Question

Give two alternative representations of the points in polar coordinates. $$\left(3, \frac{2 \pi}{3}\right)$$

Short answer

Question: Provide two alternative representations for the point (3, 2π/3) in polar coordinates. Answer: Two alternative representations for the point (3, 2π/3) in polar coordinates are (3, 8π/3) and (-3, -π/3).

Step-by-step solution

Recall polar representation formula

A point \((x, y)\) in Cartesian coordinates is represented as \((r, \theta)\) in polar coordinates where \(r\) is the distance of the point from the origin and \(\theta\) is the angle measured counterclockwise from the positive x-axis. Note that we can change the value of \(\theta\) by adding or subtracting multiples of \(2\pi\) without changing the location of the point.

Find alternative angle representations

Given the point \((3, \frac{2\pi}{3})\), we can add or subtract multiples of \(2\pi\) to the angle: Add \(2\pi\) to the angle: $$\left(3, \frac{2\pi}{3} + 2\pi\right) = \left(3, \frac{2\pi}{3} + \frac{6\pi}{3}\right) = \left(3, \frac{8\pi}{3}\right)$$ Subtract \(2\pi\) from the angle: $$\left(3, \frac{2\pi}{3} - 2\pi\right) = \left(3, \frac{2\pi}{3} - \frac{6\pi}{3}\right) = \left(3, -\frac{4\pi}{3}\right)$$

Alternative representations with negative radius

Now, we can represent the point with a negative radius and the angle shifted by \(\pi\) radians, using one of the alternative angles we've just found: Alternative angle \(\frac{8\pi}{3}\), add \(\pi\) radians: $$\left(-3, \frac{8\pi}{3} + \pi\right) = \left(-3, \frac{8\pi}{3} + \frac{3\pi}{3}\right) = \left(-3, \frac{11\pi}{3}\right)$$ Alternative angle \(-\frac{4\pi}{3}\), add \(\pi\) radians: $$\left(-3, -\frac{4\pi}{3} + \pi\right) = \left(-3, -\frac{4\pi}{3} + \frac{3\pi}{3}\right) = \left(-3, -\frac{\pi}{3}\right)$$

Final alternative representations

Considering all the alternatives, we have found two valid alternative representations for point \((3, \frac{2\pi}{3})\) in polar coordinates: - \((3, \frac{8\pi}{3})\) - \((-3, -\frac{\pi}{3})\)

Key concepts

Angle Representation

In the realm of polar coordinates, angles play a pivotal role in defining the exact location of a point. Originally represented as \(\theta\), the angle \(\theta\) in a set of polar coordinates specifies how far the point is rotated around a central point, typically the origin.

To adjust a polar coordinate \(\theta\), you can easily add or subtract multiples of \(2\pi\) (360 degrees). This method stems from the cyclical nature of circular motion around a point:

• Adding \(2\pi\) rotates the angle a full circle back to the original position.
• Subtracting \(2\pi\) achieves the same, in the opposite direction.

This means a point at an angle of \(\frac{2\pi}{3}\) remains unchanged at \(\frac{8\pi}{3}\) or \(-\frac{4\pi}{3}\), because they all point in the same direction from the origin.

Radial Distance

Radial distance in polar coordinates is like a spoke in a wheel. It tells you how far away from the origin a point is. Represented by \(r\), radial distance can be either positive or negative:

• Positive radial distance: Points outward along the direction given by the angle, \(\theta\).
• Negative radial distance: Points in the opposite direction to \(\theta\), essentially flipping the orientation.

This flipping mechanism allows us to express the same location by shifting the angle by \(\pi\) (180 degrees) when using a negative radial distance, which is effectively the opposite direction. For example, the point \( (3,\frac{2\pi}{3}) \) can be alternatively represented as \( (-3, \frac{11\pi}{3}) \). Here, the negative radial distance requires that the angle is adjusted by adding \(\pi\). This dual representation showcases the flexibility inherent in polar coordinates.

Cartesian Coordinates

To fully appreciate polar coordinates, it’s helpful to understand their relationship with Cartesian coordinates. Cartesian coordinates \( (x,y) \) describe points based on how far along the x and y axes they are. In contrast, polar coordinates express locations based on distance from the origin and angle of inclination.

The transformation between these two systems helps offer multiple representations of the same point. For a given polar coordinate \( (r, \theta) \):

• To find x-coordinate: \( x = r \cos(\theta) \)
• To find y-coordinate: \( y = r \sin(\theta) \)

Conversely, to convert back to polar coordinates from Cartesian coordinates:

• Radial distance, \( r = \sqrt{x^2 + y^2} \)
• Angle, \( \theta = \tan^{-1}(\frac{y}{x}) \)

This conversion enables exploring the same point on the plane from different perspectives and understanding how these two systems interchangeably lay out points, providing flexibility for interpreting spatial relationships.