Problem 34 - Precalculus : Building Concepts and Connections

Question

For each of the points given in polar coordinates, find two additional pairs of polar coordinates \((r, \theta),\) one with \(r>0\) and one with \(r<0\). $$\left(-3, \frac{\pi}{2}\right)$$

Step-by-Step Solution

Answer

Two additional pairs of polar coordinates for the point \((-3, \pi/2)\) that can represent the same point are \((3, 5\pi/2)\) or \((3, -3\pi/2)\) with positive radius and \((-3, 5\pi/2)\) or \((-3, -3\pi/2)\) with negative radius.

Step 1: Understand the polar coordinate system

In the polar coordinate system, each point is determined by the distance from the origin \(r\) (radius), and the angle \(\theta\) from the positive x-axis. The point is in standard position if the radius is greater than or equal to 0 and the angle \(\theta\) is between 0 and \(2\pi\). However, different combinations of \(r\) and \(\theta\) can represent the same point. If the radius is negative, the point is on the terminal side of the angle \(\theta\) but \(180^\circ\) or \(\pi\) radians away.

Step 2: Determine the polar coordinates with positive radius

The point \(-3, \frac{\pi}{2}\) is equivalent to the point \(3, \frac{5\pi}{2}\) or \(3, -\frac{3\pi}{2}\) (adding \(\pi\) i.e. \(180^\circ\) to the angle). Here the radius is positive.

Step 3: Determine the polar coordinates with negative radius

The point \(-3, \frac{\pi}{2}\) is equivalent to the point \(-3, \frac{5\pi}{2}\) or \(-3, -\frac{3\pi}{2}\) (adding \(2\pi\) i.e. \(360^\circ\) to the angle). Here, the radius remains negative.

Key Concepts

Angle

When working with polar coordinates, the angle \(\theta\) is a crucial component as it determines the direction of the point from the origin. The angle is measured from the positive x-axis in a counter-clockwise direction. This angle can be expressed in radians or degrees. It's important to remember that angles are periodic, which means if you keep adding full circle rotations (\(2\pi\) radians or 360 degrees), you reach the same direction again.

• An angle of \(\frac{\pi}{2}\) means the point is located directly upwards along the y-axis.
• An angle of \(\frac{5\pi}{2}\) is just \(\frac{\pi}{2}\) radians plus a full rotation (\(2\pi\) radians).

This property of angles helps us find different representations of the same point in polar coordinates by adding or subtracting these rotations.

Negative Radius

In polar coordinates, the radius \(r\) indicates how far a point is from the origin. A negative radius may seem unusual at first, but it has a simple geometric interpretation. When \(r\) is negative, the point is reflected through the origin to the opposite side of its original position.

• Given \((-3, \frac{\pi}{2})\), this means the point lies opposite to where it would with a positive radius along the line defined by \(\theta = \frac{\pi}{2}\).
• To graphically represent this, it is equivalent to taking a point that would lie on \(r = 3\) with \(\theta = \frac{\pi}{2}\), and flipping it to \(-3\) on the same angular path.

To obtain this negative radius, you can alter the angle by adding \(2\pi\) (or equivalently subtracting \(2\pi\)), which places the negative point back into a recognizable form in polar coordinates.

Positive Radius

When \(r\) is positive, finding a new version of a polar coordinate point is straightforward. The positive radius means you are moving along the angle \(\theta\) directly outward from the origin. To find equivalent polar coordinates, you can add or subtract any multiple of \(2\pi\) to the angle but maintain a positive \(r\).

• For example, the point \(3, \frac{5\pi}{2}\) is equivalent to \(3, \frac{\pi}{2}\) because \(\frac{5\pi}{2}\) is \(2\pi\) plus \(\frac{\pi}{2}\).
• This is like starting at the same angle and moving to the same position after one complete circle.

Therefore, keeping \(r\) positive is typically simpler when you rearrange or reframe polar coordinates, because it keeps directions straightforward without reflections or shifts across the origin.