Lesson Explainer: Polar Coordinates Mathematics • 12th Grade

In this explainer, we will learn how to define and plot points given in polar coordinates and convert between the Cartesian and polar coordinates of a point.

When we think about points in a plane, we usually think of Cartesian coordinates as this is the most prevalent coordinate system. In particular, the Cartesian coordinates of a point are used in linear motion where specifying the axis of motion is simple and where the motion will take a linear path to a particular location.

Cartesian coordinates, in two dimensions, define a position as the linear displacement from the origin in two mutually perpendicular axes. The origin is the point where the axes intersect, and the points on the plane are specified by a pair of numbers . This coordinate system also allows both positive and negative directions, relative to the origin, and each coordinate set defines a unique point in space.

Recall that when using Cartesian coordinates, we mark a point by how far along the horizontal axis it is (left or right), denoted by , and how far along the vertical axis it is (up or down), denoted by , relative to the origin.

In Cartesian coordinates, any point in space can be defined by a unique set of coordinates given by the pair of numbers . However, there are other ways of representing the position of a point in the plane using a coordinate pair; we will explore one such way known as polar coordinates. These coordinates define a position in space using a combination of radial and angular units and a vector is specified by a straight-line displacement from the origin and the angle from the positive -axis. These are known as the radial and angular coordinates , which indicate the displacement from the origin and the angular direction.

Polar coordinates provide us with an alternative way of plotting points and drawing graphs; we can often express complicated graphs using simple polar functions. For example, represents all points of displacement of one from the origin, which is the unit circle centered at the origin. In Cartesian coordinates, this is described by the curve .

Polar coordinates are naturally used in nonlinear motion, for example, if the motion involves a circular path. This makes polar coordinates useful in calculating the equations of motion for a lot of mechanical systems. It also has other real-world applications, such as in plan position indicators in radars and in gravitational fields the characteristics of a microphone and guiding industrial robots in various production applications, to name a few.

In polar coordinates, we mark a point by its displacement from the origin, denoted by , and its angle from the positive -axis, denoted by .

These are equivalent ways of defining the same point.

The modulus of the radial coordinate, , is equal to the length or distance from the origin. By the Pythagorean theorem, the distance between two points and is given by

Thus, for any points and the origin , this distance is given by

The radial coordinate is defined as

If is negative, it means that the point lies in the quadrant on the opposite side of the pole. We also note that the modulus of the radial coordinate is equal to the length since and .

Since we can form a right triangle from the coordinates, we express the sides of the triangle in terms of and .

This also allows us to express the Cartesian coordinates in terms of .

Definition: Converting from Polar Coordinates to Cartesian Coordinates

The Cartesian coordinates can be written in terms of polar coordinates as

So, if we are given a point in polar coordinates, the displacement from the origin , and the angle , we can determine the point in Cartesian coordinates, and , using these equations.

As an example, letβs convert a point from polar coordinates, in radians, to Cartesian coordinates, for an acute angle located in the first quadrant.

Example 1: Finding the Cartesian Coordinates of a Point Given in Polar Coordinates

Find the Cartesian coordinates of the point .

In this example, we want to determine the Cartesian coordinates for a particular polar coordinate located in the first quadrant, since .

Recall that polar coordinates define a point according to the displacement from the origin, denoted by , and the angular direction from the positive -axis, denoted by .

For any given point in polar coordinates , we can find the equivalent point in Cartesian coordinates, , using the formulae

Thus, substituting the radial coordinate, , and the angular coordinate, , we have and

Therefore, the given point in Cartesian coordinates is

The counterclockwise angle is taken to be positive, while the clockwise angle is negative. In previous examples and diagrams, the angular components have been acute, since the points were in the first quadrant. If a point is in another quadrant, as shown in the diagram below, its angle is not acute.

In fact, although we derived the equations for the polar coordinates from acute angles , we know that they remain true for any angle .

Letβs consider an example where we convert a point that lies in the second quadrant with a nonacute angle from polar coordinates, in degrees, to Cartesian coordinates.

Example 2: Finding the Cartesian Coordinates of a Point Given in Polar Coordinates

Given that the polar coordinates of point are , find the Cartesian coordinates of .

In this example, we want to determine the Cartesian coordinates for a particular polar coordinate located in the second quadrant.

Recall that polar coordinates define a point according to the displacement from the origin, denoted by , and the angular direction from the positive -axis, denoted by .

