Problem 8 - The Calculus with Analytic Geometry

Question

Plot the point having the given set of polar coordinates; then give two other sets of polar coordinates of the same point, one with the same value of \(r\) and one with an \(r\) having opposite sign.\(\left(\sqrt{2},-\frac{1}{4} \pi\right)\)

Step-by-Step Solution

Answer

The plot is at \(\left(\sqrt{2}, -\frac{1}{4}\pi\right)\). Equivalent coordinates: \(\left(\sqrt{2}, \frac{7}{4}\pi\right)\) and \(\left(-\sqrt{2}, \frac{3}{4}\pi\right)\).

Step 1: Understand Polar Coordinates

Polar coordinates are represented as \(r, \theta\), where \(r\) is the distance from the origin and \(\theta\) is the angle measured counterclockwise from the positive x-axis.

Step 2: Plot the Given Point

Plot the point \(\left(\sqrt{2}, -\frac{1}{4}\pi\right)\). Since \(\theta = -\frac{1}{4}\pi\), this means a rotation of \(-45^{\circ}\) or \(-\frac{\pi}{4}\) radians from the positive x-axis. The distance \(r = \sqrt{2}\) means it is \sqrt{2} units away from the origin.

Step 3: Find Equivalent Polar Coordinates with the Same r

Add \2\pi\ to the angle to get another polar coordinate for the same point: \(\left(\sqrt{2}, -\frac{1}{4}\pi + 2\pi\right) \equiv\left(\sqrt{2}, \frac{7}{4}\pi\right)\). This is because an angle of \(2\pi\) leads back to the same point.

Step 4: Find Equivalent Polar Coordinates with Opposite r

Use \ r' = -r\ and adjust the angle by \pi\ to get the opposite direction. For \(r = \sqrt{2}\), \(r' = -\sqrt{2}\). The angle must be adjusted: \(\theta' = -\frac{1}{4}\pi + \pi = \frac{3}{4}\pi\). Therefore, the coordinates are \(\left(-\sqrt{2}, \frac{3}{4}\pi\right)\).

Key Concepts

plotting points

In polar coordinates, each point is defined by a distance and an angle. The distance, noted as \(r\), represents how far the point is from the origin (the center of your coordinate system).
The angle, noted as \(\theta\), indicates the direction of the point from the positive x-axis and is measured counterclockwise.
To plot a point using polar coordinates, follow these steps:

• Identify the distance \(r\). For example, if \(r = \sqrt{2}\), the point is \(\sqrt{2}\) units away from the origin.
• Identify the angle \(\theta\). For example, if \(\theta = -\frac{1}{4}\pi\), this means you rotate -45 degrees from the positive x-axis.

Now, plot the point at the found distance and angle. Imagine a circle of radius \(\sqrt{2}\) centered at the origin and locate the point where the angle intersects this circle. This is where your point \(\left(\sqrt{2}, -\frac{1}{4}\pi \right)\) resides.

equivalent polar coordinates

Polar coordinates can have multiple representations for the same point. This is due to the circular nature of angles, where a full rotation (\(2\pi\) radians or 360 degrees) brings you back to the original point.
Here's how you can find equivalent polar coordinates for a point:

• Keep \(r\) the same and adjust \(\theta\) by adding or subtracting multiples of \(2\pi\). For instance, adding \(2\pi\) to \(-\frac{1}{4}\pi\) in our example gives \(-\frac{1}{4}\pi + 2\pi = \frac{7}{4}\pi\). Thus, \( \left(\sqrt{2}, \frac{7}{4}\pi\right)\) is an equivalent point.
• Change the sign of \(r\) and adjust \(\theta\) by \(\pi\). With \(r = -\sqrt{2}\), add \(\pi\) to the angle: \( -\frac{1}{4}\pi + \pi = \frac{3}{4}\pi\). Hence, \( \left(-\sqrt{2}, \frac{3}{4}\pi\right)\) represents the same point.

These adjustments leverage the periodic nature of angles, showing that various sets of \(r\) and \(\theta\) can pinpoint the same location in polar coordinates.

angle adjustment

Adjusting angles in polar coordinates is crucial for finding equivalent points or when changing the sign of \(r\). Here's a closer look at the process:

• Adding/subtracting \(2\pi\): This technique utilizes the fact that a full circle is \(2\pi\) radians. Adding or subtracting \(2\pi\) to/from \(\theta\) means you are making a complete loop and ending at the same angle. For example, \(\theta = -\frac{1}{4}\pi\), becoming \(-\frac{1}{4}\pi + 2\pi = \frac{7}{4}\pi\).
• Shifting \(\pi\): When the radius \(r\) changes sign, the direction flips to the opposite point on the circle. Adjust the angle by \(\pi\) radians to point in the contrary direction. With \(\theta = -\frac{1}{4}\pi\), adding \(\pi\) gives \(-\frac{1}{4}\pi + \pi = \frac{3}{4}\pi\).

These adjustments are key in transforming and interpreting different sets of polar coordinates without changing the actual point they represent.