Question

Without using technology, sketch the polar curve \(\theta=\frac{2 \pi}{3}\).

Short answer

The polar curve is a line through the origin at a 120-degree angle with the positive x-axis.

Step-by-step solution

Understanding Polar Equations

A polar equation is expressed in terms of the angle \(\theta\) and the radius \(r\). For the given exercise, the equation \(\theta=\frac{2\pi}{3}\) specifies that the angle \(\theta\) is constant, meaning it does not depend on \(r\). This differs from typical polar equations which are functions of \(r\). What we're dealing with is not a typical curve, but rather a line at a certain angle.

Visualizing the Angle

The angle \(\theta=\frac{2\pi}{3}\) is expressed in radians. Since \(\pi\) radians is equivalent to 180 degrees, \(\frac{2\pi}{3}\) radians is \(\frac{2\cdot180}{3} = 120\) degrees. This means we're looking for a line that makes a 120-degree angle with the positive x-axis.

Sketching the Polar Line

In polar coordinates, a constant \(\theta\) with any radius \(r\) draws a straight line through the origin. This line will extend indefinitely in both directions from the origin at the specified angle. In this case, from the origin, draw a line in the direction of 120 degrees, which is equivalent to moving counterclockwise from the positive x-axis. The line will pass through the origin and continue in both directions.

Draw the Curve

On polar graph paper or your sketch, start at the origin (the pole) and mark the direction corresponding to 120 degrees. To visualize this on a standard Cartesian plane, remember that 120 degrees is in the second quadrant. Draw a straight line through the origin in this direction. This line represents all points that satisfy \(\theta = \frac{2\pi}{3}\) for any radius \(r\).

Key concepts

Polar Equations

Polar equations are a fascinating way of representing curves using polar coordinates, which consist of the radius \(r\) and the angle \(\theta\). In most cases, polar equations describe a relation where \(r\) depends on \(\theta\), giving rise to various interesting shapes like spirals and roses. However, sometimes these equations may set \(\theta\) at a constant, like in our exercise with \(\theta = \frac{2\pi}{3}\).

• This particular equation doesn't determine \(r\); instead, it leads to the formation of a straight line at the specified constant angle.
• Each point on this line simply has the same angle \(\theta\) but can have different distances from the origin, represented by \(r\).

Such setups challenge our usual perception of curves as they remind us that not all relations are typical, dynamic functions of \(r\) and \(\theta\). By fixing one parameter, they simplify the scenario, guiding us directly to a geometric representation, like a ray from the origin.

Radians to Degrees Conversion

The angle \(\theta\) in polar coordinates is typically expressed in radians, but converting it to degrees can sometimes make it easier to visualize the problem, especially if you're more comfortable with the latter.

• To convert radians to degrees, we use the relation: \(\theta_{\text{degrees}} = \theta_{\text{radians}} \times \left(\frac{180}{\pi}\right)\).
• In our exercise, \(\theta = \frac{2\pi}{3}\) radians converts to: \(\frac{2\pi}{3} \times \left(\frac{180}{\pi}\right) = 120\) degrees.
• This conversion tells us that the line will be at a 120-degree angle with respect to the positive x-axis, placing it firmly in the second quadrant of the Cartesian plane, oriented counterclockwise.

Grasping this conversion is crucial as it links the conceptual and visual understanding of angles, aiding in accurately drawing them in familiar coordinate systems.

Polar Graphs

Drawing polar graphs is a unique skill that involves understanding the spatial arrangement and relationship between angle \(\theta\) and radius \(r\) on a polar plane. In the given exercise, with a fixed angle \(\theta = \frac{2\pi}{3}\), no matter the value of \(r\), the result is a line.

• Polar graph paper, which consists of circles (for \(r\)) and lines radiating outward (for \(\theta\)), provides a useful tool for sketching these lines.
• A line that cuts through the origin and crosses at a consistent angle represents the journey a point would take as \(r\) increases or decreases symmetrically about the origin.
• This line does not have a beginning or end in terms of \(r\), just like any straight line on a graph.

While Cartesian graphs heavily focus on 'x' and 'y' axes, polar graphs encourage a perspective that centers around angles and radii, a shift that opens up a new realm for visualizing mathematical equations.