Problem 12 - 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 \(-2\pi<\theta<2\pi\). \(\left(-3, \dfrac{11\pi}{6}\right)\)

Step-by-Step Solution

Answer

The two additional polar representations of the point \(\left(-3, \dfrac{11\pi}{6}\right)\) are \(\left(3, \dfrac{7\pi}{6}\right)\) and \(\left(-3, -\dfrac{\pi}{6}\right)\).

Step 1: Plot the given point

Take the given point in polar coordinates \(\left(-3, \dfrac{11\pi}{6}\right)\). Since the radius is negative, we move into the opposite direction. With the angle being \(\dfrac{11\pi}{6}\), it’s just short of a full circle which is \(2\pi\) or \(\dfrac{12\pi}{6}\). So, you end up in the first quadrant but you need to move in the opposite direction due to the negative radius. Hence, the point lands in the third quadrant.

Step 2: Find first additional representation

One different representation would be to negate the radius and add \(\pi\) (or subtract \(\pi\)) to the angle. Because it flips the direction of the radius and the adding/subtracting of \(\pi\) compensates by flipping the direction of the angle. Therefore, we get the coordinates as \(\left(3, \dfrac{11\pi}{6}-\pi\right)\) or \(\left(3, \dfrac{7\pi}{6}\right)\)

Step 3: Find second additional representation

Another representation could be to keep the original radius and add/subtract \(2\pi\) to/from the angle, since this would bring us back around to the same spot on the unit circle. Therefore, the coordinates become \(\left(-3, \dfrac{11\pi}{6}-2\pi\right)\) or \(\left(-3, -\dfrac{\pi}{6}\right)\)

Key Concepts

Plotting Polar Coordinates

Understanding how to plot polar coordinates is a fundamental skill in mathematics, especially when dealing with complex numbers, trigonometry, and calculus. Unlike the rectangular coordinate system which uses horizontal and vertical axes (x and y), polar coordinates express a point based on its distance from a reference point (the pole, usually represented as the origin in a Cartesian plane) and the angle from a reference direction (usually the positive x-axis).

To plot a point given in polar coordinates, such as \( (-3, \frac{11\pi}{6}) \), you'll undertake a two-step process. First, identify the angle by rotating counter-clockwise from the positive x-axis. With \( \frac{11\pi}{6} \) as our angle, this indicates just shy of a full \( 2\pi \) rotation, near the first quadrant. Second, since the radius is negative, we move in the opposite direction from the angle's terminal side, landing in the third quadrant.

Think about the radius as an arrow: positive numbers mean 'forward' from the terminal side of the angle, and negative indicate 'backward'. This visualization helps in accurately plotting polar coordinates and ensures a clear understanding of how points are positioned in the polar coordinate system.

Polar Representation

Polar representation is a way of expressing a point or a vector in a plane using a combination of a radius and an angle, rather than x and y coordinates. This system is particularly useful when dealing with rotations and circles.

Each point in the polar coordinate system has multiple representations. For example, you can represent a point by increasing or decreasing the angle by \( 2\pi \) radians (a full circle) and the point's position remains unchanged. Similarly, changing the sign of the radius to its opposite and adjusting the angle by \( \pi \) radians (half-circle) leads to the same point in the polar plane.

For the point \( (-3, \frac{11\pi}{6} ) \), we found two additional representations: \( (3, \frac{7\pi}{6} ) \) and \( (-3, -\frac{\pi}{6} ) \). These alternative polar coordinates demonstrate the equivalence of points located symmetrically on the unit circle, maintaining the concept of direction and distance from the origin.

Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin in the polar (or Cartesian) coordinate system. It's a powerful tool for understanding trigonometric functions and their properties.

In the context of polar coordinates, every point on the unit circle can be identified by an angle, with the radius being 1. When dealing with points not on the unit circle, like \( (-3, \frac{11\pi}{6}) \), you can still use the unit circle as a reference. The negative radius in our example simply means that you move in the opposite direction across the unit circle from where you would if the radius were positive.

Knowing your way around the unit circle is crucial not just for plotting points but for understanding the sine and cosine of angles, which correspond to the x and y coordinates of points on the unit circle. Remembering key angle measures like \( \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2} \), and their multiples can aid in quickly identifying the corresponding points and their polar representations.