Problem 18 - 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 \(-2\pi<\theta<2\pi\). \(\left(-5, -2.36\right)\)

Step-by-Step Solution

Answer

The two additional polar representations for the point \((-5, -2.36)\) are \((-5, -2.36 + 2\pi)\) and \((-5, -2.36 - 2\pi)\).

Step 1: Identify and Plot the Given Polar Coordinate

The given polar coordinate is \((-5, -2.36)\), where -5 is the distance from the origin (r) and -2.36 is the angle (theta) in radians. In polar coordinates, negative distance means the point is on the opposite side of the origin along the same line. Plot this point on a polar coordinate plane.

Step 2: Calculate First Additional Representation

For the first additional representation, since a full circle is \(2\pi\) radians, adding or subtracting multiples of \(2\pi\) will result in the same position along the circle. Try adding \(2\pi\) to the angle component. The distance, however, remains the same. By doing this, observe that the new polar representation doesn't violate the given constraints and is hence valid.

Step 3: Calculate Second Additional Representation

Calculate the second representation in a similar way as the first one, but now subtract by \(2\pi\). This, again, will give same position along the circle. Check whether it falls within the constraints of the problem, which it does. This gives the second valid additional representation.

Key Concepts

Angle in Radians

When working with polar coordinates, the angle is typically measured in radians, a unit that offers a natural way to describe angles. Here's how radians work:

• In one full circle, there are \(2\pi\) radians. This is equivalent to 360 degrees.
• A positive angle in radians rotates counter-clockwise from the positive x-axis, while a negative angle rotates clockwise.
• Converting degrees to radians can be done using the formula: \( radians = degrees \times \left(\frac{\pi}{180}\right) \).

The angle in this exercise is \(-2.36\) radians, which places it clockwise from the positive x-axis. Understanding angles in radians is crucial for practicing polar coordinates, as they determine the point's position around the circle without ambiguity.

Negative Distance

The concept of negative distance in polar coordinates can be perplexing but is an interesting feature. Here's an explanation:

• In polar coordinates, the distance, usually represented as \(r\), can be negative. This means the point lies on the line extending in the opposite direction from the usual positive distance.
• Imagine reflecting the point through the origin to the opposite side; this helps visualize where a negative \(r\) places it.
• For example, a negative \(-5\) as given in the exercise implies you consider its positive counterpart in the opposite direction.

This feature adds symmetry and extends the use of polar coordinates, making them very versatile in representation.

Polar Representation

Polar representation is a unique way to locate points using a distance and an angle. Here's what you need to know:

• Polar coordinates consist of \((r, \theta)\), where \(r\) is the radial distance from the origin, and \(\theta\) is the angle in radians as previously discussed.
• An exciting feature is how several polar coordinates can represent the same point. These alternate representations arise from rotating angles by adding or subtracting multiples of \(2\pi\) without changing the actual location of the point.
• In the exercise, you derived two additional representations by adding and subtracting \(2\pi\), which broadened the understanding of representing positions on a plane.

The flexibility of polar representation helps solve complex geometry problems and is particularly useful in fields like physics and engineering.