Question

Find the cartesian coordinates of the point whose polar coordinates are \( \left( 2, \frac{\pi}{4} \right) \).

Hint:

Use \( x = r \cos \theta \), \( y = r \sin \theta \) for polar to Cartesian conversion; know exact trigonometric values.

Answer 1

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