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

• A. $(-3,\frac{\mathrm{5<mi>\pi </mi>}}{4})$
• B. $(3,\frac{\pi }{2})$
• C. $(3,0)$
• D. $(3,\frac{\pi }{4})$

Answer: (D)

Verified step by step guidance

1. Recall that polar coordinates are given as \((r, \theta)\), where \(r\) is the 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 by adjusting \(r\) and \(\theta\) using these rules: if you add or subtract \$2\pi\( to \)\theta\(, the point remains the same; also, if you change \)r\( to \)-r\(, you must add or subtract \)\pi\( to \)\theta$ to represent the same point.
3. Check the given point \((3, \frac{\pi}{4})\) and compare it with the options by applying these rules. For example, for the point \((-3, \frac{5\pi}{4})\), note that \(-3\) is the negative of \$3\(, so add \)\pi\( to \)\frac{\pi}{4}\( to get \)\frac{5\pi}{4}$, which matches the angle given, indicating the same point.
4. For the other options, verify if the radius and angle correspond to the original point by either adding multiples of \$2\pi\( to the angle or checking if the radius sign and angle differ by \)\pi$.
5. Conclude which pairs represent the same point by confirming that the radius and angle adjustments follow the polar coordinate equivalence rules.