Problem 13 - 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 \(-2\pi<\theta<2\pi\). \(\left(0, -\dfrac{7\pi}{6}\right)\)

Step-by-Step Solution

Answer

The point \((0, -\dfrac{7\pi}{6})\) lies at the origin. Two additional polar representations within \(-2\pi<\theta<2\pi\), for the point will be \((0, \dfrac{5\pi}{6})\) and \((0, -\dfrac{19\pi}{6})\).

Step 1: Understanding Polar Coordinates

Polar coordinates provide a method of rendering graphics in terms of a distance and an angle. In polar coordinates, a point in the plane is represented by \((r, \theta)\) where \(r\) is the distance of the point from the origin and \(\theta\) is the angle from the positive x-axis measured counterclockwise. A polar coordinate with \(r=0\) is at the origin regardless of the value of \(\theta\). This indicates that the given point \((0, -7\pi/6)\) resides at the origin or center of the graph.

Step 2: Identifying the Polar Coordinates

The point is plotted at the origin of the graph. The value of \(\theta = -\dfrac{7\pi}{6}\) does not affect the location of the point because \(r=0\).

Step 3: Finding Additional Representations

Extra representations for any point \((r, \theta)\) can be discovered by either changing \(r\) to \(-r\) and adding \(\pi\) to \(\theta\), or keeping \(r\) and \(\theta\) as they are and adding any multiple of \(2\pi\) to \(\theta\). Here, since \(r=0\), additional polar coordinates for the point may only be determined by adding any multiple of \(2\pi\) to \(\theta\). These would be \((0, -7\pi/6 + 2\pi) = (0, 5\pi/6)\) and \((0, -7\pi/6 - 2\pi) = (0, -19\pi/6)\) within \(-2\pi<\theta < 2\pi\) .

Key Concepts

Plotting Polar Points

To plot points in the polar coordinate system, first understand that every point is defined by a pair, \( (r, \theta) \), where \(r\) denotes the radial distance from the origin (the pole), and \(\theta\) represents an angle measured from the positive x-axis (the polar axis) counterclockwise. When \(r = 0\), the point is at the origin, and \(\theta\) becomes irrelevant as it doesn't change the point's location. Picture a circle's radius extending out from the origin; plotting a point requires rotating this radius by \(\theta\) degrees and then marking your point \(r\) units along this line.

For the exercise at hand, the point given in polar coordinates is \( (0, -\frac{7\pi}{6}) \) which means we must start at the origin. Since \(r = 0\), the angle \(\theta\) has no effect, placing our point directly at the pole. To help visualize this process and further solidify understanding, using graph paper or software that can simulate the polar coordinate system is an effective study method. Then drawing a circle with a radius \(r\), and marking the point on the circumference at the angle \(\theta\) teaches the plotting process hands-on.

Polar Representations

In polar coordinates, there can be multiple representations for a single point, unlike the unique \( (x, y) \) notation in Cartesian coordinates. This multiplicity arises because adding \(2\pi\) radians (or 360 degrees) to the angle \(\theta\) loops us back to our initial position.

Considering the point with \(r = 0\), changing the angle doesn't affect the point's location since it's at the origin. But for a non-zero \(r\), we have two principal ways to find additional representations:

• Keep \(r\) and add multiples of \(2\pi\) to \(\theta\) to loop around the origin.
• Change \(r\) to \( -r \) and add \(\pi\) to \(\theta\), which effectively rotates the point to the opposite side of the circle.

For example, additional polar representations of \( (0, -\frac{7\pi}{6}) \) are \( (0, 5\pi/6) \) and \( (0, -19\pi/6) \) within the range \( -2\pi < \theta < 2\pi \). To solidify this concept, students can experiment with varying \(\theta\) values while plotting to see how these changes impact the position of the point on the graph.

Polar Coordinate System

The polar coordinate system is an alternative to the Cartesian (or rectangular) coordinate system. It's particularly useful for dealing with problems involving curves and angles, like those in trigonometry and complex numbers. Unlike the straight x-y grid, the polar system consists of concentric circles (representing different radii) and rays emanating from a central point (the pole) at various angles (representing different angles \(\theta\)).

This system is essential for students to grasp as it offers a unique way to describe the position of a point that can be more intuitive in certain contexts like circular movement, wave forms, and fields in physics. To help remember this system, one could think of the polar coordinates as instructions on how to get to a point by first turning to face the correct direction and then walking a straight line to the destination, akin to a treasure hunt map with steps like 'Walk five paces to the north.'