Question

Find the cartesian coordinates of the point whose polar coordinates are \( \left( \frac{1}{2}, \frac{\pi}{3} \right) \).

Hint:

Convert polar to Cartesian using \( x = r \cos \theta \), \( y = r \sin \theta \); know values of \( \cos \frac{\pi}{3} \) and \( \sin \frac{\pi}{3} \).

Answer 1

Polar to Cartesian conversion: \( x = r \cos \theta \), \( y = r \sin \theta \).
Given: \( r = \frac{1}{2} \), \( \theta = \frac{\pi}{3} \).
\[ x = \frac{1}{2} \cos \frac{\pi}{3} = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}, y = \frac{1}{2} \sin \frac{\pi}{3} = \frac{1}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{4}. \] Answer: \( \left( \frac{1}{4}, \frac{\sqrt{3}}{4} \right) \).