Problem 14 - 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. $$ \left(2,270^{\circ}\right) $$

Step-by-Step Solution

Answer

The point in polar coordinate system \((2, 270°)\) will be located 2 units below the origin.

Step 1: Understand the Polar Coordinates

The polar coordinate system is a two-dimensional coordinate system where each point on a plane is determined by a distance from a fixed point (often the origin point \((0,0)\)) and an angle from a fixed direction (often the x axis or right horizontal direction). The polar coordinates are given in the form \((r, θ)\) where \(r\) is the distance from origin to the point, while \(θ\) is the angle created by the line connecting the point to the origin with respect to the fixed direction. In this exercise, the given polar coordinates are \((2, 270°)\). Here, \(r = 2\) and \(θ = 270°\).

Step 2: Convert the Angle

The direction \(θ = 270°\) is a standard angle in geometry, pointing downward vertically if we start measuring angles from right horizontal direction. It is important to note that angles in polar coordinates are often measured in radians, and that counter-clockwise is the positive direction, but in this case, degrees have been used, which are more intuitive.

Step 3: Plot the Polar Coordinates

Now that we have interpreted \((r, θ)\) as \((2, 270°)\), the next step is to plot this on the polar coordinate plane. Start at the origin (0,0). Rotate counter-clockwise 270° from the positive x-axis (or 90° clockwise if that's easier to visualize), then move out a distance of 2 units. Mark this point.

Key Concepts

Understanding Polar Coordinates

The polar coordinate system represents points on a two-dimensional plane, diverging from the traditional Cartesian coordinates that many students are familiar with. Unlike the Cartesian system, which uses a grid of x and y coordinates, the polar system measures distances and angles from a central point, called the pole (analogous to the origin in Cartesian coordinates).

A point in polar coordinates is expressed as \( (r, \theta) \), where \( r \) is the radial distance from the pole and \( \theta \) is the angular component measured in degrees or radians from the positive x-axis, known as the polar axis. To comprehend this system, imagine reaching the desired location by first turning to face the correct direction and then walking straight ahead the distance required.

Plotting Polar Coordinates

Plotting a point using polar coordinates involves two main steps: orientation and radial placement. Begin at the pole, and rotate counterclockwise by the angle \( \theta \) to set the direction. If the angle provided is in degrees and exceeds 360° or is negative, adjust it within the range by adding or subtracting full circles (360°).

Once the proper direction is established, move outward from the pole a distance equal to \( r \) to pinpoint the exact location of the point. If \( r \) is negative, invert the direction by 180° before proceeding. This method ensures that students can plot any given polar coordinates on the plane quickly and accurately.

Converting Degrees to Radians

In mathematics and physics, angles can be measured in two primary units: degrees and radians. Radians are often preferred in higher mathematics due to their natural connection with the properties of circles. The conversion is based on the relationship that \( 360° \) is equivalent to \( 2\pi \) radians.

To convert from degrees to radians, use the simple formula \( \text{radians} = \text{degrees} \times \frac{\pi}{180°} \). For example, to convert 270° to radians, you would calculate \( 270° \times \frac{\pi}{180°} = \frac{3}{2}\pi \). Understanding this conversion is crucial as certain trigonometric functions and calculus operations in polar coordinates require the use of radians.

Geometry of Polar Coordinates

The geometry of polar coordinates is markedly different from the rectangular grid of Cartesian coordinates. In polar coordinates, straight lines can appear as spirals or circles, while shapes we see as complex in Cartesian systems, like circles, become simpler.

One elegant aspect of polar geometry is that it allows for the representation of curves and shapes that are centered around the pole, making it especially useful in fields dealing with circular or rotational phenomena. By understanding how radial distances and angles interact in this system, students can better appreciate the symmetry and unique properties of shapes within the polar coordinate plane.