Problem 14 - Precalculus with Limits

Question

In Exercises 5-18, plot the point given in polar coordinate sand find two additional polar representations of the point, using \(-2\pi<\theta<2\pi\). \(\left(0, -\dfrac{7\pi}{2}\right)\)

Step-by-Step Solution

Answer

The two additional polar representations for the given point \((0, -7\pi /2)\) are \((0, -3\pi /2)\) and \((0, \pi /2)\).

Step 1: Plot Original Point

The given point is \((0, -7\pi /2)\). Since the radial coordinate is 0, the point is at the origin.

Step 2: First Additional Representation

The first additional polar representation can be obtained by adding \(2\pi\) to the angle. This gives us the point as \((0, -7\pi /2 + 2\pi) = (0, -3\pi /2)\). This is within the required range of \(-2\pi < \theta < 2\pi\) and therefore is valid.

Step 3: Second Additional Representation

For the second additional representation, \(2\pi\) can be added to the angle from step 2, i.e., \((0,-3\pi /2 + 2\pi) = (0, \pi /2)\). This is also within the required range of \(-2\pi < \theta < 2\pi\) and is therefore another valid representation.

Key Concepts

Polar Representations

Polar coordinates allow us to pinpoint a location in a plane using a distance and an angle. Unlike the more familiar Cartesian coordinates, which use an x and y value, polar coordinates are given as \( (r, \theta) \).

• \( r \) represents the radial coordinate, which tells us how far the point is from the origin or how large the radius is.
• \( \theta \) represents the angle measured in radians from the positive x-axis.

Whereas in Cartesian coordinates a point has only one representation, in polar coordinates a point can have multiple representations. This is because adding or subtracting full rotations \( (2\pi\text{ radians}) \) to the angle results in the same location.
In the given exercise, the point at \( (0, -\frac{7\pi}{2}) \) is at the origin, since \( r = 0 \) means no matter the angle, the point stays at the origin.
By using the flexibility of polar coordinates, we can alter the angle by \( 2\pi \) to find multiple expressions of the same point. These new angles brought within the prescribed range retain the position at the origin.

Angle Range

In the context of polar coordinates, the angle \( \theta \) determines the direction from the origin. To meet mathematic conventions and specified conditions, like in the given exercise, angles are typically required to fall within a certain range. In this exercise, the range is \(-2\pi < \theta < 2\pi \), which represents a full circle and then a bit more on either side.
This range ensures clarity and consistency when interpreting the data, allowing us to anticipate which direction an angle will swing the radial coordinate.
To convert an angle outside this range, you simply add or subtract \( 2\pi \) until it fits within the desired boundaries. It's akin to winding a clock forward or backward until reaching the target time. This way, both \(-3\pi/2 \) and \( \pi/2 \) express the original point within the specified range, offering alternate representations but verifying the same polar position.

Radial Coordinate

The radial coordinate is a key component in polar coordinates, symbolized by \( r \). It specifies the distance from the origin to the point. Here's the general idea:

• If \( r > 0 \), the point is in the direction of the angle \( \theta \).
• If \( r < 0 \), the point appears opposite to the given angle.
• When \( r = 0 \), as is the case in this exercise, the point lies directly at the origin regardless of the angle \( \theta \).

Why does this matter? It shows that polar coordinates aren't just about direction but also magnitude. When plotting, \( r \) tells us how far out from the center or origin a point lands. In this exercise, since \( r = 0 \), the angle doesn't change the point's positional reality — it's always the origin.