Question

Plot each point in polar coordinates. $$\left(3.8,345^{\circ}\right)$$

Short answer

To plot \( (3.8,345^\circ) \) in polar coordinates, move 3.8 units from the origin and rotate 345 degrees counterclockwise from the positive x-axis.

Step-by-step solution

Understand Polar Coordinates

In polar coordinates, a point is defined by a pair \( (r, \theta) \) where \( r \) is the radius (distance from the origin) and \( \theta \) is the angle in degrees measured counterclockwise from the positive x-axis.

Identify the Radius and Angle

For the point \( (3.8,345^\circ) \) the radius \( r = 3.8 \) and the angle \( \theta = 345^\circ \) should be identified first.

Plot the Point

To plot the point, start from the origin. Move radially outwards to a distance of 3.8 units. Then, rotate 345 degrees counterclockwise from the positive x-axis. Where you end is the location of the point \( (3.8,345^\circ) \).

Key concepts

Plotting in Polar Coordinates

When using polar coordinates, the location of points is determined by two values: the radial distance from the origin and the angle relative to the positive x-axis. This system is especially useful in situations where the relationship between points is radial or circular, which often occurs in fields like physics and engineering.

To plot a point such as \((3.8,345^\circ)\), one would first move from the origin directly outwards along the radial line to the specified distance, which is 3.8 units in this case. Imagine stretching a ruler from the center of a circle outward, stopping at the 3.8-unit mark—that's your radius. Next, rotate the ruler counterclockwise, starting from the positive x-axis until it reaches the specified angle of 345 degrees. It's crucial to move in the counterclockwise direction, as this is the positive orientation in polar coordinates. The point at which the 3.8-unit mark on the ruler now points is the location of your point in polar space.

Visualizing with Polar Graph Paper

Using polar graph paper can make this process easier. It includes concentric circles representing radial distances and lines emanating from the center at common angles. By finding the circle that corresponds to the radial distance and tracing it until it intersects with the line representing the angle, you can accurately plot the point.

Angle of Rotation

The angle of rotation, often denoted by \(\theta\), is a critical value in determining the direction of a point in polar coordinates. This angle is measured in degrees or radians and denotes the amount of rotation from the initial side (positive x-axis) to the terminal side, which is the line passing through the point and the origin. It's measured counterclockwise from the x-axis, with angles of rotation between 0 and 360 degrees corresponding to a full circle.

For the point \((3.8,345^\circ)\), the angle of 345 degrees indicates that the rotation should begin from the positive x-axis and move almost entirely around the circle. It's essential to measure accurately and understand that slightly different angles can lead to a point being in a very different location. Moreover, angles can be given in degrees beyond 360 or even negative, in which case, standardizing them into the 0 to 360-degree range can help simplify the plotting process. This process is called 'angle normalization' and is a skill that can prevent common mistakes when plotting polar coordinates.

Common Angle References

Knowing common angles and their corresponding points on the coordinate circle, like 0, 90, 180, and 270 degrees, is very helpful. These angles correspond to the positive x-axis, positive y-axis, negative x-axis, and negative y-axis, respectively. This can serve as checkpoints for validating the correctness of your plotted point.

Radius in Polar Coordinates

In the context of polar coordinates, the radius \(r\) represents the distance from the origin (or pole) to the point. It's essentially the 'how far out' measurement that signifies how much you have to move from the center of the plane to reach your point. The radius can be any non-negative real number, with zero being at the origin itself.

For the example point \((3.8,345^\circ)\), the radius is 3.8, which means you would count out 3.8 units from the origin along a straight line, following any rotation. If you imagine a spider walking out from the center of a web, the radius is the length of the walk before the spider starts curving around to follow an angle.

Interpreting Negative Radius

Sometimes, you may encounter a negative radius. While it might initially seem counterintuitive since distance is not negative, in polar coordinates, it means the point is plotted in the opposite direction to the angle given, as though you have rotated an additional 180 degrees. For instance, a point with coordinates \((-3.8,345^\circ)\) would be plotted at the same angle but moving in the opposite direction into the coordinate space, effectively ending up at \((3.8,165^\circ)\). This concept is particularly useful for understanding the symmetry in polar graphs and for solving complex problems involving polar equations.