What are the rectangular coordinates of the point whose polar coordinates are $(3,\frac{\pi }{6})$?

• A. $(3,0)$
• B. $(\frac{3}{2},\frac{3}{\sqrt{3}})$
• C. $(0,3)$
• D. $(\frac{3}{\sqrt{3}},\frac{3}{2})$

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. To convert from polar coordinates \((r, \theta)\) to rectangular coordinates \((x, y)\), use the formulas: \(x = r \cdot \cos(\theta)\) \(y = r \cdot \sin(\theta)\)
3. Identify the given values: \(r = 3\) and \(\theta = \frac{\pi}{6}\).
4. Calculate \(x\) by substituting the values into the formula: \(x = 3 \cdot \cos\left(\frac{\pi}{6}\right)\)
5. Calculate \(y\) by substituting the values into the formula: \(y = 3 \cdot \sin\left(\frac{\pi}{6}\right)\)