Problem 41 - 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 \(r<0\). $$\left(1.3, \frac{3 \pi}{4}\right)$$

Step-by-Step Solution

Answer

The additional pair of polar coordinates with \(r > 0\) is: \((1.3, \frac{11\pi}{4})\). The additional pair of polar coordinates with \(r < 0\) is: \((-1.3, \frac{7\pi}{4})\)

Step 1: Determine the Additional Pair with \(r > 0\)

This can be found by just adding \(2\pi\) to the initial angle while keeping the radius the same. Thus, the new point is: \((r, \theta + 2\pi) = (1.3, \frac{3\pi}{4} + 2\pi) = (1.3, \frac{11\pi}{4})\)

Step 2: Determine the Additional Pair with \(r < 0\)

This can be found by adding \(\pi\) to the original angle to turn 180 degrees and making the radius negative. As a result, the new point is: \((-r, \theta + \pi) = (-1.3, \frac{3\pi}{4} + \pi) = (-1.3, \frac{7\pi}{4})\)

Key Concepts

Angle Addition

When working with polar coordinates, adding angles allows you to represent a point in different ways. If you have a polar point \((r, \theta)\), you can add full rotations to the angle without changing the location.

• The standard rotation is \(2\pi\) because it completes one full circle (360 degrees).
• Adding \(2\pi\) to an angle keeps the radius positive, effectively maintaining its direction.

For example, starting with \((1.3, \frac{3\pi}{4})\), if you add \(2\pi\), the point becomes \((1.3, \frac{11\pi}{4})\). This new angle represents the same point but takes the full circle into account, allowing us to identify it with a different angle.

Positive Radius

A positive radius \(r > 0\) in polar coordinates means that the point is measured outward from the origin in the direction given by \(\theta\). It's the most direct representation of a point.

• To keep the radius positive, you can adjust the angle by whole revolutions (like \(2\pi\)).
• This method ensures the radius remains positive while the angle adapts.

As seen in \((1.3, \frac{3\pi}{4})\), by keeping \(r = 1.3\) and adjusting \(\theta\) with \(2\pi\), the point \((1.3, \frac{11\pi}{4})\) preserves the positive radius.

Negative Radius

With a negative radius \(r < 0\), the point is mirrored across the origin, pointing in the opposite direction.

• To express this, you add \(\pi\) to the angle, effectively rotating the direction by 180 degrees.
• This flips the point while making the radius negative, fundamentally changing its direction.

For instance, from \((1.3, \frac{3\pi}{4})\), changing to \((-1.3, \frac{7\pi}{4})\) involves making the radius negative while adding \(\pi\) to the angle. This transformation ensures that the point is correctly represented on the opposite side of the original direction.