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 find the polar coordinates of the point whose Cartesian coordinates are (-1, -1), we can follow these steps:

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

### Step 2: Determine the Quadrant
Since both x and y are negative, the point (-1, -1) lies in the third quadrant.

### Step 3: Calculate the Radius (R)
The radius \( R \) in polar coordinates is calculated using the formula:
\[ R = \sqrt{x^2 + y^2} \]
Substituting the values:
\[ R = \sqrt{(-1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

### Step 4: Calculate the Angle (θ)
The angle \( \theta \) can be found using the tangent function:
\[ \tan(\theta) = \frac{y}{x} = \frac{-1}{-1} = 1 \]
The angle whose tangent is 1 is \( \frac{\pi}{4} \) radians. However, since we are in the third quadrant, we need to add \( \pi \) to this angle:
\[ \theta = \pi + \frac{\pi}{4} = \frac{5\pi}{4} \]

### Step 5: Write the Polar Coordinates
The polar coordinates are given by \( (R, \theta) \):
\[ \text{Polar Coordinates} = \left( \sqrt{2}, \frac{5\pi}{4} \right) \]

### Step 6: Check for Equivalent Angles
Since angles can be represented in multiple ways, we can also express \( \frac{5\pi}{4} \) in a negative direction. This can be done by subtracting \( 2\pi \):
\[ \frac{5\pi}{4} - 2\pi = \frac{5\pi}{4} - \frac{8\pi}{4} = -\frac{3\pi}{4} \]
Thus, the angle can also be represented as \( -\frac{3\pi}{4} \).

### Final Answer
The polar coordinates of the point (-1, -1) are:
\[ \left( \sqrt{2}, -\frac{3\pi}{4} \right) \]

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