Problem 16 - Algebra and Trigonometry Real Mathematics, Real People

Question

Plot the point given in polar coordinates and find three additional polar representations of the point, using \(-2 \pi<\boldsymbol{\theta}<\mathbf{2} \pi\) $$\left(-5 \sqrt{2}, \frac{2 \pi}{3}\right)$$

Step-by-Step Solution

Answer

The point (-5√2, 2π/3) in polar coordinates represents the same location as the points (-5√2, -4π/3), (-5√2, -8π/3), and (-5√2, 8π/3). All these points are equivalent as they represent the same location.

Step 1: Understanding the Point in Polar Coordinates

Firstly, identify the given polar coordinates. The coordinates are given as (-5√2, 2π/3), where the first term -5√2 is the length of the radius and the second term 2π/3 is the angle formed with the polar axis (which is normally the positive x-axis). Since the radius is negative, the point is located in the opposite direction of the positive θ.

Step 2: Plotting Polar Coordinates

Plotting the polar coordinates involves determining the angle and the radius. Given the angle of 2π/3 radians, mark this angle on the positive X-axis. As the radius is negative, move -5√2 units in the opposite direction to mark the point.

Step 3: Find Additional Polar Representations

To find additional polar representations, add or subtract multiples of 2π from the angle θ. This changes the angle, but the point will remain at the same location because a full rotation around the circle is 2π radians. Therefore, the three additional polar representations could be: (-5√2, 2π/3 - 2π), (-5√2, 2π/3 - 4π), (-5√2, 2π/3 + 2π). These simplify to (-5√2, -4π/3), (-5√2, -8π/3), and (-5√2, 8π/3).

Key Concepts

Plotting Polar Points

Understanding how to plot polar points is essential in fields such as mathematics, physics, and engineering. Unlike the Cartesian coordinate system which uses a grid of x and y values, polar coordinates express the location of points with a distance and an angle from a fixed point, known as the pole (equivalent to the origin in Cartesian coordinates).
The process begins by identifying the angle, usually denoted as \( \theta \), and measuring it counterclockwise from the polar axis (positive x-axis). Following this, the distance (radius \( r \) from the pole is measured along the angle's line. If \( r \) is positive, you move in the direction of the angle; if \( r \) is negative, as in the exercise provided, it signifies movement in the opposite direction.
For example, with the point \( (-5 \sqrt{2}, \frac{2 \pi}{3}) \), one would measure an angle of \( \frac{2 \pi}{3} \) radians from the polar axis, then move \( -5 \sqrt{2} \) units in the reverse direction. This results in locating the point on the opposite side of the pole. This system can be counterintuitive, but with practice, it offers a powerful way to represent points in a plane, especially when dealing with circular or radial systems.

Polar Representations

When dealing with polar representations, it is crucial to grasp that a single point can have an infinite number of polar representations. This is because the angle component can always be adjusted by adding or subtracting multiples of \( 2\pi \) (full circle rotations) without changing the point's location in the plane.

Equivalent Polar Coordinates

One can find alternative representations of a polar point by altering the angle \( \theta \) by multiples of \( 2\pi \) radians. In our exercise, the point \( (-5 \sqrt{2}, \frac{2 \pi}{3}) \) can also be represented as \( (-5 \sqrt{2}, \frac{2 \pi}{3} - 2\pi) \), \( (-5 \sqrt{2}, \frac{2 \pi}{3} - 4\pi) \), and \( (-5 \sqrt{2}, \frac{2 \pi}{3} + 2\pi) \). Recognizing this property helps in simplifying complex polar equations or converting polar coordinates to Cartesian coordinates when necessary.
Furthermore, polar representations are particularly helpful in physics for analyzing systems with rotational symmetry, like the motion of celestial bodies, or in engineering when designing components such as gears and turbines.

Radian Measure

The concept of radian measure is a central aspect of trigonometry and calculus and is fundamental in the understanding of polar coordinates. A radian is a unit of angular measure used in mathematics to express angles based on the radius of a circle.

Definition of a Radian

One radian is the angle created at the center of a circle by an arc whose length is equal to the radius of the circle. This implies that a full circular rotation, which is \( 2\pi \) radians, corresponds to the circumference of the circle.

Using Radians to Measure Angles

Radians offer a more direct measurement of angles than degrees due to their relationship with the circle's radius, which simplifies calculations involving arc lengths, sector areas, and angular velocities. In our example, the angle \( \frac{2 \pi}{3} \) radians represents a counterclockwise rotation from the polar axis and amounts to one-third of a full revolution (as \( 2\pi \) radians equals a full circle). Understanding radian measure is not only essential for working with polar coordinates but also intertwines with other areas of mathematics and physics, facilitating the study of periodic functions, waves, and harmonic motion.