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

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

Answer: (D)

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 have multiple representations by adjusting \(r\) and \(\theta\) using these properties: \((r, \theta)\) is the same as \((r, \theta + 2k\pi)\) for any integer \(k\), and also \((r, \theta)\) is the same as \((-r, \theta + (2k+1)\pi)\) for any integer \(k\).
3. Given the point \((3, \frac{\pi}{4})\), to find an equivalent point with a positive radius, add multiples of \$2\pi\( to the angle \)\theta\(. For example, adding \)2\pi\( (which is \)\frac{8\pi}{4}\() to \)\frac{\pi}{4}\( gives \)\frac{9\pi}{4}\(, so \)(3, \frac{9\pi}{4})$ represents the same point.
4. To find an equivalent point with a negative radius, change \(r\) to \(-3\) and add \(\pi\) (or an odd multiple of \(\pi\)) to the angle. For example, \((-3, \frac{\pi}{4} + \pi) = (-3, \frac{5\pi}{4})\) represents the same point.
5. Check each given option by applying these transformations to see which coordinate pair matches the original point \((3, \frac{\pi}{4})\).