Convert Rectangular Coordinates To Polar Coordinates and vice versa - Pre-Calculus

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

Therefore, our radius is .

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

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

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.

Therefore, the answer is .

To convert from rectangular coordinates to polar coordinates :

Using the rectangular coordinates given by the question,

The polar coordinates are

To convert from rectangular coordinates to polar coordinates :

Using the rectangular coordinates given by the question,

The polar coordinates are

To convert from rectangular coordinates to polar coordinates :

Using the rectangular coordinates given by the question,

The polar coordinates are

To convert polar coordinates to rectangular coordinates ,

Using the information given in the question,

The rectangular coordinates are

To convert polar coordinates to rectangular coordinates ,

Using the information given in the question,

The rectangular coordinates are

To convert polar coordinates to rectangular coordinates ,

Using the information given in the question,

The rectangular coordinates are

To convert polar coordinates to rectangular coordinates ,

Using the information given in the question,

The rectangular coordinates are

To convert polar coordinates to rectangular coordinates ,

Using the information given in the question,

The rectangular coordinates are

To convert polar coordinates to rectangular coordinates ,

Using the information given in the question,

The rectangular coordinates are

To convert polar coordinates to rectangular coordinates ,

Using the information given in the question,

The rectangular coordinates are

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

.

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:

If the angle in the polar coordinates is , that means it's in the second quadrant, and away from the x-axis.

This means that the x-coordinate will be negative, and the y-coordinate will be positive:

We can find the x-coordinate using cosine:

multiply both sides by 7

, however we know that the x-coordinate is negative, so we'll use -5.66.

We can find the y-coordinate using sine:

multiply both sides by 7

the 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

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 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.