Problem 32 - Algebra and Trigonometry

Question

Plot each point given in polar coordinates. $$ \left(-3,-\frac{3 \pi}{4}\right) $$

Step-by-Step Solution

Answer

Plot the point 3 units away at an angle of \(\frac{\pi}{4}\) from the positive x-axis.

Step 1: Understand Polar Coordinates

Polar coordinates are represented as \(r, \theta\), where \(r\) is the radius and \(\theta\) is the angle in radians measured from the positive x-axis.

Step 2: Identify the Given Coordinates

The given coordinates are \(-3, -\frac{3 \pi}{4}\).\(r\) is the radius which is -3, and \(\theta\) is the angle which is \(-\frac{3 \pi}{4}\).

Step 3: Adjust the Radius and Angle

Typically, the radius \(r\) should be positive. When the radius is negative, the point lies in the direction opposite to the given angle. Convert the negative radius to positive: \(r = |-3| = 3\). Adjust the angle \(\theta\) by adding \pi\ (since a negative radius means adding \pi\ to the angle): \(-\frac{3 \pi}{4} + \pi = -\frac{3 \pi}{4} + \frac{4 \pi}{4} = \frac{\pi}{4}\).

Step 4: Plot the Point

With \(r = 3\) and \(\theta = \frac{\pi}{4}\), plot the point on the polar coordinate plane by rotating an angle of \(\frac{\pi}{4}\) (45 degrees) from the positive x-axis and moving a distance of 3 units from the origin.

Key Concepts

Radius Adjustment

In polar coordinates, the radius (\r) determines the distance from the origin to the point. Normally, we use positive values for the radius. However, sometimes the radius may be given as a negative value.
When the radius is negative, the point lies directly opposite the given angle. To properly place this point on the polar plane:

• Change the radius to its absolute value (make it positive).
• Adjust the angle by adding \( \pi \) to it. This effectively points in the opposite direction.

For example, given the coordinates \(-3, -\frac{3 \pi }{4} \), the radius adjustment involves transforming \(-3 \) to \(3 \). Then, we modify the angle by adding \ \pi \:
\( -\frac{3 \pi }{4} + \pi = \frac{\pi}{4} \). Now, we have \( (3, \frac{\pi}{4}) \). This represents a point 3 units away from the origin at an angle of \( \frac{\pi}{4} \) radians.

Angle Conversion

Angles in polar coordinates are typically measured in radians. To understand how to convert angles, remember:

• Positive angles are measured counterclockwise from the positive x-axis.
• Negative angles are measured clockwise from the positive x-axis.
• For converting negative angles to positive, add \2 \pi \ until positive.

In this example of \(- \frac{3 \pi }{4} \), converting to positive includes adding \ \pi \ due to the radius adjustment. This results in an angle of \ \frac{\pi}{4} \, corresponding to 45 degrees counterclockwise.
Always make sure your final angle is within the standard range of \ 0 \ to \ 2 \pi \ after adjustments.

Plotting Points

Plotting points in polar coordinates involves both distance and angle. Let's break it down:
• With the adjusted radius \ (r = 3) \ and angle \ (\frac{\pi}{4}) \, start from the origin.
• The angle \( \frac{\pi}{4} \) means you rotate 45 degrees counterclockwise from the positive x-axis.
• From there, measure a distance of 3 units along that angle direction.
These steps will place your point precisely on the polar coordinate grid. Recall that polar plotting is inherently radial and depends heavily on the angular direction as well as the radius from the origin.