Problem 90 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. If \(\left(r, \theta_{1}\right)\) and \(\left(r, \theta_{2}\right)\) represent the same point in the polar coordinate system, then \(\theta_{1}=\theta_{2}+2 \pi n\) for some integer \(n\).

Step-by-Step Solution

Answer

The statement is True. In the polar coordinates system, an angle \(\theta\) is periodic with period \(2 \pi\), which means that adding \(2 \pi n\) (where \(n\) is any integer) to an angle doesn't change the position of the point. Hence, if two sets of polar coordinates \((r, \theta_{1})\) and \((r, \theta_{2})\) represent the same point, then \(\theta_{1}=\theta_{2}+2 \pi n\), for some integer \(n\).

Step 1: Verification of the statement

Let's take \((r, \theta_{1})\) and \((r, \theta_{2})\) as two representations of the same point in polar coordinates. According to the problem, \(\theta_{1}=\theta_{2}+2 \pi n\). In the polar coordinate system, a full rotation around the origin is \(2\pi\) radians. This means, if you add \(2 \pi\) to an angle, you're still pointing to the same direction. That implies \((r, \theta_{1} + 2\pi n)\) and \((r, \theta_{2})\) are identical for any integer \(n\) because adding full turn(s) doesn't change the position of the point.

Step 2: Concluding the analysis

In summary, the statement is indeed true: If two different sets of polar coordinates, \((r, \theta_{1})\) and \((r, \theta_{2})\), represent the same point, then \(\theta_{1}=\theta_{2}+2 \pi n\), where \(n\) is an integer. This is because the angle \(\theta\) is periodic with period \(2 \pi\), thus adding full rotation(s) \(2 \pi n\) doesn't change the position of the point.

Key Concepts

Angle Periodicity

In polar coordinates, angle periodicity is a key property. A full circle around a point spans an angle of \(2\pi\) radians. This principle comes into play when you add \(2\pi\) or repeated increments to any initial angle. Even if the angle appears different, with these increments, it still points in the same direction. This is why in polar coordinates, angles are viewed as periodic. Whenever you go around a full circle, you're back where you started. To understand it better, picture a clock's hands. Each full rotation amounts to \(2\pi\) radians. Moving around the clock face more than once simply adds more full rotations, but the hands would still end in the same position on every full circle.

Radian Measure

The radian is a standard unit of angular measure. Understanding radians is key for mastering polar coordinates. It provides an intuitive way to relate angle measures to the circle's radius. A radian essentially measures the angle based on the radius of the circle. In a full circle of 360 degrees, we have \(2\pi\) radians, representing one complete round around the circle's circumference. That means \(1\) radian is equivalent to about \(57.3\) degrees.Why use radians, then? Dealing with trigonometric functions or calculus often involves radians. As they relate directly to circle properties, many mathematical formulas become easier to express when using radians rather than degrees. So understanding the radian measure in terms of \(\theta\) in polar coordinates is essential.

Coordinate System

The polar coordinate system offers a unique way to identify points. Unlike Cartesian coordinates that use \(x\) and \(y\) to position points in a plane, polar coordinates use a distance and an angle. This system is based on two key components:

• Radius \((r)\): The distance from the origin (or pole) to the point.
• Angle \((\theta)\): The direction from the positive x-axis to the point.

Instead of horizontal and vertical measures, the location of a point is marked by how far out and in which direction from the pole it lies.One characteristic feature of polar coordinates is their flexibility in representing the same point in multiple ways. Because angles are periodic, as explained earlier, a single point can have infinitely many polar representations. This is what makes the polar coordinate system especially useful in fields such as physics and engineering, where circular symmetry and periodic behavior are common.