# Precalculus : Convert Rectangular Coordinates To Polar Coordinates

## Example Questions

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### Example Question #1 : Convert Rectangular Coordinates To Polar Coordinates

Convert the rectangular coordinates  to polar form with an angle between  and .

Explanation:

We must first recall that the polar coordinates of a point are expressed in the form , where  is the radius (or the distance from the origin to the point) and  is the angle formed between the positive x-axis to the radius.

The radius can be calculated using the distance formula.

Our first point is the origin  and our second point is the one in question

Therefore, substituting gives us

We can find our angle  using the formula

Substituting the coordinates of our point  gives

We can use our knowledge or a chart or calculator to determine that the angle that gives this tangent value is  or .  Since we want a postive angle less than , we need to go with the latter option.

Therefore, the polar coordinates of our point are

### Example Question #2 : Convert Rectangular Coordinates To Polar Coordinates

Convert  to polar coordinates.

Explanation:

Write the Cartesian to polar conversion formulas.

Substitute the coordinate point to the equations to solve for .

Ensuring that  is located the first quadrant, the correct angle is zero.

### Example Question #3 : Convert Rectangular Coordinates To Polar Coordinates

Convert the following rectangular coordinates to polar coordinates:

Explanation:

To convert from rectangular coordinates to polar coordinates :

Using the rectangular coordinates given by the question,

The polar coordinates are

### Example Question #4 : Convert Rectangular Coordinates To Polar Coordinates

Convert the rectangular coordinates to polar coordinates:

Explanation:

To convert from rectangular coordinates to polar coordinates :

Using the rectangular coordinates given by the question,

The polar coordinates are

### Example Question #5 : Convert Rectangular Coordinates To Polar Coordinates

Convert the rectangular coordinates to polar coordinates:

Explanation:

To convert from rectangular coordinates to polar coordinates :

Using the rectangular coordinates given by the question,

The polar coordinates are

### Example Question #6 : Convert Rectangular Coordinates To Polar Coordinates

Convert the rectangular coordinates to polar coordinates:

Explanation:

To convert from rectangular coordinates to polar coordinates :

Using the rectangular coordinates given by the question,

The polar coordinates are

### Example Question #7 : Convert Rectangular Coordinates To Polar Coordinates

How could you express in polar coordinates?

Explanation:

These rectangular coordinates form a right triangle whose side adjacent to the angle is 7.5, and whose opposite side is 4. This means we can find the angle using tangent:

This would be the angle if these coordinates were in the first quadrant. Since both x and y are negative, this point is in the third. We can adjust the angle by adding , giving us .

Now we just need to find the radius - this will be the hypotenuse of the triangle:

take the square root of both sides

So, our polar coordinates are

### Example Question #8 : Convert Rectangular Coordinates To Polar Coordinates

Which coordinates would not describe a point at ?

Explanation:

Plotting the point listed gives this triangle:

Using Pythagorean Theorem or just knowing that this is a Pythagorean Triple, we get that the hypotenuse/ radius in polar coordinates is 5.

To find that angle, we can use tangent:

That's the angle in quadrant I. This point is in quadrant IV, so we can figure out the angle by subtracting from :

We can also find the corresponding angle in quadrant II by subtracting from , and the corresponding angle in quadrant III by adding , giving us these angles:

The point originally converted to polar coordinates is , so we know that works.

If the radius is negative, we want the angle to be at 2.4981, so the point works.

Since our angle 5.6397 is exactly 0.6435 radians below the x-axis, the point will work.

Similarly, the negative version of 2.4981 would be -3.7851, so

works.

The one that does not work has a positive 5 radius and 0.6435 as the angle, which would be located in quadrant I.

### Example Question #9 : Convert Rectangular Coordinates To Polar Coordinates

Which polar coordinates conicide with the rectangular point ?

Explanation:

Since the x and y coordinates indicate the same distance, we know that the triangle formed has two angles measuring .

The ratio of the legs to the hypotenuse is always , so since the legs both have a distance of 6, the hypotenuse/ radius for our polar coordinates is .

Since the x-coordinate is negative but the y-coordinate is positive, this angle is located in the second quadrant.

, so our angle is .

This makes our coordinates

.

### Example Question #10 : Convert Rectangular Coordinates To 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.

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