Problem 183 - Calculus Volume 3

Question

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

Step-by-Step Solution

Answer

The curve is a straight line at an angle of \( \frac{2\pi}{3} \) from the positive x-axis, passing through the origin.

Step 1: Understanding the Polar Equation

The equation given is \( \theta = \frac{2\pi}{3} \). This is a polar equation, where \( \theta \) is the angle from the positive x-axis in the polar coordinate system. It specifies a direction for all points on the curve.

Step 2: Identify the Nature of the Curve

In polar coordinates, \( \theta = \frac{2\pi}{3} \) represents a line, since \( \theta \) is fixed while the radial distance \( r \) can vary. This line passes through the origin and makes an angle of \( \frac{2\pi}{3} \) with the positive x-axis.

Step 3: Sketch the Polar Curve

To sketch, start from the origin (the pole). Draw a line that makes an angle of \( \frac{2\pi}{3} \) or 120 degrees with the positive x-axis. Since \( r \) is not specified, this line extends infinitely in both directions from the origin.

Key Concepts

Polar Equation

In the realm of mathematics, polar coordinates offer an alternative way of defining the position of points besides the usual Cartesian coordinates. Here, a polar equation is an expression involving the polar coordinates: the radius \( r \) and the angle \( \theta \). Rather than using horizontal and vertical distances to locate a point, polar equations use

• the distance from a reference point, called the origin or pole, and
• the angle from a reference direction, typically the positive x-axis.

For example, in the given equation \( \theta = \frac{2\pi}{3} \), \( \theta \) represents a fixed angle. As a polar equation, it doesn't directly provide a unique value for \( r \). Instead, it implies a set of all points that share this angle but have any radial distance from the origin.

Angle Representation

Angle representation in polar coordinates is crucial to understanding polar equations and graphing. In standard position, angles are measured from the positive x-axis. They can be expressed in radians or degrees, with radians often used in mathematical equations for precision.

• One full rotation around the origin is \( 2\pi \) radians, equivalent to 360 degrees.
• An angle of \( \frac{2\pi}{3} \) represents 120 degrees when converted from radians.

This angle representation allows us to accurately determine the direction of a line or curve. For instance, in our exercise, the angle \( \theta = \frac{2\pi}{3} \) dictates that the line lies 120 degrees counterclockwise from the positive x-axis. By consistently using angles, it's easy to correlate different positions around the pole.

Sketching Polar Curves

Sketching polar curves can initially seem challenging, but understanding the basic components simplifies the process. With the polar equation \( \theta = \frac{2\pi}{3} \), we are instructed to draw a line, but it's essential to know how to interpret and sketch it.

• Start at the pole, which is the central point from which measurements are taken.
• Locate the angle, in this case, \( \frac{2\pi}{3} \) or 120 degrees, from the positive x-axis.
• Since the radial distance \( r \) is not specified, draw the line extending infinitely in both directions from the origin.

Sketching polar curves becomes intuitive with practice. Visualizing these angles and how they relate to the x-axis helps in accurately rendering the curves as intended. By understanding the role of \( \theta \) in defining directions, you can more readily sketch different kinds of polar curves.