# Precalculus : Convert Polar Equations To Rectangular Form and vice versa

## Example Questions

### Example Question #41 : Convert Polar Equations To Rectangular Form And Vice Versa

Convert the rectangular equation into polar form:

Explanation:

Recall that

Substitute that into the equation.

Recall that,

### Example Question #42 : Convert Polar Equations To Rectangular Form And Vice Versa

How could you write the equation in polar coordinates?

Explanation:

To convert from rectangular to polar, use the equivalent forms  and . Substituting these in, we get:

divide both sides by r

divide both sides by to get this equation in terms of r=

Note that we could simplify this a little bit if we wanted to but that wasn't one of the choices.

### Example Question #43 : Convert Polar Equations To Rectangular Form And Vice Versa

How could you express the rectangular equation in polar form?

Explanation:

To convert from rectangular to polar, we can substitute in and . Our equation now becomes:

square both sides to remove the radical

Now we can see that in terms of r, this is a quadratic. We can solve using the quadratic formula if we subtract everything from the right side and get our equation equal to 0:

Put our coefficents a, b, and c into the quadratic formula:

multiplying yields 1, so this now becomes:

we can simplify this knowing the trigonometric identity that

### Example Question #44 : Convert Polar Equations To Rectangular Form And Vice Versa

Which equation is the polar equivalent of the rectangular quadratic ?

Explanation:

To convert from rectangular to polar, we can substitute and . That gives us:

We can see that this is a quadratic in terms of r, so to solve, just like any other quadratic, we want to subtract everything from the right side so that it is equal to 0.

Now we can use the quadratic formula to solve for r:

we can simplify using the trig identity

to get rid of the fraction in the denominator, multiply top and bottom by 2

### Example Question #41 : Convert Polar Equations To Rectangular Form And Vice Versa

How would you write the equation as a polar equation?

Explanation:

This simple rectangular equation represents a circle centered at the origin with radius 3,

since .

The way to write that in polar form is just .

### Example Question #46 : Convert Polar Equations To Rectangular Form And Vice Versa

Write in polar form.

Explanation:

To convert from rectangular to polar, substitute and :

factor out r

divide

### Example Question #47 : Convert Polar Equations To Rectangular Form And Vice Versa

Write the equation in polar form.

Explanation:

To convert from rectangular to polar, substitute in and :

factor out r

this gives us a trivial answer of r = 0, and a second answer found by setting the second [more interesting] answer equal to zero:

### Example Question #48 : Convert Polar Equations To Rectangular Form And Vice Versa

Write in polar form.

Explanation:

To convert, substitute and

factor out r

This gives us the trivial answer r = 0, but also another answer from setting the second factor equal to zero:

multiply by 2

### Example Question #49 : Convert Polar Equations To Rectangular Form And Vice Versa

Write in polar form.

Explanation:

To convert, substitute and

divide both sides by r

The answer choice appears in a slightly different order,

, but these are equivalent expressions.

### Example Question #50 : Convert Polar Equations To Rectangular Form And Vice Versa

Convert to polar form

Explanation:

Re-arrange the left side so that is next to , and factor the 4 out on the right side :

Make the substitutions and :

take the square root of both sides

divide both sides by r

subtract from both sides