Polar Coordinate System: Videos & Practice Problems

In the polar coordinate system, points are represented as ordered pairs (r, θ), where r is the distance from the pole and θ is the angle from the polar axis. To plot points, locate θ first, then count r units from the pole. Negative r values indicate counting in the opposite direction. Multiple ordered pairs can represent the same point by adding or subtracting multiples of 2π to θ or changing r to negative and adding π to θ. Understanding these concepts is crucial for mastering polar coordinates and their applications in various mathematical contexts.

Intro to Polar Coordinates

In mathematics, the polar coordinate system offers a unique way to represent points using the ordered pair (r, θ), where r denotes the distance from the pole (the origin) and θ represents the angle measured from the polar axis (analogous to the positive x-axis in Cartesian coordinates). This system is particularly useful when dealing with circular shapes and angles, as it allows for a more intuitive understanding of positions in a circular context.

To plot a point in polar coordinates, one must first identify the angle θ. This angle is measured counterclockwise from the polar axis. For example, if θ is π/3 radians, you would locate this angle on the polar coordinate system. After identifying the angle, the next step is to measure the distance r from the pole. If r is positive, you move outward from the pole along the line defined by θ. If r is negative, you would move in the opposite direction along the same line, effectively reflecting the point across the pole.

When working with polar coordinates, it is essential to understand how to interpret both positive and negative values for r and θ. For instance, a point given as (4, π/3) indicates moving 4 units away from the pole at an angle of π/3 radians. Conversely, a point like (5, -π/3) requires measuring -π/3 radians clockwise from the polar axis before moving 5 units away from the pole.

Additionally, when the r value is negative, such as in the case of (-3, π/6), the process involves first locating the angle π/6 and then counting 3 units in the opposite direction from the pole. This method of reflection helps visualize the point's location in relation to the pole.

Understanding these principles is crucial for effectively plotting points in the polar coordinate system and recognizing how they relate to their Cartesian counterparts. As you practice, pay close attention to the signs of both r and θ to ensure accurate plotting.

Intro to Polar Coordinates Example 1

In polar coordinates, points are defined by a radius (r) and an angle (θ). When plotting these points, it’s essential to first identify the angle before determining the position based on the radius. Angles can be expressed in both radians and degrees, which is crucial for accurate plotting.

For example, consider point A, which has coordinates (5, 60°). To plot this, convert 60 degrees to radians, which is \frac{\pi}{3} . Start by locating the angle of \frac{\pi}{3} on the polar coordinate system. From this angle, move outward along the line to a distance of 5 units to find point A.

Next, point B is represented as (3, 90°). The angle of 90 degrees corresponds to \frac{\pi}{2} radians. After locating this angle, extend 3 units along the line to reach point B.

Point C is given as (0, - \frac{5\pi}{3} ). The negative angle indicates a clockwise measurement. First, locate - \frac{5\pi}{3} , which brings you back to the same line as point A. However, since the radius is 0, point C remains at the origin, or the pole.

Finally, point D is defined by the coordinates (2, \frac{7\pi}{3} ). The angle \frac{7\pi}{3} represents a full rotation plus an additional \frac{\pi}{3} . After completing the rotation, move 2 units outward to find point D.

By understanding how to convert between degrees and radians and how to interpret the radius and angle, you can effectively plot points in polar coordinates. Practice with various points to strengthen your skills in this coordinate system.

Plot the point on the polar coordinate system.
$(5,210°)$

• A.
• B.
• C.
• D.

Answer: (A)

Plot the point on the polar coordinate system.
$(-3,-90°)$

• A.
• B.
• C.
• D.

Answer: (C)

Plot the point on the polar coordinate system.
$(6,-\frac{11\pi}{6})$

• A.
• B.
• C.
• D.

Answer: (A)

Plot the point on the polar coordinate system.
$(-2,\frac{2\pi}{3})$

• A.
• B.
• C.
• D.

