Question

Plot each point given in polar coordinates, and find other polar coordinates \((r, \theta)\) of the point for which: (a) \(r>0, \quad-2 \pi \leq \theta<0\) (b) \(r<0, \quad 0 \leq \theta<2 \pi\) (c) \(r>0, \quad 2 \pi \leq \theta<4 \pi\) $$ (2, \pi) $$

Short answer

(2, -\pi), (-2, 0), (2, 3 \pi)

Step-by-step solution

Understand the Given Polar Coordinate

The given polar coordinate is \( (2, \pi) \). This means that the point is at a radius of 2 units from the origin and at an angle of \( \pi \) radians from the positive x-axis.

Plot the Given Point

On the polar coordinate system, locate the angle \( \pi \). From the origin, move 2 units outward along this angle. Mark this point.

Find Coordinates for Part (a)

For part (a), we need to find another set of coordinates where \((r > 0, -2\pi \leq \theta < 0)\)\. \( \pi \) is equivalent to \(-\pi \) in the range \([-2 \pi, 0)\)\. So, \( (2, -\pi) \) is another coordinate for the point.

Find Coordinates for Part (b)

For part (b), we need to find coordinates where \((r < 0, 0 \leq \theta < 2 \pi)\)\. The negative radius means we go the opposite direction. \((2, \pi)\) can be written as \((-2, 0)\) because an angle of \( \pi \) makes \(-2\) point in the opposite direction\.

Find Coordinates for Part (c)

For part (c), we need coordinates where \((r > 0, 2 \pi \leq \theta < 4 \pi)\)\. Angle \pi\ gets repeated every \(2 \pi \) period. Therefore, \pi\ can be represented as \(3 \pi \). Hence, \( (2, 3 \pi) \) is another coordinate for the point.

Key concepts

Coordinate Transformation

To understand different coordinate systems, you must first become familiar with transforming coordinates. In this exercise, we are transforming points from the rectangular coordinate system (x, y) to the polar coordinate system (r, θ). In rectangular coordinates, points are defined in relation to the x and y axes. In polar coordinates, points are defined by their distance from the origin (radius, r) and the angle (θ) they make with the positive x-axis. Transforming coordinates helps visualize and solve problems more efficiently, particularly when the problem has rotational symmetry.

Polar Plot

A polar plot represents data points in the polar coordinate system, where each point is given by a radius (r) and angle (θ). To plot a point: first identify the angle, then move outward from the origin for the specified radius. In this exercise, the point (2, π) means a radius of 2 units at an angle of π radians (180 degrees).
With the polar plot, you can easily represent points with the same radius but different angles, emphasizing their symmetry around the origin.

Radian Measurement

Radians are a unit of angle measurement commonly used in mathematics. Unlike degrees, which divide a circle into 360 parts, radians divide a circle based on π (pi). One full circle is 2π radians, meaning π radians is half a circle (180 degrees).
Understanding radians and their relationship with π is crucial for problems like this one. Here, knowing π radians equals 180 degrees helps you place the point correctly on the polar plot and convert angles effectively.

Angle Conversion

Converting and finding equivalent angles is key to working with polar coordinates. Angles can often be represented in different ways but still specify the same direction. For instance, π radians can also be described as -π radians because they point in the same direction. Similarly, angles beyond 2π (like 3π) still map onto the same point due to the periodic nature of circles.
In this exercise:

• For part (a), π equates to -π, both representing the same direction.
• For part (b), an angle of π with a negative radius converts the point (2, π) to (-2, 0) by moving in the opposite direction.
• For part (c), π + 2π is equivalent to 3π, thus (2, 3π) is a valid representation of the original point.

Negative Radius

A negative radius in polar coordinates changes the direction. Instead of moving the specified distance along the given angle, you move in the exact opposite direction. For instance, a point (2, π) can be represented as (-2, 0).
The angle theorem helps this: an angle of π essentially lets us switch from positive to negative radius to find the same location by toggling the direction. This flexibility makes polar coordinates extremely useful for various applications.