Given the polar coordinates $(3,\frac{\pi }{4})$, what are the corresponding Cartesian coordinates?

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

Answer: (C)

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 to Cartesian coordinates \((x, y)\), use the formulas: \(x = r \cdot \cos(\theta)\) and \(y = r \cdot \sin(\theta)\).
3. Substitute the given values \(r = 3\) and \(\theta = \frac{\pi}{4}\) into the formulas: \(x = 3 \cdot \cos\left(\frac{\pi}{4}\right)\) and \(y = 3 \cdot \sin\left(\frac{\pi}{4}\right)\).
4. Recall that \(\cos\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\), so rewrite the expressions as \(x = 3 \cdot \frac{\sqrt{2}}{2}\) and \(y = 3 \cdot \frac{\sqrt{2}}{2}\).
5. Simplify the expressions to get the Cartesian coordinates: \(\left( \frac{3\sqrt{2}}{2}, \frac{3\sqrt{2}}{2} \right)\).