Answer: (D)

Determining Different Coordinates for the Same Point

In the study of polar coordinates, it's essential to understand that a single point can be represented by multiple ordered pairs. This concept is similar to coterminal angles in the unit circle, where angles like \( \frac{\pi}{6} \) and \( \frac{13\pi}{6} \) point to the same location. In polar coordinates, a point is defined by an ordered pair \( (r, \theta) \), where \( r \) is the radius and \( \theta \) is the angle. To find different representations of the same point, we can apply the same principles used for coterminal angles.

For any point represented as \( (r, \theta) \), we can generate additional ordered pairs by adding or subtracting multiples of \( 2\pi \) to the angle \( \theta \). This can be expressed mathematically as:

\( (r, \theta) \) is coterminal with \( (r, \theta + 2\pi n) \) for any integer \( n \).

Additionally, if we want to change the sign of \( r \) (making it negative), we must adjust the angle \( \theta \) by adding \( \pi \) to it. This ensures that the point remains in the same location. The relationship can be summarized as:

\( (r, \theta) \) is equivalent to \( (-r, \theta + \pi) \).

For example, starting with the point \( (4, \frac{\pi}{6}) \), we can find another representation by keeping \( r \) the same and adjusting \( \theta \). If we subtract \( 2\pi \) from \( \frac{\pi}{6} \), we get:

\( (4, \frac{\pi}{6} - 2\pi) = (4, -\frac{11\pi}{6}) \).

To find a representation with a negative \( r \), we set \( r = -4 \) and keep \( \theta \) within the interval \( [0, 2\pi] \). The original angle \( \frac{\pi}{6} \) is already in this range, but we need to adjust it for the negative radius:

\( (-4, \frac{\pi}{6} + \pi) = (-4, \frac{7\pi}{6}) \).

When working with polar coordinates, it's crucial to remember these transformations to accurately find multiple ordered pairs that correspond to the same point. This understanding not only enhances your grasp of polar coordinates but also strengthens your overall mathematical skills in dealing with angles and their representations.

Plot the point $(3,\frac{\pi}{2})$ & find another set of coordinates, $(r,θ)$, for this point, where:
(A) $r≥0,2π≤θ≤4π$,
(B) $r≥0,-2π≤θ≤0$,
(C) $r≤0,0≤θ≤2π$.

• A. $(3,\frac{5\pi}{2}),(-3,-\frac{3\pi}{2}),(-3,\frac{3\pi}{2})$
• B. $(3,\frac{5\pi}{2}),(3,-\frac{3\pi}{2}),(-3,\frac{3\pi}{2})$
• C. $(-3,\frac{5\pi}{2}),(-3,-\frac{3\pi}{2}),(-3,\frac{\pi}{2})$
• D. $(3,\frac{5\pi}{2}),(3,-\frac{3\pi}{2}),(-3,\frac{\pi}{2})$

Answer: (B)

Determining Different Coordinates for the Same Point Example 2

In this exercise, we explore the concept of polar coordinates, specifically focusing on the point represented by the coordinates \((-2, \frac{5\pi}{3})\). In polar coordinates, a point is defined by a radius \(r\) and an angle \(\theta\). Here, the radius is negative, indicating that we will plot the point in the opposite direction of the angle.

To begin, we identify the angle \(\theta = \frac{5\pi}{3}\). This angle corresponds to a position on the polar graph. However, since the radius \(r = -2\) is negative, we move in the direction opposite to \(\frac{5\pi}{3}\). By counting 2 units in that direction, we find the point \((-2, \frac{5\pi}{3})\) on the graph.

Next, we can generate additional sets of coordinates for the same point by manipulating the angle while keeping the radius consistent. One method is to add multiples of \(2\pi\) to the angle. For instance, adding \(2\pi\) to \(\frac{5\pi}{3}\) gives us:

\[\frac{5\pi}{3} + 2\pi = \frac{5\pi}{3} + \frac{6\pi}{3} = \frac{11\pi}{3}\]

This results in the coordinates \((-2, \frac{11\pi}{3})\), which represents the same point on the graph. Continuing this process, we can add another \(2\pi\) to \(\frac{11\pi}{3}\):

\[\frac{11\pi}{3} + 2\pi = \frac{11\pi}{3} + \frac{6\pi}{3} = \frac{17\pi}{3}\]

Thus, we obtain another set of coordinates: \((-2, \frac{17\pi}{3})\).

