Question

Find the Cartesian co-ordinates of the point whose polar co-ordinates are \( \left( \sqrt{2}, \frac{\pi}{4} \right) \).

Hint:

To convert from polar to Cartesian coordinates, use the formulas \( x = r \cos{\theta} \) and \( y = r \sin{\theta} \).

Answer 1

We are given the polar coordinates \( r = \sqrt{2} \) and \( \theta = \frac{\pi}{4} \), and we need to convert these into Cartesian coordinates. The formula for converting from polar to Cartesian coordinates is: \[ x = r \cos{\theta}, \quad y = r \sin{\theta} \] Step 1: Substituting the given values for \( r \) and \( \theta \): \[ x = \sqrt{2} \cos{\left(\frac{\pi}{4}\right)} = \sqrt{2} \cdot \frac{1}{\sqrt{2}} = 1 \] \[ y = \sqrt{2} \sin{\left(\frac{\pi}{4}\right)} = \sqrt{2} \cdot \frac{1}{\sqrt{2}} = 1 \] Step 2: Therefore, the Cartesian coordinates are: \[ (x, y) = (1, 1) \]