For any given point in polar coordinates , we can find the equivalent point in Cartesian coordinates, , using the formulae

Thus, substituting the radial coordinate, , and the angular coordinate, , we have and

Therefore, the point in Cartesian coordinates is

So far, we have seen examples of how to convert a point from polar coordinates to Cartesian coordinates, using trigonometry. But what if we wanted to do the reverse, that is, convert a point from Cartesian coordinates into polar coordinates?

Letβs begin by recalling the equations expressing the components of the Cartesian coordinates, and , in terms of the components of the polar coordinates, and :

Using these, we want to write the components of polar coordinates, and , in terms of the components of the Cartesian coordinates, and , no matter which quadrant the point lies in.

If we take the square of each of these and add them up, by using the Pythagorean identity, we can eliminate and show that these satisfy

Also, when we divide the equation for by the equation for , we can cancel the that appears, for , to obtain

We note that this only holds true for . We have a special case when , or in the Cartesian coordinates. For this, we have which leads to , , or . We can ignore the case , since this would imply , which corresponds to the origin , which in polar coordinates is denoted by , for any angle .

Thus, or , which correspond to the -axis. These angles place the point on the -axis and one possible representation for the radial coordinate (for ) is equal to the absolute value of the -coordinate, . A representation of the polar coordinates is , for , and , for .

So, if , we have the following equation to determine the angle :

The range for the inverse tangent function is when the domain of the tangent function is restricted to the same interval, known as a principal branch. This is to ensure that the tangent function is one to one so that the inverse tangent function evaluates to a single value, known as the principal value.

Thus, as long as , we can take the inverse tangent of both sides of the equation to obtain

The angular coordinates correspond to the first and fourth quadrants, or the quadrants where .

In the next example, we will convert a point from Cartesian coordinates to polar coordinates, in degrees.

Example 3: Converting Coordinates into Polar Coordinates

Convert to polar coordinates. Give the angle in degrees and round your answer to one decimal place.

In this example, we want to determine the polar coordinates, in degrees, for the particular Cartesian coordinates.

Recall that polar coordinates define a point according to the displacement from the origin, denoted by , and the angular direction from the positive -axis, denoted by .

The Cartesian coordinates can be expressed in terms of the polar coordinates as

Now, letβs find the polar coordinates of the point by using the graphical representation directly with the definition. The radial coordinate is just the displacement from the origin to the point , which we can find by using the Pythagorean theorem on the right triangle, as shown the diagram. We want to find the hypotenuse of this triangle using

Since the point is located in the first quadrant, the angular coordinate in polar coordinates will be the positive counterclockwise angle from the positive -axis, as shown in the diagram. We can form a right triangle with angle and sides of lengths 2 and 3. Since is an acute angle, we can write this in terms of the sides using the inverse tangent as

We could have also arrived at this answer by using the fact that we can convert the Cartesian coordinate located in the first quadrant into the polar coordinates by using

This gives the same radial and angular coordinate after substituting and .

Thus, to one decimal place, the polar coordinates are

In the previous example, we converted a point in Cartesian coordinates that lies in the first quadrant into polar coordinates. As seen in this example, we can compute the angular coordinate of a point by using in the first and fourth quadrants. However, this is no longer the case if the point lies in the second or third quadrant.

For the second and third quadrants, a value of , in radians, or , in degrees, must be added to or subtracted from the angle , to adjust the angular coordinate so that the point lies in the correct quadrant. This does not affect the tangent function itself since we have the identities or, more generally,

To see this, consider a point in Cartesian coordinates that lies in the second quadrant, , with and .

The angular coordinate for this point in polar coordinates will be the positive counterclockwise angle from the positive -axis and angle is measured from the negative -axis, as shown in the diagram. We can form a right triangle with angle and sides of lengths and . Since is an acute angle, we can write this in terms of the sides using the inverse tangent as

We also have , and substituting angle , we can rearrange this to find where, for the last equality, we used the fact that the tangent function, and hence the inverse tangent function, is an odd function. Since we have , this is equivalent to

In the next example, we will find the polar coordinates of a particular point that lies in the second quadrant.

Example 4: Converting Coordinates into Polar Coordinates

Convert to polar coordinates. Give the angle in radians and round your answer to two decimal places.

In this example, we want to determine the polar coordinates , in radians, for the particular Cartesian coordinates .

Recall that polar coordinates define a point according to the displacement from the origin, denoted by , and the angular direction from the positive -axis, denoted by .

