Question

Plot the points with the given polar coordinates. (a) \(A=P(2,3 \pi)\) (b) \(B=P(1,-\pi)\) (c) \(C=P(1,2)\) (d) \(D=P(1 / 2,5 \pi / 6)\)

Short answer

Plot all points using their polar coordinates on a polar grid.

Step-by-step solution

Understanding Polar Coordinates

Polar coordinates are represented as \((r, \theta)\), where \(r\) is the radial distance from the origin and \(\theta\) is the angle measured from the positive x-axis.

Plot Point A

For point \(A = P(2, 3\pi)\), the radial distance is 2, and the angle is \(3\pi\). Normally, \(\pi\) represents 180 degrees, so \(3\pi\) is a full rotation (360 degrees) plus an additional half rotation (180 degrees), ending on the left side of the horizontal axis. Since the radial distance is positive, plot the point two units in the direction of \(3\pi\).

Plot Point B

For point \(B = P(1, -\pi)\), the radial distance is 1, and the angle is \(-\pi\), which corresponds to 180 degrees in the clockwise direction, bringing us to the left side of the x-axis. Plot the point one unit from the origin towards the left.

Plot Point C

For point \(C = P(1, 2)\), the radial distance is 1, and the angle is 2 radians. Since 2 radians is approximately 114.6 degrees (using the conversion factor \(\frac{180}{\pi}\)), plot the point one unit from the origin in this direction.

Plot Point D

For point \(D = P\left(\frac{1}{2}, \frac{5\pi}{6}\right)\), the radial distance is \(\frac{1}{2}\), and the angle is \(\frac{5\pi}{6}\). This is equivalent to 150 degrees, which is in the second quadrant. Plot the point half a unit from the origin in this direction.

Key concepts

Radial Distance

In the realm of polar coordinates, "radial distance" refers to how far a point is from the origin, also known as the pole. This is denoted by the variable \(r\). If you imagine a point starting at the origin, it will move outward in a straight line to a distance represented by \(r\).

• A positive radial distance indicates that the point is situated in the direction dictated by the angle \(\theta\).
• A negative radial distance reverses the point's direction, positioning it opposite to \(\theta\).
• For example, with point \( A = P(2, 3\pi) \), the radial distance is 2, which means the point is 2 units from the origin.

Understanding radial distance is crucial for accurately placing points in a coordinate system and is the bedrock of plotting polar coordinates.

Angle Measurement

Angle measurement in polar coordinates describes the direction of a point from the positive x-axis. This angle, represented by \(\theta\), determines how far to rotate from this horizontal line before moving a distance \(r\).

• The angle is typically expressed in radians, where \( 2\pi \) radians equate to a full circle or 360 degrees.
• Angles can be positive, indicating a counterclockwise rotation, or negative, denoting a clockwise path.
• For instance, in point \(B = P(1, -\pi)\), the angle \(-\pi\) suggests a rotation of 180 degrees clockwise, landing directly left of the origin.

Mastering angle measurement helps in predicting exactly where a point will land around the circle, indispensable for plotting accurately.

Plotting Points

To plot points using polar coordinates, combine the radial distance and angle measurement to pinpoint their exact location. It's like drawing a line from the origin and then moving straight to where the point resides.

• First, determine the angle \(\theta\) on a circle centered at the origin.
• Next, trace along this angle until reaching the radial distance \(r\).
• Each point's plot reflects both a direction and a distance, such as point \(C = P(1, 2)\), placed one unit from the origin at an angle of 2 radians, or about 114.6 degrees.

Every plotted point creates a coordinate system, enhancing comprehension of spatial relationships in polar form.

Full Rotation and Half Rotation

Understanding full and half rotations is pivotal in navigating the polar coordinate system. A full rotation occurs when a point makes a complete circle, turning 360 degrees or \(2\pi\) radians.

• Half rotation is a 180-degree turn, equivalent to \(\pi\) radians, flipping the point to the opposite side of the circle.
• Points such as \(A = P(2, 3\pi)\) illustrate this, where \(3\pi\) encompasses a full \(2\pi\) rotation plus an extra \(\pi\), ultimately positioning the point opposite the start.
• Grasping these rotations aids in determining if a point moves beyond initial surroundings or remains within a quadrant.

Visualizing rotations helps maintain orientation in the polar plane and addresses how various angles alter a point's position.