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

## Example Questions

### Example Question #21 : Convert Polar Equations To Rectangular Form And Vice Versa

Write the equation for  in rectangular form

Explanation:

First multiply both sides by cosine

Now we can make the substitutions and

square both sides

multiply both sides by the denominator

distribute on the left side

subtract from both sides

factor out y squared

divide both sides by

take the square root

### Example Question #22 : Convert Polar Equations To Rectangular Form And Vice Versa

Which is equivalent to in rectangular form?

Explanation:

Multiply both sides by r squared

Now we can substitute and

square both sides

### Example Question #23 : Convert Polar Equations To Rectangular Form And Vice Versa

Which is equivalent to in rectangular form?

Explanation:

First multiply both sides by r:

Now make the substitutions and :

subtract x squared and y squared from both sides

square both sides

### Example Question #24 : Convert Polar Equations To Rectangular Form And Vice Versa

Write in rectangular form

Explanation:

multiply both sides by r:

Now we can make the substitutions , , and :

If we subtract the terms on the right from both sides, we can complete the square twice and make this into a regular circle equation:

To complete the square for x, add 1, since .

To complete the square for y, add since .

re-write the left side as binomials squared:

### Example Question #25 : Convert Polar Equations To Rectangular Form And Vice Versa

Write the equation for in rectangular form.

Explanation:

Multiply both sides by r:

Now we can substitute in , , and .

if we subtract both terms on the right from both sides, we can complete the square twice to get this into the normal form for a circle.

To complete the square for x, add 1 to both sides, since

To complete the square for y, add 16 to both sides, since :

Re-write the left side as two binomials squared

### Example Question #26 : Convert Polar Equations To Rectangular Form And Vice Versa

Write the equation for in rectangular form

Explanation:

To convert, make the substitutions and :

square both sides

multiply both sides by

subtract  from both sides

subtract from both sides

factor our

divide both sides  by

take the square root of both sides

### Example Question #27 : Convert Polar Equations To Rectangular Form And Vice Versa

Convert the equation to rectangular form

Explanation:

Multiply both sides by :

distribute

multiply both sides by r

convert to rectangular by making the substitutions , , and :

subtract 8x from both sides and combine like terms

complete the square for x by adding 16 to both sides

divide both sides by 16

### Example Question #28 : Convert Polar Equations To Rectangular Form And Vice Versa

Convert  to polar coordinates.

Explanation:

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.

### Example Question #29 : Convert Polar Equations To Rectangular Form And Vice Versa

Convert  to polar form.

Explanation:

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.

### Example Question #30 : Convert Polar Equations To Rectangular Form And Vice Versa

Convert the rectangular equation to polar form:

Explanation:

Because  and , substitute those values in the rectangular form.

Now, expand the equation.

Subtract  from both sides.

Recall the trigonometric identity

Factor the equation.