The Cartesian coordinates can be expressed in terms of the polar coordinates as

Now, letβs find the polar coordinates of the point by using the graphical representation directly with the definition. The radial coordinate is just the displacement from the origin to the point , which we can find from using the Pythagorean theorem on the right triangle, as shown the diagram. We want to find the hypotenuse of this triangle using

Since the point is located in the second quadrant, the angular coordinate in polar coordinates will be the positive counterclockwise angle from the positive -axis and angle is measured from the negative -axis, as shown in the diagram. We can form a right triangle with angle and sides of lengths 2 and 5. Since is an acute angle, we can write this in terms of the sides using the inverse tangent as

We also have , and substituting the angle , we can rearrange this to find

We could have also arrived at this answer by using the fact that we can convert the Cartesian coordinate located in the second quadrant into the polar coordinates by using

This gives the same radial and angular coordinate after substituting and .

Thus, the polar coordinates rounded to two decimal places is given by

For the third quadrant we can show, in a similar way, that we have to subtract from to get the angular coordinate in the correct quadrant.

Polar coordinates are not unique and there are many ways of representing the same point. As an example, letβs determine the polar coordinates of the point in Cartesian coordinates.

The radial coordinate is the displacement from the origin, which we can determine using the Pythagorean theorem on the right triangle with sides of length 1 and angle . In particular, which is a particular choice for the radial coordinates with , but we could also use .

There are many ways to express the angular coordinate . One is the positive counterclockwise angle from the positive -axis, which from the right triangle gives us the tangent in terms of the ratio of the opposite and adjacent side:

Since the point lies in the first quadrant and is an acute angle in the diagram, we can find the angle directly from the inverse tangent:

Thus, a polar coordinate to describe the point is .

Another polar coordinate can be found if we use the negative clockwise angle from the positive -axis which would give an equivalent point as . In fact, if we make a full revolution from this point, in either the clockwise or counterclockwise direction, we return back to the same point. So, another representation would be .

This shows a key difference when using polar coordinates, as it allows for an infinite number of coordinate pairs to describe any given point. This is because we can add any integer multiple of a full revolution ( or ) to the angular coordinate to get an equivalent point in polar coordinates. This follows because the trigonometric functions, which are used to define polar coordinates, are themselves periodic.

The radial coordinate can also be positive or negative, and when negative radial coordinates are used, the angular coordinate places the location in the opposite quadrant from the intended point, though typically we keep the radial coordinate positive and modify the angle accordingly by adding or subtracting or from to place the location in the opposite quadrant.

These equivalence conditions can be summarized as follows.

Definition: Periodicity Condition for Polar Coordinates

If describes polar coordinates, then we can express the equivalent polar coordinates as and for any .

In other words, we add an even integer multiple of (or ) for positive radial coordinates and an odd integer multiple of (or ) for negative radial coordinates. For the second condition, we usually consider the points as

Now, letβs look at an example where we have to find multiple equivalent representations of a polar coordinate.

Example 5: Multiple Representations of Polar Coordinates

Which of the ordered pairs , , , and does not describe the position of point in the diagram?

In this example, we want to find the equivalent representations of the same polar coordinate, in degrees, and determine which of the given points does not describe the position of that point.

Recall that polar coordinates define a point according to the displacement from the origin, denoted by , and the angular direction from the positive -axis, denoted by .

These coordinates are not unique, since we can add any integer multiple of a full revolution () to the angular coordinate to get an equivalent point in polar coordinates. In particular,

Another way to have an equivalent representation is using a negative radius which places the point in the opposite quadrant and is equivalent to adding or subtracting half a revolution (); taking the periodicity condition into account, this can be written as

From the diagram, we can read the polar coordinates of , which has a radial coordinate 4 and angular coordinate and thus the polar coordinate .

Since the angular coordinate is periodic by , we can add and subtract a full revolution to find equivalent representations; in particular, and

Therefore, two equivalent representations are the polar coordinates and .

The remaining point lies in the first quadrant, but the angle given is outside the range and we can find the angle within this range by subtracting from the angle to write the point in a standardized form. Hence, the equivalent point is

While the radial coordinate is the same, this is clearly different from point , since it is in a completely different quadrant from the angular coordinate. We can also plot this point in relation to point .

Another way to check whether two points in polar coordinates are coincident or not is by calculating the distance between the two points. If the distance is zero, then the two points are coincident, while a nonzero value indicates that they are not.

Hence, the ordered pair that does not describe point in the diagram is

Thus, due to these equivalences in writing the polar coordinates, in order to express polar coordinates in a standardized form with and in radians or in degrees, we may have to adjust the value of the angular coordinate .

