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

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

Answer: (A)

Verified step by step guidance

1. Recall that a point in polar coordinates is given by \((r, \theta)\), 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 can have multiple polar coordinate representations. One key property is that \((r, \theta)\) and \((-r, \theta + \pi)\) represent the same point because moving \(r\) units in direction \(\theta\) is the same as moving \(-r\) units in direction \(\theta + \pi\) (opposite direction).
3. Given the original point \((3, \frac{\pi}{4})\), apply the property to find an equivalent point with a negative radius: change \(r\) to \(-3\) and add \(\pi\) to the angle \(\frac{\pi}{4}\), resulting in \(\left(-3, \frac{\pi}{4} + \pi\right)\).
4. Simplify the angle: \(\frac{\pi}{4} + \pi = \frac{\pi}{4} + \frac{4\pi}{4} = \frac{5\pi}{4}\), so the equivalent point is \(\left(-3, \frac{5\pi}{4}\right)\).
5. Check other options by adding or subtracting multiples of \$2\pi$ to the angle or by using the negative radius property to see if they represent the same point as the original coordinates.