Given the point with polar coordinates $(3,\frac{\pi }{4})$, which of the following polar coordinate pairs labels the same point?

• A. $(-3,\frac{\pi }{4})$
• B. $(3,\frac{9\pi }{4})$
• C. $(3,\frac{5\pi }{4})$
• D. $(3,\frac{\pi }{2})$

Answer: (B)

Verified step by step guidance

1. Recall that 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.
2. Understand that the same point in polar coordinates can be represented by adding or subtracting full rotations of \(2\pi\) to the angle \(\theta\), or by changing the sign of \(r\) and adjusting the angle by \(\pi\) (180 degrees).
3. Given the original point \((3, \frac{\pi}{4})\), to find an equivalent point with the same \(r\) but a different angle, add multiples of \(2\pi\) to \(\frac{\pi}{4}\). For example, \(\frac{\pi}{4} + 2\pi = \frac{\pi}{4} + \frac{8\pi}{4} = \frac{9\pi}{4}\).
4. Check if the given options match this pattern. The point \((3, \frac{9\pi}{4})\) represents the same location because the angle \(\frac{9\pi}{4}\) is coterminal with \(\frac{\pi}{4}\) (they differ by \(2\pi\)).
5. Verify other options by either adding or subtracting \(2\pi\) or by considering the negative radius rule: \((r, \theta)\) is equivalent to \((-r, \theta + \pi)\). This helps confirm which pairs label the same point.