To explore further, we can change the radius \(r\) from negative to positive. By negating the radius, we have \(r = 2\). To maintain the same point, we must adjust the angle by adding \(\pi\) to the original angle:

\[\frac{5\pi}{3} + \pi = \frac{5\pi}{3} + \frac{3\pi}{3} = \frac{8\pi}{3}\]

This gives us the coordinates \((2, \frac{8\pi}{3})\), which also represents the same point on the graph.

In summary, we have derived three sets of coordinates for the same point: \((-2, \frac{5\pi}{3})\), \((-2, \frac{11\pi}{3})\), and \((-2, \frac{17\pi}{3})\), along with \((2, \frac{8\pi}{3})\). This illustrates the concept that a single point in polar coordinates can be represented by infinitely many coordinate pairs, depending on the manipulation of the angle and radius.

Plot the point $(5,-\frac{\pi}{3})$, then identify which of the following sets of coordinates is the same point.

• A. $(-5,-\frac{\pi}{3})$
• B. $(-5,\frac{\pi}{3})$
• C. $(-5,\frac{2\pi}{3})$
• D. $(-5,\frac{5\pi}{3})$

Answer: (C)

Plot the point $(-3,-\frac{\pi}{6})$, then identify which of the following sets of coordinates is the same point.

• A. $(-3,\frac{11\pi}{6})$
• B. $(-3,\frac{5\pi}{6})$
• C. $\left(3,\frac{11\pi}{6}\right)$
• D. $(3,\frac{\pi}{6})$

Answer: (A)

Here’s what students ask on this topic:

What are polar coordinates and how do they differ from Cartesian coordinates?

Polar coordinates represent points using a distance from a reference point (the pole) and an angle from a reference direction (the polar axis). They are written as (r, θ), where r is the radius and θ is the angle. In contrast, Cartesian coordinates use perpendicular axes (x and y) to define points as (x, y). The key difference is that polar coordinates use a radial distance and angle, while Cartesian coordinates use horizontal and vertical distances.

How do you plot a point in the polar coordinate system?

To plot a point in the polar coordinate system, follow these steps: First, locate the angle θ on the polar grid, measured counterclockwise from the polar axis. Next, from the pole (origin), move r units along the direction of θ. If r is positive, move outward; if r is negative, move in the opposite direction. For example, to plot (4, π/3), find π/3 on the grid and move 4 units outward from the pole.

How can multiple ordered pairs represent the same point in polar coordinates?

In polar coordinates, a single point can have multiple representations. This is achieved by adding or subtracting multiples of 2π to the angle θ, or by changing the sign of r and adding π to θ. For example, the point (4, π/6) can also be represented as (4, 13π/6) or (-4, 7π/6). These transformations leverage coterminal angles and reflections to map back to the same point.

What is the significance of negative r values in polar coordinates?

Negative r values in polar coordinates indicate that the point is located in the direction opposite to the angle θ. To plot a point with a negative r, first locate the angle θ, then move |r| units in the direction opposite to θ. For example, to plot (-3, π/6), find π/6 on the grid and move 3 units in the opposite direction. This effectively reflects the point over the pole.

How do you convert Cartesian coordinates to polar coordinates?

To convert Cartesian coordinates (x, y) to polar coordinates (r, θ), use the following formulas: r = √(x² + y²) and θ = arctan(y/x). The radius r is the distance from the origin to the point, and θ is the angle formed with the positive x-axis. Ensure to adjust θ based on the quadrant of the point. For example, the Cartesian point (3, 4) converts to polar coordinates as (5, arctan(4/3)).