Problem 4 - Precalculus

Question

Plot indicated point in a polar coordinate system. $$\left(1, \frac{2 \pi}{3}\right)$$

Step-by-Step Solution

Answer

Plot a point 1 unit from the origin at an angle of \(\frac{2\pi}{3}\) radians.

Step 1: Understand Polar Coordinates

Polar coordinates are defined by a distance from the origin, called the radius, denoted by \( r \), and an angle \( \theta \), measured in radians or degrees from the positive x-axis.

Step 2: Identify the Radius

In the given polar coordinate \((1, \frac{2\pi}{3})\), the first value, 1, represents the radius \(r\). This indicates that the point is 1 unit away from the origin.

Step 3: Identify the Angle

The second value, \(\frac{2\pi}{3}\), represents the angle \(\theta\). This angle is measured counter-clockwise from the positive x-axis.

Step 4: Convert the Angle (Optional)

Although not required, you can convert \(\frac{2\pi}{3}\) radians to degrees for better visualization. Since \(\pi\) radians equal 180 degrees, \(\frac{2\pi}{3} \) is equivalent to 120 degrees.

Step 5: Plot the Point

Draw a polar coordinate system with a circle centered at the origin. Locate the angle \(\frac{2\pi}{3}\) or 120 degrees counter-clockwise from the positive x-axis. From the origin, measure outwards to a radius of 1 unit. Mark the point at this location.

Step 6: Verify the Plot (Optional)

Check that the point lies on a circle with radius 1, and that the line from the origin to the point makes an angle of \(\frac{2\pi}{3}\) with the positive x-axis.

Key Concepts

Radius in Polar Coordinates

In polar coordinates, the radius is a fundamental concept that helps in determining the position of a point in a 2D plane. The radius, often denoted by \( r \), represents the distance from the origin (the center of the coordinate system) to the point we are interested in.
In the exercise, the radius is given as 1, which means our point lies exactly one unit away from the origin. This understanding of radius differs from Cartesian coordinates where positions are defined using horizontal and vertical distances.

• The radius is always a non-negative number.
• If the radius is zero, it means the point is at the origin itself.
• The value of the radius directly influences how far the point will be marked from the center of the polar system.

Understanding the idea of the radius in polar coordinates helps in plotting points accurately and gives a clear picture of their positions relative to the origin.

Radians and Polar Coordinates

Radians are one of the primary units for measuring angles, especially in mathematics and physics. In polar coordinates, angles are typically expressed in radians rather than degrees.
This is because radians relate directly to the geometry of circles, making calculations involving circular paths more straightforward.
For the exercise, the angle given is \( \frac{2\pi}{3} \), specifying a rotational direction and magnitude from the positive x-axis.

• One full revolution around a circle is \( 2\pi \) radians, which equals 360 degrees.
• \( \pi \) radians is equivalent to 180 degrees.
• \( \frac{2\pi}{3} \) radians converts to 120 degrees, offering a clear understanding when visualizing on a circle.

By grasping how radians work within the context of polar coordinates, students can effectively plot and measure angles, enhancing their understanding of circular and rotational movements.

Understanding Angle Measurement in Polar Coordinates

Angle measurement in polar coordinates is crucial for determining the direction in which we plot the point from the origin. Unlike Cartesian coordinates, where direction is typically based on x and y values, polar coordinates use angles to indicate direction.
In our exercise, the angle provided is \( \frac{2\pi}{3} \), giving a specific line of direction that moves counter-clockwise from the positive x-axis.

• Positive angles are measured counter-clockwise.
• Negative angles are measured clockwise.
• Angles are generally measured with reference to the positive x-axis, starting from 0 radians.

This concept helps in plotting points as it allows visualization of both the distance and the precise direction of the point's location relative to the origin.