To convert from rectangular coordinates to polar coordinates :

Using the rectangular coordinates given by the question,

The polar coordinates are 

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

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.

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

.

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.

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 . 

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

To convert from rectangular coordinates to polar coordinates :

Using the rectangular coordinates given by the question,

The polar coordinates are