Problem 39 - 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 \(r<0\). $$\left(-\frac{11}{7},-5 \pi\right)$$

Step-by-Step Solution

Answer

The two additional pairs of polar coordinates for the given point are \((11/7, -4\pi)\) with \(r>0\) and \((-11/7, -3\pi)\) with \(r<0\).

Step 1: Analyze the provided Polar Coordinates

The given polar coordinates are \((-11/7, -5\pi)\). The radial distance, \(r\), is negative which means the point is in the direction opposite to that indicated by \(-5\pi\).

Step 2: Find the Polar Coordinates with positive r

To get a new set of polar coordinates for the same point with \(r>0\), the angle \(\theta\) has to be adjusted. Since the original angle is \(-5\pi\) and the points is opposite the direction of \(\theta\), by adding \(\pi\) to the angle, the new angle is \(-4\pi\). Hence, the new polar coordinates with \(r>0\) are \((11/7, -4\pi)\).

Step 3: Find the Polar Coordinates with negative r

To find a second pair of polar coordinates for the same point with \(r<0\), the angle \(\theta\) is adjusted. By adding \(2\pi\) to the original angle (-5\pi), the new angle is \(-3\pi\). Hence, the new polar coordinates with \(r<0\) are \((-11/7, -3\pi)\).

Key Concepts

Radial Distance

The concept of radial distance is fundamental in polar coordinate systems. It represents how far a point is from the origin, similar to the radius of a circle. But, unlike a circle that always has a positive radius, in polar coordinates, the radial distance, denoted as 'r', can be positive or negative. A positive 'r' indicates a point that is in the direction from the origin following the angle \theta, whereas a negative 'r' suggests the point is in the opposite direction.

In the exercise given, the point has a radial distance of \( - \frac{11}{7} \), which is negative. This implies that the point lies in the opposite direction of the angle given by \( -5 \pi \) radians. In the context of polar coordinates, the sign of 'r' can be 'flipped' by adding \( \pi \) to the angle, which effectively 'turns us around' by half a revolution, pointing us in the opposite direction, but towards the same location if we then take 'r' as positive.

Angle Adjustment in Polar Coordinates

Angles in polar coordinates define the direction from the origin to the point. When dealing with angle adjustments, we must consider that angles are periodic with period \(2\pi\), meaning that adding or subtracting multiples of \(2\pi\) results in an angle that corresponds to the same direction. This concept is essential for the exercise, where we need to find alternative polar coordinates representing the same point.

In the solution, we adjust the angle by adding \( \pi \) to find the coordinates with a positive 'r'. This is achieved by understanding that adding \( \pi \) radians (or 180 degrees) to an angle points us in the exact opposite direction. Here, the initial angle is \( -5 \pi \), and by adding \( \pi \) we obtain \( -4 \pi \). Likewise, to find another set with a negative 'r', we can add \(2\pi\) instead, thereby completing a full revolution and maintaining the same direction as the original angle, resulting in a new angle of \( -3\pi \).

Converting Polar Coordinates

Converting between polar coordinates involves changing either the angle \(\theta\) or the radial distance 'r' while maintaining the same point's location. This process aids in finding equivalent expressions for points in the polar coordinate system, which can be highly useful in various mathematical and engineering applications.

As shown in the exercise, the conversion process may include finding a positive 'r' when originally presented with a negative one, which intrinsically involves angle adjustment. After the adjustment in our given example, we get two new pairs of coordinates: \( \left( \frac{11}{7}, -4 \pi \right) \) with a positive radial distance, and \( \left( -\frac{11}{7}, -3 \pi \right) \) maintaining a negative radial distance. These conversions are important not only for exercises but also for real-world problems where different forms of the same polar coordinates are needed for analyses, simulations, or when applying boundary conditions in physical scenarios.