Question

Graph the polar function on the given interval. \(\theta=\pi / 6, \quad=1 \leq r \leq 2\)

Short answer

Draw a line segment from r=1 to r=2 at angle \(\theta=\frac{\pi}{6}\).

Step-by-step solution

Identify the Key Elements of the Polar Function

We are given a polar function in the form of \(r(\theta) = r\) with two parameters: \theta = \frac{\pi}{6}\ and \1 \leq r \leq 2\. This suggests that for this constant angle \theta\, we are looking to graph the radial distance \(r\) between 1 and 2.

Understand the Interval for r

The interval \(1 \leq r \leq 2\) indicates that the radial distance from the origin will start at 1 and extend to 2. This means the graph will be a line segment which originates 1 unit away from the origin and ends at 2 units.

Graph the Polar Line Segment at the Given Angle

On the polar coordinate system, locate the angle \(\theta = \frac{\pi}{6}\). This angle is 30 degrees and is measured from the positive x-axis. From this angle, draw a line segment that starts at the point (r = 1) and ends at the point (r = 2). This represents the graph of the polar function over the given interval.

Verify the Graph

Check that the line segment correctly extends from a distance of 1 to 2 units from the origin, maintaining the fixed angle of \(\frac{\pi}{6}\). The polar graph should be a straight line situated in the correct position within its radial constraint.

Key concepts

Polar Function

Polar functions are fundamental in understanding how polar coordinates work. They describe the relationship between the angle, usually denoted by \( \theta \), and the radial distance, \( r \). Unlike Cartesian coordinates, which use \( (x, y) \) points to define positions on a plane, polar functions utilize angles and distances from the origin.

In the exercise, the polar function is \( r(\theta) = r \) with a constant angle, \( \theta = \frac{\pi}{6} \). This means that the radial distance \( r \) changes within the given interval \( 1 \leq r \leq 2 \). Polar functions are particularly useful for dealing with circular and spiral shapes, as well as for problems involving directional data and cyclic patterns.

• Understand that \( \theta \) dictates the direction from the origin.
• The value of \( r \) specifies how far along that direction the point lies.

Polar functions open up a versatile world of mathematical applications, allowing for unique interpretations of spatial data.

Graphing Polar Equations

When it comes to graphing polar equations, the process involves plotting points based on angle and radial distance. This is slightly different from graphing in the Cartesian system, where we plot based on \( x \) and \( y \) coordinates.

For the provided polar function \( \theta=\frac{\pi}{6} \) where \( 1 \leq r \leq 2 \), the task is to draw a line on the polar grid.

• Start by finding \( \theta = \frac{\pi}{6} \), which is equivalent to 30° on the polar coordinate grid.
• Identify the range for \( r \) and plot points from \( r = 1 \) to \( r = 2 \) along this angle.
• Finally, draw a straight line segment connecting these points.

The graph will be a visible line against the background of circles (for radial measurements) and lines (for angles), forming the structure of the polar coordinate system.

Radial Distance

Radial distance in polar coordinates is the distance of a point from the origin, also known as the pole. In the context of polar graphs, it plays a crucial role as the primary dimension of measurement, much like "radius" in a circle. The given exercise specifies a radial distance within \( 1 \leq r \leq 2 \).

This specific instruction implies:

• The graph starts at 1 unit away from the origin.
• Ends at 2 units away, creating a definite line segment.

Radial distance is consistent and clear, facilitating the representation of symmetrical and rotational patterns that would be cumbersome to handle in Cartesian systems. It's essential to visualize radial distances accurately when graphing to ensure the graph aligns with the intended polar function parameters.

Angle Measurement in Polar Coordinates

In polar coordinates, angle measurement is essential to orient the radial distance in the correct direction. It's typically measured in radians but can also be understood in degrees. The exercise uses \( \theta = \frac{\pi}{6} \), which is equivalent to 30 degrees.

Understanding how angle measurement functions in polar graphs involve:

• Identifying the angle relative to the positive x-axis.
• Using this angle to align the radial distance correctly.

In the polar coordinate system, the angle defines the direction, and the radial distance determines how far in that direction the point is located. For our line segment graph, \( \theta = \frac{\pi}{6} \) ensures that the entire line remains oriented at a 30-degree angle, providing a spatial constant that anchors the variable radial distances.