Problem 11

Question

Plot the point given in polar coordinates and then give three different expressions for the point such that (a) $r<0$ and $0 \leq \theta \leq 2 \pi$ (b) $r>0$ and $\theta \leq 0$ (c) $r>0$ and $\theta \geq 2 \pi$ $$ \left(-1, \frac{2 \pi}{3}\right) $$

Step-by-Step Solution

Verified

Answer

Three expressions are: (a) $(-1, \frac{2\pi}{3})$, (b) $(1, \frac{-4\pi}{3})$, (c) $(1, \frac{8\pi}{3})$.

1Step 1: Understanding Polar Coordinates

The given point is in polar coordinates, denoted as $(-1, \frac{2\pi}{3})$. This means the radius $r = -1$ and the angle $\theta = \frac{2\pi}{3}$. To plot this, move 1 unit in the opposite direction of the angle $\frac{2\pi}{3}$. This is equivalent to moving 1 unit at angle $\pi + \frac{2\pi}{3}$.

2Step 2: Convert to Cartesian Coordinates

Converting the polar coordinates $(r, \theta)$ to Cartesian coordinates $(x, y)$ using the formulas $x = r\cos(\theta)$ and $y = r\sin(\theta)$. Here, $x = -1 \times \cos\left(\frac{2\pi}{3}\right)$ and $y = -1 \times \sin\left(\frac{2\pi}{3}\right)$, which simplifies to $x = \frac{1}{2}$ and $y = \frac{\sqrt{3}}{2}$.

3Step 3: Expression for r<0 and 0 ≤ θ ≤ 2π

Since the radius is already negative $r = -1$, we need to find the equivalent angle within $[0, 2\pi]$. The given angle is already $\frac{2\pi}{3}$, which satisfies the condition. Thus, one expression is $(-1, \frac{2\pi}{3})$.

4Step 4: Expression for r>0 and θ≤0

For $r > 0$, we take the positive form $r = 1$ and adjust the angle. The equivalent negative angle for $\frac{2\pi}{3}$ is $\frac{-4\pi}{3}$, obtained by subtracting $2\pi$. So another expression is $(1, \frac{-4\pi}{3})$.

5Step 5: Expression for r>0 and θ≥2π

Here, with $r > 0$, take $r = 1$ again and find an equivalent angle greater than or equal to $2\pi$. We add $2\pi$ to $\theta = \frac{2\pi}{3}$; thus $\theta = \frac{8\pi}{3}$. Hence, the last expression is $(1, \frac{8\pi}{3})$.

Key Concepts

Polar to Cartesian ConversionAngle TransformationRadius Sign Change

Polar to Cartesian Conversion

The journey from polar to Cartesian coordinates can feel like a magical transformation in the world of mathematics. When dealing with polar coordinates, like in the pair $(-1, \frac{2\pi}{3})$, the concept involves two components:
The radius, represented by $r$, and the angle, denoted by $\theta$. To convert these into Cartesian coordinates, we use the following formulas:

$x = r \cos(\theta)$
$y = r \sin(\theta)$

In our example, $x = -1 \times \cos\left(\frac{2\pi}{3}\right)$, which calculates to $x = \frac{1}{2}$. Similarly, for $y = -1 \times \sin\left(\frac{2\pi}{3}\right)$, the result is $y = \frac{\sqrt{3}}{2}$.
This combination $(\frac{1}{2}, \frac{\sqrt{3}}{2})$ now represents the same point in the Cartesian coordinate system, offering a more intuitive understanding, especially when dealing with straight lines and rectangular layouts.

Angle Transformation

Transforming angles in polar coordinates is crucial when wanting to express them differently while maintaining their geometric consistency. For the given point $\left(-1, \frac{2\pi}{3}\right)$, understanding angle transformation helps us find different expressions while adhering to specific conditions on the angle.

Positive and Negative Angles

Angles can have both positive and negative equivalents. In Step 4, transforming $\frac{2\pi}{3}$ to fit the condition $\theta \leq 0$, requires finding its negative equivalent. By subtracting $2\pi$, we acquire $\frac{-4\pi}{3}$, maintaining its direction but in a negative scope.

Angles Greater than $2\pi$

Similarly, angles can be wrapped around to exceed $2\pi$, signifying a complete revolution and more. Therefore, adding $2\pi$ to $\frac{2\pi}{3}$ gives $\theta = \frac{8\pi}{3}$, as demonstrated in Step 5.
This enrichment of angles provides multiple viewpoints, each strategically beneficial under different circumstances, yet representing an identical point on a polar graph.

Radius Sign Change

In the realm of polar coordinates, the radius plays a vital role in defining a point's location via its magnitude and direction. When the radius is negative, such as $r = -1$, it indicates a position diametrically opposite to the given angle.

From Negative to Positive Radius

Converting a negative radius to a positive one involves an axial adjustment. For $\left(-1, \frac{2\pi}{3}\right)$, changing $r$ to positive requires altering the direction of the angle by $\pi$ radians. "Flipping" across the origin adjusts our angle such that $r = 1$, and the new angular equivalent becomes crucial, such as $\theta = \frac{-4\pi}{3}$ or $\theta = \frac{8\pi}{3}$, depending on additional criteria.
These transformations unfold new expressions for the same point, demonstrating versatile approaches within polar systems, each complementing vector illustrations and rotational symmetry.

Problem 11