The polar coordinates of the point whose cartesian coordinates are (-1, -1) is

Options

A. `(sqrt(2),(pi)/(4))`
B. `(sqrt(2),(3pi)/(4))`
C. `(sqrt(2),-(pi)/(4))`
D. `(sqrt(2),-(3pi)/(4))`

Answer

To convert the Cartesian coordinates (-1, -1) into polar coordinates (r, θ), we follow these steps:

### Step 1: Identify the Cartesian Coordinates
The given Cartesian coordinates are:
- x = -1
- y = -1

### Step 2: Calculate r
The formula for r in polar coordinates is given by:
\[ r = \sqrt{x^2 + y^2} \]

Substituting the values of x and y:
\[ r = \sqrt{(-1)^2 + (-1)^2} \]
\[ r = \sqrt{1 + 1} \]
\[ r = \sqrt{2} \]

### Step 3: Calculate θ
The angle θ can be found using the formula:
\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]

Substituting the values of y and x:
\[ \theta = \tan^{-1}\left(\frac{-1}{-1}\right) \]
\[ \theta = \tan^{-1}(1) \]

The value of \( \tan^{-1}(1) \) is \( \frac{\pi}{4} \).

### Step 4: Determine the Correct Quadrant
The point (-1, -1) is located in the third quadrant, where both x and y are negative. In the third quadrant, the angle is given by:
\[ \theta = \pi + \frac{\pi}{4} \]
\[ \theta = \frac{4\pi}{4} + \frac{\pi}{4} \]
\[ \theta = \frac{5\pi}{4} \]

### Step 5: Write the Polar Coordinates
Now that we have both r and θ, we can express the polar coordinates:
\[ \text{Polar Coordinates} = \left( \sqrt{2}, \frac{5\pi}{4} \right) \]

### Final Answer
The polar coordinates of the point whose Cartesian coordinates are (-1, -1) are:
\[ \left( \sqrt{2}, \frac{5\pi}{4} \right) \]

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