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Chapter 6: Problem 4

Plot the point on a polar coordinate system. $$\left(4, \frac{7 \pi}{6}\right)$$

Short Answer

Expert verified
The point with polar coordinates \((4, \frac{7 \pi}{6})\) can be plotted in a polar coordinate system by first moving out 4 units along the positive x-axis (the radius), then turning to form an angle slightly more than \(\pi\) (or 180掳) from there in the counter-clockwise direction (which is \(\frac{7 \pi}{6}\)).

Step by step solution

01

Understand the polar coordinates given

Here, the polar coordinates are \((4, \frac{7 \pi}{6})\). Here, \(r = 4\) is the radius or the distance from the origin. \(\theta = \frac{7 \pi}{6}\) represents the angle measured from the positive x-axis. We will use these two values to plot the point.
02

Plot the radius

Start from the origin of the graph and move out 4 units along the positive x-axis. This represents the radius 'r' in the polar coordinates.
03

Plot the angle

From the point plotted at radius 4, we need to represent the angle \(\frac{7 \pi}{6}\) which is slightly more than \(\pi\). Remember that \(\pi\) represents half a rotation around the origin. Therefore, \(\frac{7 \pi}{6}\) represents a bit more than half a rotation from the positive x-axis in the counter-clockwise direction. Thus, draw an angle slightly larger than \(\pi\) or 180掳 in the counter-clockwise direction.
04

Plot the point

Now, from the origin, move along the angle \(\frac{7 \pi}{6}\) until you reach 4 units of distance (which is our radius r). The point where you stop is the location of the point in polar coordinates.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radius
The concept of the radius in polar coordinates is essential. The radius, denoted as \( r \), represents the distance from the origin to the point on the polar plane. Unlike Cartesian coordinates, where points are described using horizontal and vertical distances, polar coordinates rely on the radius and angle. In the given problem, the radius \( r = 4 \) tells us precisely how far we have to move from the origin before considering the angle. This establishes the fundamental piece of where the point lies in the polar system.
Angle Measurement
Angle measurement in polar coordinates is crucial to locating a point. Angles are usually described in terms of radians rather than degrees. The angle \( \theta \) is measured from the positive x-axis in a counter-clockwise direction. In the example given, \( \theta = \frac{7\pi}{6} \) is our angle. This angle informs us about the direction in which we need to move from the initial position along the x-axis after having moved the specified radius. It鈥檚 the angular position that, along with the radius, positions the point in the polar coordinate system.
Plotting Points
Plotting points using polar coordinates involves understanding both the radius and the angle. To plot the point \( (4, \frac{7\pi}{6}) \), start by drawing a straight line from the origin along the positive x-axis to represent the radius of 4. This step establishes the distance component. Then, apply the angle \( \frac{7\pi}{6} \).
  • First draw the initial line to the radius of 4.
  • Second, rotate it by the angle \( \frac{7\pi}{6} \).
This point will be located in the third quadrant of the polar grid, as \( \frac{7\pi}{6} \) is slightly more than half a full circle, which places it past the negative x-axis.
Radians
Radians are the standard unit for measuring angles in mathematics, particularly within polar coordinates. A full circle is \( 2\pi \) radians, and half a circle is \( \pi \) radians. This scales intuitively, as \( \pi \) radians equals 180 degrees, and \( 2\pi \) radians is the full 360 degrees rotation.
  • For the given angle \( \frac{7\pi}{6} \), we know it's more than \( \pi \), or 180 degrees.
  • Thus, it is slightly past the midpoint of a full circle.
This helps in understanding where the point lies on the polar graph, giving clear guidance on its angular position relative to the angle measurements commonly seen in circles.

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