Question

The Cartesian coordinates $(x,y)$ of a point $A$ with polar coordinates $(4,\pi/4)$ is

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

Hint:

At $\pi/4$, sine and cosine are equal, so $x = y$ for all points at that angle.

Answer 1

The Correct Option is D

Convert polar coordinates to Cartesian using:
$x = r\cos\theta$, $y = r\sin\theta$.
[4pt] For $(r,\theta) = (4,\pi/4)$:
$\cos(\pi/4) = \sin(\pi/4) = \frac{\sqrt{2}}{2}$.
[4pt] So,
$x = 4 \cdot \frac{\sqrt{2}}{2} = 2\sqrt{2}$,
$y = 4 \cdot \frac{\sqrt{2}}{2} = 2\sqrt{2}$.