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: Understanding the Question:
We are given a point in the 2D plane defined by its polar coordinates $(r, \theta)$ and need to convert it into standard rectangular Cartesian coordinates $(x, y)$.

Step 2: Key Formula or Approach:
The conversion equations from polar coordinates $(r, \theta)$ to Cartesian coordinates $(x, y)$ are:
$$x = r \cos \theta$$ $$y = r \sin \theta$$

Step 3: Detailed Explanation:
From the problem, we have:
$r = \sqrt{2}$
$\theta = \frac{\pi}{4}$
Substitute these values into the conversion formulas:
$$x = \sqrt{2} \cos\left(\frac{\pi}{4}\right)$$ $$y = \sqrt{2} \sin\left(\frac{\pi}{4}\right)$$ We know the exact trigonometric values for $\frac{\pi}{4}$ ($45^\circ$):
$\cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$
$\sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}$
Perform the multiplication:
$$x = \sqrt{2} \times \frac{1}{\sqrt{2}} = 1$$ $$y = \sqrt{2} \times \frac{1}{\sqrt{2}} = 1$$

Step 4: Final Answer:
The Cartesian coordinates are $(1, 1)$, matching option (D).