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) \(r>0, \quad 2 \pi \leq \theta<4 \pi\). $$ \left(-3,-\frac{\pi}{4}\right) $$

Short answer

a) (3, -\frac{5\pi}{4}) b) (-3, \frac{7\pi}{4}) c) (3, \frac{11\pi}{4})

Step-by-step solution

Plot the Point

Plot the point \((-3, -\frac{\pi}{4})\) in polar coordinates. Since \(-r\) implies reversing the radius direction, add \pi\ to \theta\. The point converts to \(3, \frac{3\pi}{4}\).

Finding Coordinates for Condition (a)

For \(r > 0, -2\pi \leq \theta < 0\): Subtract \2\pi\ to keep \theta\ within the desired range. Result: \(3, \frac{3\pi}{4} - 2\pi = 3, -\frac{5\pi}{4}\).

Finding Coordinates for Condition (b)

For \(r < 0, 0 \leq \theta < 2\pi\): Change the sign of \r\ and add \pi\ to \theta\ to revert \(-r, -\frac{\pi}{4} = -3, ( \frac{7\pi}{4} - 2\pi + \pi)= -3, \frac{7\pi}{4} + 0\), which is equivalent to \( -3, \frac{7\pi}{4})\) as -4 \pi\theta<2\theta

Finding Coordinates for Condition (c)

For \(r > 0, 2\pi \leq \theta < 4\pi\): Add \2\pi\ to \theta\ to move to the desired range. \3, \frac{3\pi}{4} + 2\pi = 3, \frac{11\pi}{4}\.

Key concepts

plotting points

In polar coordinates, plotting points involves a different approach than Cartesian coordinates. Instead of using an x-axis and y-axis, we use a radius and an angle. The radius, denoted as \(r\), is the distance from the origin. The angle, denoted as \(\theta\), specifies the direction from the origin.
To place a point like \((-3, -\frac{\pi}{4})\), follow these steps:

• Start at the origin.
• Move in the direction of -\(\frac{\pi}{4}\). This angle points down and to the right.
• Since \(r\) is negative, consider it as \(3\) and move in the exact opposite direction of -\(\frac{\pi}{4}\), which places the point in the left and up direction instead of the right and down.

The point \((-3, -\frac{\pi}{4})\) effectively moves to \((3, \frac{3\pi}{4})\). This concept simplifies plotting negative radii. Also, once the point is plotted, we can transform the coordinates to match other conditions.

coordinate transformation

When transforming coordinates in polar notation, we often manipulate the angle \(\theta\) to fit a specific range. In our exercise, we needed different versions of the point \((3, \frac{3\pi}{4})\).
Condition (a): \(r > 0\) and \(-2\pi \leq \theta < 0\)
To adjust \(\frac{3\pi}{4}\) within the range, we subtract \(2\pi\) (one full circle), yielding \(\frac{3\pi}{4} - 2\pi = -\frac{5\pi}{4}\). Hence, the coordinates become \((3, -\frac{5\pi}{4})\).
Condition (b): \(r < 0\) and \(0 \leq \theta < 2\pi\)
Here, we need negative \(r\), so \(r = -3\). Adding \(\pi\) to the angle gives us \(\theta = \frac{7\pi}{4}\). Thus, the coordinates convert to \((-3, \frac{7\pi}{4})\).
Condition (c): \(r > 0\) and \(2\pi \leq \theta < 4\pi\)
For this condition, we add \(2\pi\) to \(\frac{3\pi}{4}\), resulting in \(\frac{3\pi}{4} + 2\pi = \frac{11\pi}{4}\). Thus, the polar coordinates become \((3, \frac{11\pi}{4})\). These transformations showcase the flexibility of polar coordinates in adapting to various ranges.

polar notation

Understanding polar notation is key to working with polar coordinates. Unlike Cartesian coordinates, polar notation focuses on the radius and angle. The point \((r, \theta)\) helps us describe locations based on how far and in which direction from the origin.
For instance, \(r = 3\) tells us to move 3 units from the origin, and \(\theta = \frac{3\pi}{4}\) tells us in which direction. Polar notation is especially useful in contexts like circular movements or rotations.
A positive radius \(r > 0\) means moving outward from the origin, while a negative radius \(r < 0\) means the opposite direction. The angle \(\theta\) can be adjusted by adding or subtracting multiples of \(2\pi\), effectively using circular symmetry.
Moreover, you can easily switch between polar and Cartesian forms using transformations:

• From polar to Cartesian: \(x = r \cos(\theta)\), \(y = r \sin(\theta)\)
• From Cartesian to polar: \(r = \sqrt{x^2 + y^2}\), \(\theta = \arctan\left(\frac{y}{x}\right)\)

Leaning on these relationships equips you with tools to handle a variety of problems involving polar coordinates.