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: Write the conversion.
$x = r\cos\theta$ and $y = r\sin\theta$, with $r = \sqrt2$ and $\theta = \tfrac{\pi}{4}$.

Step 2: Use the $45^\circ$ values.
$\cos\tfrac{\pi}{4} = \sin\tfrac{\pi}{4} = \tfrac{1}{\sqrt2}$.

Step 3: Multiply.
$$x = \sqrt2 \cdot \tfrac{1}{\sqrt2} = 1, \quad y = \sqrt2 \cdot \tfrac{1}{\sqrt2} = 1$$
\[ \boxed{(1,\ 1)} \]