Plot the point $(5,-\frac{\pi}{3})$, then identify which of the following sets of coordinates is the same point.

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

Answer: (C)

Verified step by step guidance

1. Step 1: Understand the polar coordinate system. A point in polar coordinates is represented as (r, θ), where r is the radial distance from the origin and θ is the angle measured counterclockwise from the positive x-axis.
2. Step 2: Analyze the given point (5, -π/3). Here, r = 5 indicates the distance from the origin, and θ = -π/3 represents the angle measured clockwise (negative direction) from the positive x-axis.
3. Step 3: Convert the angle -π/3 into its equivalent positive angle by adding 2π (a full rotation). This gives θ = 2π - π/3 = 5π/3. This step ensures the angle is expressed in the standard range [0, 2π).
4. Step 4: Reflect the point across the origin to find equivalent coordinates with a negative radius. For a point (r, θ), the equivalent point with negative radius is (-r, θ + π). Here, (-5, θ + π) = (-5, 5π/3 + π). Simplify the angle: 5π/3 + π = 2π + 2π/3 = 2π/3.
5. Step 5: Compare the transformed coordinates (-5, 2π/3) with the given options. The correct answer is the set of coordinates that matches (-5, 2π/3).