The polar co-ordinates of the point whose cartesian co-ordinates are `(sqrt(2), sqrt(2))`, are

Options

A. `(2, (7pi)/(4))`
B. `(2, (5pi)/(4))`
C. `(2, (3pi)/(4))`
D. `(2, (pi)/(4))`

Answer

To find the polar coordinates of the point whose Cartesian coordinates are \((\sqrt{2}, \sqrt{2})\), we will follow these steps:

### Step 1: Understand the relationship between Cartesian and Polar coordinates
Polar coordinates \((r, \theta)\) can be derived from Cartesian coordinates \((x, y)\) using the formulas:
- \(r = \sqrt{x^2 + y^2}\)
- \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\)

### Step 2: Identify the Cartesian coordinates
Here, the Cartesian coordinates are given as:
- \(x = \sqrt{2}\)
- \(y = \sqrt{2}\)

### Step 3: Calculate \(r\)
Using the formula for \(r\):
\[ r = \sqrt{x^2 + y^2} = \sqrt{(\sqrt{2})^2 + (\sqrt{2})^2} = \sqrt{2 + 2} = \sqrt{4} = 2 \]

### Step 4: Calculate \(\theta\)
Using the formula for \(\theta\):
\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{\sqrt{2}}{\sqrt{2}}\right) = \tan^{-1}(1) \]
The angle whose tangent is 1 is:
\[ \theta = \frac{\pi}{4} \text{ radians} \]

### Step 5: Write the polar coordinates
Now that we have both \(r\) and \(\theta\), we can express the polar coordinates as:
\[ (r, \theta) = (2, \frac{\pi}{4}) \]

### Final Answer
The polar coordinates of the point whose Cartesian coordinates are \((\sqrt{2}, \sqrt{2})\) are \((2, \frac{\pi}{4})\).

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