answer with the correct one above
Convert the polar equation into rectangular form.

Convert the polar equation into rectangular form.
Recall that 

Multiply both sides by
to get rid of the fraction.

Recall that
.


Recall that
Multiply both sides by to get rid of the fraction.
Recall that .
Compare your answer with the correct one above
Convert the polar equation into rectangular form.

Convert the polar equation into rectangular form.

Start by multiplying both sides by
.

Now, isolate the
to one side.

Square both sides.

Recall that
and that
.


Start by multiplying both sides by .
Now, isolate the to one side.
Square both sides.
Recall that and that
.
Compare your answer with the correct one above
Convert the rectangular equation to polar form:

Convert the rectangular equation to polar form:
Because
and
, substitute those values in the rectangular form.


Now, expand the equation.

Subtract
from both sides.


Recall the trigonometric identity 

Factor the equation.

Because and
, substitute those values in the rectangular form.
Now, expand the equation.
Subtract from both sides.
Recall the trigonometric identity
Factor the equation.
Compare your answer with the correct one above
Convert the rectangular equation to polar form:

Convert the rectangular equation to polar form:
Recall that 
Substitute that into the equation.

Recall that,


Recall that
Substitute that into the equation.
Recall that,
Compare your answer with the correct one above
Convert the rectangular equation to polar form:

Convert the rectangular equation to polar form:
Recall that
.
Substitute that into the equation.

Now, isolate the
on one side.

Recall that,


Recall that .
Substitute that into the equation.
Now, isolate the on one side.
Recall that,
Compare your answer with the correct one above
Convert the rectangular equation into polar form.

Convert the rectangular equation into polar form.
Recall that 
Substitute that into the equation.

Recall that,


Recall that
Substitute that into the equation.
Recall that,
Compare your answer with the correct one above
Convert the rectangular equation into polar form:

Convert the rectangular equation into polar form:
Recall that 
Substitute that into the equation.

Recall that,


Recall that
Substitute that into the equation.
Recall that,
Compare your answer with the correct one above