Polar to Cartesian Conversion

The question asks for the Cartesian coordinates $(x, y)$ corresponding to the polar coordinates $(r, \theta) = (4, \frac{\pi}{4})$.

The conversion formulas from polar coordinates $(r, \theta)$ to Cartesian coordinates $(x, y)$ are:

• $x = r \cos(\theta)$
• $y = r \sin(\theta)$

Calculating Cartesian Coordinates

Substitute the given values $r = 4$ and $\theta = \frac{\pi}{4}$ into the formulas:

• Calculate x-coordinate:

  $x = 4 \cos(\frac{\pi}{4})$
  Since $\cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$,
  $x = 4 \times \frac{\sqrt{2}}{2}$
  $x = 2\sqrt{2}$

• Calculate y-coordinate:

  $y = 4 \sin(\frac{\pi}{4})$
  Since $\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$,
  $y = 4 \times \frac{\sqrt{2}}{2}$
  $y = 2\sqrt{2}$

Therefore, the Cartesian coordinates are $(2\sqrt{2}, 2\sqrt{2})$.

Final Answer Check

Comparing the calculated coordinates $(2\sqrt{2}, 2\sqrt{2})$ with the given options, Option D matches the result.