Problem 13 - Algebra and Trigonometry

Question

Use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. $$ \left(3,90^{\circ}\right) $$

Step-by-Step Solution

Answer

The point (3, 90°) in the polar coordinate system is 3 units distant from the origin and is located directly upwards along the positive y-axis.

Step 1: Understanding Polar Coordinates

Polar coordinates are represented by the format (r, θ), with r being the radial distance from the origin, and θ representing the angle in degrees from the positive x-axis and towards y-axis.

Step 2: Identifying the Polar Coordinates

The given polar coordinates are (3, 90°). This means the point is 3 units away from the origin, and at an angle of 90° from the positive x-axis.

Step 3: Plotting the Polar Coordinate

To plot the point (3, 90°) on the polar coordinate system, start at the origin . Measure an angle of 90° from the positive x-axis towards the y-axis. Move 3 units along this angle. The point where you end up is the point (3, 90°).

Key Concepts

Radial Distance

In polar coordinates, the radial distance is crucial. It tells us how far the point is from the origin. Think of it as the length of a straight line from the center, or origin, to the point. In our example, the radial distance is 3 units. This means the point is exactly three steps away from the origin.

• The radial distance is always a non-negative value.
• It determines how far we move away from the central point.
• In mathematical terms, it's denoted as "r" in the polar coordinate representation (r, θ).

This concept is like the radius of a circle, where the origin acts as the center of the circle you're drawing around it.

Angle in Degrees

The angle in polar coordinates provides the direction for our radial distance. It's measured in degrees, giving us the orientation from a standard position.

• In polar coordinates, it is denoted as θ.
• An angle of 0° starts at the positive x-axis.
• The angle increases counterclockwise from the x-axis.

In our example, the angle is 90°. This means from the starting point on the positive x-axis, we move counterclockwise in a perpendicular direction upwards, similar to the y-axis in a Cartesian plane. Understanding the angle helps to position the radial distance correctly.

Positive X-axis

The positive x-axis is the reference line in polar coordinates for measuring angles. By convention, this is the starting line from which all angles are measured.

• The positive direction extends to the right on a typical graphing chart.
• Angles are measured from this axis, moving in a counterclockwise direction.
• It acts like the "base" line or 0° line when plotting points with polar coordinates.

For plotting a point, you start at the positive x-axis and measure out the angle specified. For a 90° angle, you rotate from this axis upwards. This axis is the indispensable yardstick for angles in polar coordinates.

Origin

The origin is the central point in polar coordinates. It is the equivalent of the (0,0) point in a Cartesian system. All calculations and movements are measured from this point.

• It acts as the starting point for both the radial distance and the angle.
• In polar coordinates, the origin implies a radial distance "r" of zero.
• This point is key because it's where the radial distance is counted from.

In our task, starting at the origin, you move according to the given radial distance and the angle to find the location of your point. The origin is fundamental because it anchors the entire polar coordinate system.