The polar co-ordinates of the point whose cartesian co-ordinates are `((1)/(sqrt(2)), (-1)/(sqrt(2)))`, are

Options

A. `(1, (7pi)/(4))`
B. `(1, (3pi)/(4))`
C. `(1, (5pi)/(4))`
D. `(1, (pi)/(4))`

Answer

To find the polar coordinates of the point whose Cartesian coordinates are \((\frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}})\), we can follow these steps:

### Step 1: Identify the Cartesian Coordinates
The Cartesian coordinates given are:
\[ x = \frac{1}{\sqrt{2}}, \quad y = -\frac{1}{\sqrt{2}} \]

### Step 2: Calculate the Radius \( R \)
The radius \( R \) in polar coordinates is calculated using the formula:
\[ R = \sqrt{x^2 + y^2} \]
Substituting the values of \( x \) and \( y \):
\[ R = \sqrt{\left(\frac{1}{\sqrt{2}}\right)^2 + \left(-\frac{1}{\sqrt{2}}\right)^2} \]
Calculating the squares:
\[ R = \sqrt{\frac{1}{2} + \frac{1}{2}} = \sqrt{1} = 1 \]

### Step 3: Calculate the Angle \( \theta \)
To find the angle \( \theta \), we use the tangent function:
\[ \tan(\theta) = \frac{y}{x} \]
Substituting the values of \( y \) and \( x \):
\[ \tan(\theta) = \frac{-\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}} = -1 \]
The angle whose tangent is \(-1\) corresponds to \( \theta \) in the fourth quadrant. The reference angle where \( \tan(\theta) = 1 \) is \( \frac{\pi}{4} \). Therefore, in the fourth quadrant:
\[ \theta = 2\pi - \frac{\pi}{4} = \frac{8\pi}{4} - \frac{\pi}{4} = \frac{7\pi}{4} \]

### Step 4: Write the Polar Coordinates
The polar coordinates are given by:
\[ (R, \theta) = (1, \frac{7\pi}{4}) \]

### Final Answer
Thus, the polar coordinates of the point are:
\[ \boxed{(1, \frac{7\pi}{4})} \]