The conventions we use take the counterclockwise angle as positive and clockwise as negative. For the first and second quadrants, the angle is in the positive counterclockwise direction from the positive -axis, while for the third and fourth quadrants, the angle is in the negative clockwise direction from the positive -axis.

We can summarize what we have covered so far in a definition, which can be used to convert a point from one coordinate system to another in any quadrant.

Definition: Converting from Cartesian Coordinates to Polar Coordinates

One possible representation of the polar coordinates with can be expressed in terms of Cartesian coordinates as for in radians or in degrees.

The origin in Cartesian coordinates, with and , has a special case where the polar coordinates can be represented by or for any angle .

This information can be communicated efficiently using the following diagram.

Another way to express polar coordinates for all quadrants is if we instead define with polar coordinates . In other words, we can define the angular coordinate for all by the inverse tangent function without having to add or subtract or , but instead by changing the sign of the radial coordinate for specific quadrants. In particular, for the first and fourth quadrants where , we have , and for the second and third quadrants where , we have . This works because of the property where which is equivalent to the definition of for the second and third quadrant (), where we have to add or subtract from the inverse tangent.

As an example, letβs consider the points , , , and in Cartesian coordinates, where each point is located in the different quadrants, as shown in the graph. We want to determine the polar coordinates of these points in a standardized way and in degrees with .

The radial coordinate for these points in polar coordinates will be the same, since and we take for the standardized form.

The difference will be with the angular coordinate , since this will determine the direction and hence which quadrant the point in polar coordinates will lie in.

The point is in the first quadrant and we can determine the angular coordinate from the general formula as

As expected, this angle is acute, as , which places the point in the first quadrant, and it is positive, which represents the counterclockwise direction from the positive -axis.

The point is in the second quadrant and the angular coordinate is

As expected, we have , which places the point in the second quadrant, and it is positive, which represents the counterclockwise direction from the positive -axis.

The point is in the third quadrant and the angular coordinate is

As expected, we have , which places the point in the third quadrant, and it is negative, which represents the clockwise direction from the positive -axis.

Finally, the point is in the fourth quadrant and the angular coordinate is

As expected, we have , which places the point in the fourth quadrant, and it is negative, which represents the clockwise direction from the positive -axis.

Thus, a possible representation of the polar coordinates, accurate to two decimal places, of the points in Cartesian coordinates is given by

We could have also used a negative radius to represent a point in the opposite quadrant as ; that is, is equivalent to , and is equivalent to .

Now, letβs consider an example where we convert a point, in the fourth quadrant, from Cartesian coordinates to polar coordinates, in radians.

Example 6: Converting Coordinates into Polar Coordinates

Represent the point with Cartesian coordinates in terms of polar coordinates.

In this example, we want to determine the polar coordinates , in radians, for the particular Cartesian coordinates .

Recall that polar coordinates define a point according to the displacement from the origin, denoted by , and the angular direction from the positive -axis, denoted by . We use the conventions where angle is the positive counterclockwise angle in the first and second quadrants and the negative clockwise angle in the third and fourth quadrants, and all the options are given in this form. We also restrict the angles to , in order to write them in a standardized form.

The Cartesian coordinates can be expressed in terms of the polar coordinates as

The polar coordinates are not unique and there are equivalent ways of describing the same point, since the trigonometric functions used to define them are periodic.

Now, letβs find the polar coordinates of the point in a standardized form by using the definition. The radial coordinate is just the displacement from the origin to the point , which we can find by using the Pythagorean theorem on the right triangle, as shown the diagram. We want to find the hypotenuse of this triangle using

Since the point is located in the fourth quadrant, the angular coordinate in polar coordinates will be the negative clockwise angle from the positive -axis, as shown in the diagram. The positive measure of this angle is and thus .

We can form a right triangle with angle and sides of length 1. Since is an acute angle, we can write this in terms of the sides using the inverse tangent as

Thus, we have . We could have also arrived at this answer by using the fact that we can convert the Cartesian coordinate located in the fourth quadrant into the polar coordinates by using

This gives the same radial and angular coordinate after substituting and .

Thus, the polar coordinates are

This is option B.

We can also use a polar grid for polar coordinates, which is represented as a series of concentric circles radiating out from the pole, or the origin of the coordinate plane. The first coordinate is radius or the length of the line segment from the pole and angle indicates the direction. Consider the point in polar coordinates as shown in the diagram with the polar grid.

