https://file.aiquickdraw.com/mathbot/file/1729805147114p01v0mmj.jpg

The problem asks you to find two other pairs of polar coordinates that represent the same point as \((9, \frac{\pi}{4})\), one with \(r > 0\) and the other with \(r < 0\).

### Step 1: Finding another pair with \(r > 0\)
The radius \(r = 9\) stays the same, but you can adjust the angle by adding or subtracting any multiple of \(2\pi\). In this case, to maintain \(r > 0\), you can add \(2\pi\) to the angle \(\frac{\pi}{4}\).

\[
\theta = \frac{\pi}{4} + 2\pi = \frac{\pi}{4} + \frac{8\pi}{4} = \frac{9\pi}{4}
\]

Thus, the second set of coordinates is:
\[
(9, \frac{9\pi}{4})
\]

### Step 2: Finding another pair with \(r < 0\)
When \(r < 0\), you must add or subtract an odd multiple of \(\pi\) to the angle to point in the opposite direction. So, you subtract \(\pi\) from the angle:

\[
\theta = \frac{\pi}{4} - \pi = \frac{\pi}{4} - \frac{4\pi}{4} = -\frac{3\pi}{4}
\]

With \(r = -9\), the third set of coordinates is:
\[
(-9, -\frac{3\pi}{4})
\]

Thus, the two additional polar coordinates are:
1. \((9, \frac{9\pi}{4})\)
2. \((-9, -\frac{3\pi}{4})\)

Would you like a more detailed explanation or help with another step?

Here are 5 related questions to deepen your understanding:
1. What is the relationship between adding \(2\pi\) to the angle and the position of the point?
2. How does reversing the radius change the angle in polar coordinates?
3. Why is \(r = 9\) unchanged when adding multiples of \(2\pi\) to the angle?
4. How would the Cartesian coordinates change for each polar pair?
5. What would be the next polar pair if you subtracted \(2\pi\) instead of adding it?

**Tip:** In polar coordinates, angles are periodic, so adding or subtracting multiples of \(2\pi\) represents the same direction.

Find two other pairs of polar coordinates that represent the same point as (9, π/4), one with r > 0 and the other with r < 0.

Learn how to find two additional pairs of polar coordinates for the point (9, π/4). This problem involves the concept of angle periodicity in polar coordinates and how to handle positive and negative radius values. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation