# Precalculus : Convert Rectangular Coordinates To Polar Coordinates and vice versa

## Example Questions

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### Example Question #75 : Polar Coordinates

Which polar-coordinate point is not the same as the rectangular point ?

Explanation:

Plotting this point creates a triangle in quadrant I:

Using our knowledge of Special Right Triangles, we can conclude that the angle is and the radius/hypotenuse of this triangle is . Our polar coordinates are therefore , so we can eliminate that as a choice since we know it works.

Looking at the unit circle [or just the relevant parts] can give us a sense of what happens when the angles and/or the radii are negative:

Now we can easily see that the angle would correspond with our angle of , so  works.

We can see that if our radius is negative we'd want to start off at the angle , so the point  works.

As we can see from looking at this excerpt from the unit circle, another way of writing the angle would be to write , so the point  works.

The only one that does not work would be  because that would place us in quadrant II rather than I like we want.

### Example Question #76 : Polar Coordinates

Which of the following is a set of polar coordinates for the point with the rectangular coordinates

Explanation:

The relation between polar coordinates and rectangular coordinates is given by  and .

You can plug in each of the choices for  and  and see which pair gives the rectangular coordinate .

The answer turns out to be .

Alternatively, you can find  by the equation

, thus

.

As for finding , you can use the equation

, and since

.

Thus, the polar coordinate is

### Example Question #77 : Polar Coordinates

Convert the point to polar form

Explanation:

First, find r using pythagorean theorem,

Then we can find theta by doing the inverse tangent of y over x:

Since this point is in quadrant II, add 180 degrees to get

### Example Question #78 : Polar Coordinates

Convert the following rectangular coordinates to polar coordinates: