Given the polar coordinate $(-6,\frac{\pi }{12})$, which of the following points in rectangular coordinates represents the same location?

• A. $(-6cos(\frac{\pi }{12}),-6sin(\frac{\pi }{12}\left)\right)$
• B. $(6cos(\frac{\pi }{12}),6sin(\frac{\pi }{12}\left)\right)$
• C. $(6cos(\frac{\pi }{12}+\pi ),6sin(\frac{\pi }{12}+\pi \left)\right)$
• D. $(-6cos(\frac{\pi }{12}+\pi ),-6sin(\frac{\pi }{12}+\pi \left)\right)$

Answer: (C)

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. To convert from polar coordinates \((r, \theta)\) to rectangular coordinates \((x, y)\), use the formulas: \(x = r \cos(\theta)\) \(y = r \sin(\theta)\)
3. In this problem, the polar coordinate is given as \((-6, \frac{\pi}{12})\). Notice that the radius \(r\) is negative, which means the point lies in the direction opposite to the angle \(\theta\).
4. A negative radius \(r\) can be handled by adding \(\pi\) to the angle \(\theta\) and using the positive radius \(|r|\). So, rewrite the point as \((6, \frac{\pi}{12} + \pi)\).
5. Now apply the conversion formulas with \(r = 6\) and \(\theta = \frac{\pi}{12} + \pi\): \(x = 6 \cos\left(\frac{\pi}{12} + \pi\right)\) \(y = 6 \sin\left(\frac{\pi}{12} + \pi\right)\) This gives the rectangular coordinates representing the same location.