Question

If the polar co-ordinates of a point are $\left(\sqrt{2}, \frac{\pi}{4}\right)$, then its Cartesian co-ordinates are

• A. $(2, 2)$
• B. $(1, -1)$
• C. $(\sqrt{2}, \sqrt{2})$
• D. $(1, 1)$

Hint:

An angle of $\frac{\pi}{4}$ means the point lies exactly on the line $y = x$ in the first quadrant. This instantly eliminates any options where the $x$ and $y$ coordinates are not identical and positive!

Answer 1

The Correct Option is D

Step 1: Note the polar data.
The point is given as $\left(r,\theta\right)=\left(\sqrt{2},\frac{\pi}{4}\right)$ and we want Cartesian $(x,y)$.
Step 2: Recall the conversion.
$x=r\cos\theta$ and $y=r\sin\theta$.
Step 3: Values at $\frac{\pi}{4}$.
$\cos\frac{\pi}{4}=\frac{1}{\sqrt{2}}$ and $\sin\frac{\pi}{4}=\frac{1}{\sqrt{2}}$.
Step 4: Find $x$.
$x=\sqrt{2}\times\frac{1}{\sqrt{2}}=1$.
Step 5: Find $y$.
$y=\sqrt{2}\times\frac{1}{\sqrt{2}}=1$.
Step 6: State the point.
The Cartesian coordinates are $(1,1)$, which is option (4). \[ \boxed{(x,y)=(1,1)} \]