Question

Plot each point given in polar coordinates. $$ \left(-3,-\frac{\pi}{2}\right) $$

Short answer

The point in Cartesian coordinates is \(0, 3\).

Step-by-step solution

Understand Polar Coordinates

Polar coordinates are given as \(r, \theta\), where \(r\) is the radius (distance from the origin) and \(\theta\) is the angle measured from the positive x-axis. In this case, the polar coordinates are \(-3, -\frac{\pi}{2}\).

Interpret the Radius

The radius \(r = -3\) indicates the point is 3 units away from the origin, but in the opposite direction of the angle given by \(-\frac{\pi}{2}\).

Find the Angle

The angle \(-\frac{\pi}{2}\) corresponds to \(-90^{\circ}\), which points directly down along the negative y-axis.

Adjust for Negative Radius

Instead of moving 3 units down (which is what \(-\frac{\pi}{2}\) would typically indicate), move 3 units up because of the negative sign on the radius. This places the point on the positive y-axis.

Plot the Point

Move 3 units up from the origin. The coordinates in Cartesian form will be \((0, 3)\).

Key concepts

radius

The radius in polar coordinates is essentially the distance from the origin to the point. It is denoted by the variable \(r\). When \(r\) is positive, the point lies in the direction indicated by the angle. However, if \(r\) is negative, the point will be in the opposite direction of the angle.
In our example, the radius is \(-3\). This means the point lies 3 units away from the origin, but we will plot it in the opposite direction of the given angle. Understanding how to handle negative radii is crucial in mastering polar coordinates.

angle

In polar coordinates, the angle \(\theta\) tells us the direction from the positive x-axis where the radius is directed. It is usually measured in radians or degrees. A positive angle typically moves counterclockwise, while a negative angle moves clockwise.
In our example of \(\left(-3, -\frac{\pi}{2}\right)\), the angle \(-\frac{\pi}{2}\) or \(-90^{\circ}\) points straight down along the negative y-axis. It's essential to understand angles as they dictate the direction in which you measure the radius.

Cartesian coordinates

Polar coordinates \((r, \theta)\) can be converted into Cartesian coordinates \((x, y)\) to better understand their positions on a standard coordinate plane. The formulas used are:
\[ x = r \cos(\theta) \]
\[ y = r \sin(\theta) \]
In our example, given \(r = -3\) and \(\theta = -\frac{\pi}{2}\):
\[ x = -3 \cos\left(-\frac{\pi}{2}\right) = 0\]
\[ y = -3 \sin\left(-\frac{\pi}{2}\right) = 3\]
This yields Cartesian coordinates of \((0, 3)\). Plotting this point can provide a visual understanding of the polar coordinates given.

negative radius

A negative radius in polar coordinates may initially seem counterintuitive, but it simply means moving in the opposite direction of the given angle. To plot a point with a negative radius, calculate where you would normally place it with a positive radius and then reverse the direction.
For the point \(\left(-3, -\frac{\pi}{2}\right)\), if we had a positive radius, we would move 3 units down along the negative y-axis. Since the radius is negative, we move 3 units in the opposite direction, which is upwards along the positive y-axis. This concept is important for accurately plotting points in polar coordinates.