Question

Plot each point given in polar coordinates, and find other polar coordinates \((r, \theta)\) of the point for which: (a) \(r>0, \quad-2 \pi \leq \theta<0\) (b) \(r<0, \quad 0 \leq \theta<2 \pi\) (c) \(r>0, \quad 2 \pi \leq \theta<4 \pi\) $$ \left(-2,-\frac{2 \pi}{3}\right) $$

Short answer

(a) (2, -\frac{5\pi}{3}), (b) (-2, \frac{4\pi}{3}), (c) (2, \frac{7\pi}{3})

Step-by-step solution

Understanding the Given Coordinates

The given coordinates are in polar form: \(r = -2, \theta = -\frac{2 \pi}{3}\). This means the point is currently 2 units away from the origin in the direction that lies \(-\frac{2 \pi}{3}\) radians (or -120 degrees) in the standard position.

Converting to Positive r

To convert to positive \(r\), add \pi\ (or 180 degrees) to \theta\ to change the direction. The new coordinates are \(r = 2\) and \(\theta = \pi + \left(-\frac{2 \pi}{3}\right) = \frac{\pi}{3}\).

Adjusting \(θ\) for given conditions

For condition (a), since \theta must be between \-2 \pi\ and \0\, subtract \2 \pi\ from the newly found \theta\: \( \theta = \frac{\pi}{3} - 2 \pi = -\frac{5 \pi}{3}\). Verify that \theta\ is in the required range.

Negative r with Positive θ

For condition (b), \r\ must be negative and \theta\ must be between \0\ and \2 \pi\. Since the point was already given with \r = -2\, use the \theta\ as it is \(\theta = -\frac{2 \pi}{3}\). Add \(2 \pi\) to bring \theta\ within the correct range: \( \theta = -\frac{2 \pi}{3} + 2 \pi = \frac{4 \pi}{3}\).

Positive r with θ in [2π, 4π)

For condition (c), use the positive \r = 2\ and find \theta\ in the interval \2 \pi \leq \theta < 4 \pi\. Add 2π to the original positive \theta\ from step 2: \( \theta = \frac{\pi}{3} + 2 \pi = \frac{7 \pi}{3}\).

Key concepts

radian measure

In polar coordinates, angles are often expressed in radians. A radian is a way of measuring angles based on the radius of a circle.
One complete revolution around a circle corresponds to an angle of 2π radians.
To convert an angle from degrees to radians, use the conversion factor \(\text{radians} = \frac{\text{degrees} \times \text{π}}{180}\).
Likewise, you can convert radians to degrees by using the reverse conversion: \(\text{degrees} = \frac{\text{radians} \times 180}{\text{π}}\).

For example, in the given problem, the angle \(-\frac{2π}{3}\) radians can be converted to degrees by multiplying:
\(-\frac{2 \times 180}{3} = -120^\circ\).

Understanding radian measure is crucial for working with polar coordinates because it helps you to visualize and adjust angles quickly.
It also simplifies many mathematical operations in calculus and trigonometry.

coordinate conversion

Converting between polar and Cartesian coordinates is a fundamental skill in mathematics.
In polar coordinates, a point is defined by a radius (r) and an angle (θ).
To convert polar coordinates (r, θ) to Cartesian coordinates (x, y), use the formulas:
\x = r \times \text{cos}(θ)\ and \y = r \times \text{sin}(θ)\.

For instance, the polar coordinates \(-2, -\frac{2π}{3}\)\ can be converted to Cartesian coordinates as follows:
\x = -2 \times \text{cos}(-\frac{2π}{3}) = -2 \times -\frac{1}{2} = 1\ and \y = -2 \times \text{sin}(-\frac{2π}{3}) = -2 \times -\frac{\text{√3}}{2} = \text{√3}\.

Conversely, to convert Cartesian coordinates (x, y) to polar coordinates (r, θ), use:
\r = \text{√}(x^2 + y^2)\ and \θ = \text{atan2}(y, x)\, where atan2 is the arctangent function that takes into account the signs of x and y to determine the correct quadrant for θ.
Remember, when converting coordinates, the value of θ may need to be adjusted to fall within the specified interval.

polar plot

A polar plot allows you to visualize points based on their polar coordinates.
In this plot, the position of a point is determined by its distance from the origin (r) and the angle (θ) from the positive x-axis.
Polar plots are excellent for representing periodic functions and spiral shapes.

Let's plot the given polar coordinates \(-2, -\frac{2π}{3}\).
First, convert the negative radius into a positive radius by adding π to the angle:
\r = 2\ and \θ = \frac{π}{3}\.

For negative and positive r values:

• Condition (a): To find the new angle for r > 0 within \-2π ≤ θ < 0\, the coordinates are \(2, -\frac{5π}{3}\).


• Condition (b): To find θ with r < 0 within 0 ≤ θ < 2π, the coordinates are \(-2, \frac{4π}{3}\).


• Condition (c): To find θ within \2π ≤ θ < 4π\ with r > 0, the coordinates are \(2, \frac{7π}{3}\).

Using these steps, you can accurately plot points and understand their positions in the polar coordinate system. Understanding how to read and plot these points helps in various fields like engineering, physics, and navigation.