Switching Coordinates Examples

The rectangular coordinates of the point P are (4, 3). What are the polar coordinates of this point?

Graph the point P and draw the triangle:

We need to find r and θ.

We use the Pythagorean theorem to find the hypotenuse:

To find θ we use the definition of the tangent function.

Therefore

The polar coordinates of P are

(r, θ) = (5, 0.644).

The polar coordinates of the point Q are . What are the rectangular coordinates of Q?

It's always to wise to start by graphing the point and drawing a triangle:

.

We need to find x and y. To find y we use

to get

Therefore

To find x we use

to get

Therefore

The rectangular coordinates of Q are

If the point in question is on one of the axes, there's no need to draw a triangle because we can look and see what's going on.

What are the polar coordinates of the point with rectangular coordinates (5, 0)?

The point lies on the x-axis: 

Since the point is on the positive side of the x-axis, we can take θ = 0. Since the point is a distance of 5 from the origin, r must be 5. The polar coordinates are (r, θ) = (5, 0). They're exactly the same as the rectangular coordinates!

Translate the rectangular coordinates (-4, -4) into polar coordinates.

This point is in the third quadrant:

When we translate into polar coordinates, r should be positive and θ should be between π and .

Here's the useful triangle:

We find the magnitude r by finding the hypotenuse of the triangle:

Then we find the direction θ by finding the reference angle α and adding π:

We have

therefore

The polar representation of the point is

This is in the correct quadrant, so it's a perfectly reasonable answer.

Translate the polar coordinates into rectangular coordinates.

This point is in the fourth quadrant, which means the rectangular coordinates should have x > 0 and y < 0.

We know that

We do mean θ here, not some reference angle.

Therefore

y = r sin θ.

and

x = r cos θ.

To find x and y, we plug the values and into these equations.

The rectangular coordinates are