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

## Example Questions

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### Example Question #51 : Polar Coordinates

Convert the polar coordinates  to rectangular form.

Explanation:

We begin by recalling that polar coordinates are expressed in the form , where  is the radius (the distance from the origin to the point) and  is the angle formed between the postive x-axis and the radius.

We can find our x-coordinate and y-coordinate in rectangular form quite easily by keeping in mind two equations.

or

or

Substituting in both of these gives respectively

Therefore, the rectangular coordinates of our point are

### Example Question #111 : Polar Coordinates And Complex Numbers

Convert the polar coordinates to rectangular coordinates:

Explanation:

To convert polar coordinates  to rectangular coordinates ,

Using the information given in the question,

The rectangular coordinates are

### Example Question #51 : Polar Coordinates

Convert the polar coordinates to rectangular coordinates:

Explanation:

To convert polar coordinates  to rectangular coordinates ,

Using the information given in the question,

The rectangular coordinates are

### Example Question #1 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Convert the polar coordinates to rectangular coordinates:

Explanation:

To convert polar coordinates  to rectangular coordinates ,

Using the information given in the question,

The rectangular coordinates are

### Example Question #51 : Polar Coordinates

Convert the polar coordinates to rectangular coordinates:

Explanation:

To convert polar coordinates  to rectangular coordinates ,

Using the information given in the question,

The rectangular coordinates are

### Example Question #1 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Convert the polar coordinates to rectangular coordinates:

Explanation:

To convert polar coordinates  to rectangular coordinates ,

Using the information given in the question,

The rectangular coordinates are

### Example Question #121 : Polar Coordinates And Complex Numbers

Convert the polar coordinates to rectangular form:

Explanation:

To convert polar coordinates  to rectangular coordinates ,

Using the information given in the question,

The rectangular coordinates are

### Example Question #8 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

Convert the polar coordinates to rectangular coordinates:

Explanation:

To convert polar coordinates  to rectangular coordinates ,

Using the information given in the question,

The rectangular coordinates are

### Example Question #1 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

How could you express  in rectangular coordinates?

Explanation:

The polar coordinates given have an angle of but a negative radius, so our coordinates are located in quadrant III.

This means x and y are both negative. You can figure out these x and y coordinates using trigonometric ratios, or since the angle is , special right triangles.

The hypotenuse of this triangle is 5, but in the special right triangle it's 2, so we know we're multiplying each side by .

That makes the x-coordinate or adjacent side be

and the y-coordinate or opposite side be

.

In this case, once again, both are negative, so our answer is

.

### Example Question #2 : Convert Rectangular Coordinates To Polar Coordinates And Vice Versa

How could you express  in rectangular coordinates?

Round to the nearest hundredth.

Explanation:

In order to determine the rectangular coordinates, look at the triangle representing the polar coordinates:

We can see that both x and y are positive. We can figure out the x-coordinate by using the cosine:

multiply both sides by 10.

We can figure out the y-coordinate by using the sine:

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