Question

Plot each of the following points on the polar plane. a. \(\left(2, \frac{\pi}{4}\right)\) b. \(\left(-3, \frac{2 \pi}{3}\right)\) c. \(\left(4, \frac{5 \pi}{4}\right)\)

Short answer

Plot points based on their angle and radius, adjusting for negative radii.

Step-by-step solution

Understanding Polar Coordinates

Polar coordinates are given in the form \((r, \theta)\), where \(r\) is the distance from the origin, and \(\theta\) is the angle from the positive x-axis. When \(r\) is negative, the point is plotted in the opposite direction of the angle.

Plot Point (2, \(\frac{\pi}{4}\))

Locate \(\frac{\pi}{4}\) on the polar plane, which is equivalent to 45 degrees. From the origin, measure a distance of 2 units along this angle direction, and mark the point where it ends.

Plot Point (-3, \(\frac{2\pi}{3}\))

Locate \(\frac{2\pi}{3}\) on the polar plane, which corresponds to 120 degrees. Measure 3 units in the opposite direction of this angle because the radius is negative. Mark the point.

Plot Point (4, \(\frac{5\pi}{4}\))

Find \(\frac{5\pi}{4}\) on the polar plane, equivalent to 225 degrees. From the origin, move 4 units along this direction and place the point.

Key concepts

Plotting Points in Polar Coordinates

In polar coordinates, every point is described using a pair \((r, \theta)\), where \(r\) indicates the radius or distance from the origin, and \(\theta\) signifies the angle with respect to the positive x-axis. This is different from Cartesian coordinates that use \((x, y)\) to define a position.
Plotting points in polar coordinates is like using a combination of a compass and a protractor.

• First, determine the angle \(\theta\) on your polar plane, which looks like concentric circles with lines radiating outwards, similar to a pie chart.
• After identifying the angle, measure the distance \(r\) from the origin along the direction of the angle.
• Marking the position depends on whether \(r\) is positive or negative. A negative \(r\) means moving backwards from the angle direction, while a positive \(r\) follows the direction of the angle.

This method allows for an intuitive way to plot positions that might feel like fitting pieces into a circular puzzle.

Understanding the Polar Plane

The polar plane is a different way to present a grid compared to the traditional Cartesian plane. It consists of concentric circles radiating from a center point, known as the pole, and radial lines emanating like spokes on a wheel.

• The pole acts like the origin in a Cartesian plane. It is the center from which all measurements are made.
• Each circle on the polar plane represents a constant distance \(r\) from the pole. These circles help visualize the radius component of polar coordinates.
• The radial lines, often at specific angles, help visualize the angular component \(\theta\).

When visualizing points on a polar plane, it's as if you're graphing on a circular grid. This system is especially helpful in problems involving circular or spiral patterns.

Angle Measurement in Polar Coordinates

Angles in polar coordinates are measured from the positive x-axis, which serves as the reference direction. The standard unit used is radians, though degrees can often provide more intuitive comprehension.

• One full rotation around a circle is \(2\pi\) radians or 360 degrees.
• The angle \(\theta\) dictates the direction in which you plot your point starting from the positive x-axis. For example, \(\frac{\pi}{4}\) or 45 degrees starts at the x-axis and moves into the first quadrant.
• Negative angles move clockwise, opposite the direction of positive angles.

Understanding angle measurement is essential since it directs how points are plotted and influences the radial symmetry found in polar coordinate systems. This contrasts with the Cartesian system, where direction is dictated by changes in the \(x\) or \(y\) values.