# Precalculus : Polar Coordinates and Complex Numbers

## Example Questions

### Example Question #11 : Convert Polar Equations To Rectangular Form And Vice Versa

Convert the polar equation into rectangular form.

Explanation:

Start by multiplying both sides by .

Now, isolate the  to one side.

Square both sides.

Recall that  and that .

### Example Question #12 : Convert Polar Equations To Rectangular Form And Vice Versa

Convert the polar equation into rectangular form:

Explanation:

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

### Example Question #13 : Convert Polar Equations To Rectangular Form And Vice Versa

What would be the rectangular equation form for the polar equation ?

Explanation:

To convert from polar coordinates to rectangular coordinates, know that r is the hypotenuse of a right triangle with legs x and y, so .

The cosine of theta is this triangle's adjacent side over the hypotenuse r, so . Making these substitutions into we get:

square the right side to simplify

square both sides to remove the radical

multiply both sides by the right denominator

take both sides to the power

subtract from both sides

take the square root

### Example Question #14 : Convert Polar Equations To Rectangular Form And Vice Versa

Which is equivalent to in rectangular form?

Explanation:

To convert from polar form to rectangular form, substitute in , , and . Equivalently, and :

Substituting these into the original polar equation, we get:

multiply the second two fractions

now multiply these fractions

square both sides

multiply both sides by the denominator

### Example Question #15 : Convert Polar Equations To Rectangular Form And Vice Versa

Write the equation in rectangular form

Explanation:

To convert to rectangular form, it is easiest to first multiply both sides by r:

Now we can make the substitutions and :

We want to solve for y, so subtract x squared from both sides:

now take the square root of both sides

### Example Question #16 : Convert Polar Equations To Rectangular Form And Vice Versa

Convert to rectangular form

Explanation:

First, multiply both sides by the denominator:

multiply both sides by r

Now we can make the substitutions  and :

subtract y from both sides

square both sides

subtract y squared from both sides

we are trying to get this in the form of y=, so subtract from both sides

divide both sides by

simplify

or

### Example Question #17 : Convert Polar Equations To Rectangular Form And Vice Versa

Convert the equation to rectangular form

Explanation:

First, multiply both sides by the denominator:

multiply both sides by r

To convert, make the substitutions , , and

subtract y from both sides

square both sides

subtract y squared from both sides

we want to get y by itself, so subtract from both sides

divide both sides by

### Example Question #18 : Convert Polar Equations To Rectangular Form And Vice Versa

Which rectangular equation is equivalent to ?

Explanation:

First multiply both sides by the denominator:

multiply both sides by r

distribute

Now we can make the substitutions , and :

distribute

combine like terms

subtract 2 x squared from both sides

We want to complete the square on the right, so factor our the -2:

to complete the square, add inside the parentheses. This multiplied by the -2 outside the parentheses is , so this means we're actually subtracting from both sides:

multiply both sides by 8

### Example Question #19 : Convert Polar Equations To Rectangular Form And Vice Versa

Which is the rectangular form for ?

Explanation:

First multiply both sides by the right denominator:

multiply both sides by r

distribute

Now we can start to convet to rectangular by making the substitutions , , and :

combine like terms:

subtract y from both sides, and re-order this in decending order of powers of y:

this is a quadratic, so we can use the quadratic equation to get y by itself:

The answer choice that works is

### Example Question #20 : Convert Polar Equations To Rectangular Form And Vice Versa

Write the equation for in rectangular form

Explanation:

Multiply both sides by the right denominator:

multiply both sides by r

Now we can substitute in and to start converting to rectangular form:

subtract x from both sides

square both sides

multiply both sides by 4

subtract x squared from both sides

take the square root of both sides