This grid can also be used in real-world applications in navigation. For example, if a sailboat encounters rough weather over 12 km from the port and is blown off course by a strong wind, we can use this polar grid with polar coordinates to indicate the location of the sailboat to the coast guard.

Next, letβs look at how we can use a graph with a polar grid to determine the polar coordinates of a particular point.

Example 7: Graphing Polar Coordinates

Consider the points plotted on the graph.

Write down the polar coordinates of , giving the angle in the range .

In this example, we want to determine the polar coordinates of a particular point, specified on a polar grid.

Recall that polar coordinates define a point according to the displacement from the origin, denoted by , and the angular direction from the positive -axis, denoted by . A polar grid for polar coordinates is represented as a series of concentric circles radiating out from the pole, or the origin of the coordinate plane.

From the graph with the polar grid, each circle increases its radius by 0.5 from the last, starting from and ending at . In other words, the radius is for , with the concentric circle starting from the one closest to the origin.

The point lies in the fourth quadrant and on the second concentric circle that has radius .

Each segment or ray represents an angle of , starting from and going all around the circle.

The point is in between the ray of argument and that of as marked on the graph.

The point is on the ray of argument

But this is the counterclockwise angle from the positive -axis. Since we require to be in the range and the angle is taken clockwise for this quadrant, the negative direction from the positive -axis, we have to subtract to get

Equivalently, we can also find the angle graphically, since has 3 segments of angle in the clockwise direction and will be negative:

Thus, the polar coordinates of are given by

For any two points in polar coordinates, we can also calculate the distance between the points in polar coordinates using the following definition.

Definition: Distance between Two Points in Polar Coordinates

The distance between two points and in polar coordinates is given by

We can derive this from the distance between two points and in Cartesian coordinates, given by

We can use this to find the distance between two points and in polar coordinates by converting the Cartesian to polar using for . Thus, the distance in polar coordinates is

Now, by applying the Pythagorean identity and the angle difference formula for cosine given by we can rewrite the expression for the distance as

Returning to the example with the sailboat, suppose the locations of the port and the sailboat are and , respectively, as depicted on the diagram. To specify the polar coordinates in a standardized form, we can modify the angle of the sailboat, as the second angle, , is not in the range . The polar coordinate is the same as .

Thus, the sailboat is located at and the port is located at . We can calculate the distance between the port and the sailboat if we substitute , and , :

We should note that for this distance formula, we do not actually need to write the polar coordinates in a standardized form first as it works for any angle , since the cosine function is periodic. Using the coordinates for the sailboat and for the port directly, we can substitute , and , to obtain the same result:

Thus, the distance between the port and the sailboat, accurate to two decimal places, is given by

In the next example, we will look at how to determine the distance between two points in polar coordinates.

Example 8: Finding the Distance between Two Polar Coordinates

Find the distance between the polar coordinates and . Give your answer accurate to two decimal places.

In this example, we want to find the distance between two polar coordinates, expressed in radians.

Recall that polar coordinates define a point according to the displacement from the origin, denoted by , and the angular direction from the positive -axis, denoted by .

The distance between two points and in polar coordinates is given by

We can substitute , and , :

Therefore, the distance accurate to two decimal places is 2.12.

Now, how can we tell whether two points and are coincident (i.e., they describe the same point)? From the periodicity conditions of equivalent points in polar coordinates, they are coincident when or for any . We can express both of these as a single condition: for any .

Another way to check if two points in polar coordinates are coincident or not is by calculating the distance between the two points. If the distance is zero, then the two points are coincident, while a nonzero value indicates that they are not. We can check this by using the distance formula in polar coordinates, by substituting and :

Since we have , the distance becomes

As expected, the distance between two coordinates that describe the same point is zero. For a nonzero distance, the two points would not describe the same point.

Key Points

• A polar coordinate is of the form , which denotes the displacement from the origin and the angle from the positive -axis.
• The conventions we use take the counterclockwise angle as positive and clockwise as negative. For the first and second quadrants, the angle is in the positive counterclockwise direction from the positive -axis, while for the third and fourth quadrants, the angle is in the negative clockwise direction from the positive -axis.
• To convert from polar to Cartesian , we use
• To convert from Cartesian to polar , one possible representation is given by for in radians or in degrees.
• We can find equivalent polar coordinates by adding or subtracting any integer multiple of a full revolution ( or ), or by using a negative radius which places the coordinate in the opposite quadrant, which is the same as adding or subtracting half a revolution ( or ) from . In particular, we have
• The distance between two points in polar coordinates and is given by